Unsupervised Testing Strategies for ASR Google
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Unsupervised Testing Strategies for ASR Brian Strope, Doug Beeferman, Alexander Gruenstein, Xin Lei Google, Inc. bps, dougb, alexgru, xinlei @google.com Abstract This paper describes unsupervised strategies for estimating relative accuracy differences between acoustic models or language models used for automatic speech recognition. To test acoustic models, the approach extends ideas used for unsupervised discriminative training to include a more explicit validation on held out data. To test language models, we use a dual interpretation of the same process, this time allowing us to measure differences by exploiting expected ‘truth gradients’ between strong and weak acoustic models. The paper shows correlations between supervised and unsupervised measures across a range of acoustic model and language model variations. We also use unsupervised tests to assess the non-stationary nature of mobile speech input. Index Terms: speech recognition, unsupervised testing, nonstationary distributions 1. Introduction Current commercial speech recognition systems can use years of unsupervised data to train relatively large, discriminatively optimized, acoustic models (AM). Similarly, web-scale text corpora for estimating language models (LM) are often available online, and unsupervised recognition results themselves can provide an additional source of LM training data. Since there is no human transcription in any of these steps, the remaining use for manual human transcription is for generating test sets, as a final sanity check for validating system parameters and models. In this paper, we augment that strategy with unsupervised evaluations and begin the discussion of whether eventually we might be able to get rid of the need for any explicit human transcription. The motivation for human transcription for testing is obvious. Despite steady advances and relative commercial successes, it is generally accepted that humans are much more accurate transcribers than automatic speech recognition systems [1]. While there are a few notable exceptions where machines were more accurate than humans [2], human transcription accuracy is so much better, we use it unquiestioningly as our best approximate for absolute truth. But there are equally obvious disadvantages to relying on human transcription. While it may feel premature, accepting human performance as absolute truth imposes an upper bound on accuracy. The absolute truth is not absolute, and so we’ll eventually have to figure out how to beat it. In fact with our current processes and tasks, below, we show that human transcribers can be only comparable in accuracy to current ASR systems. Absolute truth is already a problem. In response, we are improving transcription processes, but also considering unsupervised ways to augment traditional testing. Another obvious disadvantage of human transcription is that the tests themselves have to be limited in size and type. Even in a commercially successful research lab, getting extensive tests across every combination of speaker and channel type, recognition context, language, and time period is prohibitive. But a detailed characterization of those types of variations could help prioritize efforts. Similarly when tests are unsupervised, it is easier to update development and evaluation sets to avoid problems related to stale, over-fit tests. This is mostly an empirical paper. The next section describes some of the experiments we ran trying to assess our existing human transcription accuracy. Then we describe the generalizations of unsupervised discriminative training that enable a new evaluation strategy. Next the paper includes evaluations that show correlations between supervised and unsupervised tests, and concludes with unsupervised tests that start to characterize the non-stationary distribution of spoken data coming through Google mobile applications. 2. Problems with human transcriptions Recent efforts have begun to consider human transcription accuracy in the context of increased efficiency. These studies have generally shown that depending on the amount of effort, and the task, individual word error rates can vary from 2-15% [3, 4]. Ef- ficiency pressures on human transcription can lead to transcription noise and bias. 2.1. Early experiments Over the last few years we have seen several simple experiments not work: we have added matched data to our language models and seen error rates get worse; we have added unsupervised acoustic modeling data matched to a new fielded acoustic condition, and seen the error rates on new matched tests go up, but surprisingly, error rates on an old test, with slightly mismatched conditions, go down. For each of these, after tediously examining errors, we found the problem was that we typically “seed” our transcription process with the recognition result from the field. Mostly as a matter of expedience; it is easier for the transcriber to hit return than to type “home depot in palo alto california” yet again, and it can improve reliability since retyping can be error prone. But the power of the suggested transcription is also enough to bias the transcribers into rubber-stamping some of the fielded recognition results. When the transcriber rubber-stamps an error we potentially get penalized twice. The baseline gets credit where it should not, and a new system that corrects that error is falsely penalized for adding an error. The surprising improvement noted on the older, slightly mis-matched test happened because the transcriptions for the older test were seeded with transcriptions from an older system, decorrelating some of the transcription bias with the current baseline. In this case, transcription bias toward the baseline model was a bigger effect than the change in acoustics. Copyright © 2011 ISCA 28-31 August 2011, Florence, Italy INTERSPEECH 2011 16852.2. Multiple attempts To measure the human transcription accuracy more directly we started sending the same data for multiple attempts at human transcription, and we intentionally reduced the quality of our starting seeds to move any bias away from our best systems. For one test we sent 200K Voice Search utterances to be transcribed twice. Ignoring trivial differences like spaces, apostrophes, function words, and others, half of the transcripts agreed, which implies a sentence transcription accuracy of 71%, assuming independence of the attempts. Similarly when we sent the remaining 100K utterances, where transcriptions did not agree, back for two more attempts, we were still left with about 10% of the original set with 4 distinct human transcriptions. Again assuming independence, 10% disagreement in 4 attempts is consistent with 68% accuracy for each attempt. But we believe our system has a sentence accuracy higher than 70%. Looking through the errors many of the problems are related to cultural references, popular names, and businesses that are not obvious to everyone. The cultural and geographic requirements of the voice search task may be unusually difficult. It combines short utterances and wide open semantic contexts to generate surprisingly unfamiliar sounding speech. Finding ways to bring the correct cultural context to the transcriber is another obvious path to pursue. 3. Generalizing unsupervised discriminative training While some published results considered unsupervised maximum likelihood estimation of model parameters [5], many systems use unsupervised discriminative optimization, directly using recognizer output as input [6]. Cynically we might ask what we are learning if we are using the recognition result as truth for discriminatively optimizing its parameters. It is hard to imagine that we can fix the errors it makes, when we use the model to generate truth. But when we look into the details of commonly used discriminative training techniques based on maximum mutual information, we see that the LM used to generate competing hypotheses is not the same LM used to generate truth. To improve the generalization of discriminative training, we use a unigram to describe the space of potential errors [7], but a trigram or higher to give us transcription truth with unsupervised training. One interpretation of unsupervised discriminative training for acoustic models is that we are using the difference between a weak unigram and a relatively stronger trigram to give us a known improvement in relative truth. We do not know that the strong-LM (trigram) result is absolutely correct, we only know that it is better than the result with the weak LM (unigram). When there is a difference, if we can move toward the results of the strong-LM system by changing acoustic model parameters, then we are building a more accurate AM, that also helps with the final system using a stronger LM. With this interpretation, the AM learns from the ‘truth gradient’ between the strong and weak LMs. 3.1. Unsupervised AM testing Extending unsupervised discriminative AM training to unsupervised AM testing involves retesting the criterion used during training in a new test context. More prescriptively, we sample a new set of live data from production logs, and take the recognition result from the fielded system using a strong AM and LM as assumed truth. Then we re-recognize the same data using multiple strong acoustic models and a weak LM. If one of the systems using a weak LM can better approximate the system using a strong LM, then at a minimum, we can say that it is doing a better job of generalizing our training criteria to new data. More directly, we have evidence that one of the strong acoustic models could be more accurate than the rest. For scoring we are assuming truth from the fielded system, not a human transcriber. Therefore, when reporting unsupervised testing results, we count traditional word error rates, but because there is no human transcription, we report it as a word difference rate (WDR), to highlight that, for example, in the case of unsupervised AM tests, it is the word differences between the systems with the strong and weak LM. 3.2. Unsupervised LM testing To use the same strategy for LM testing we reverse the roles of the AM and the LM. For better generalization of discriminative AM testing, we used a weak LM to generate more competing alternates. That establishes a truth gradient that generally changes around 1/3 of the words. The dual for LM testing is to use a weak AM instead. To get a truth gradient of a similar magnitude with our systems, we backed off to a context-dependent acoustic model that uses around 1/10th the number of parameters of our strong models, and only uses maximum likelihood training. Then as above, we test with multiple strong LMs and assume that the LM that can move the results of the system using the weak AM closest to the results of the production system (with the strong AM), is the most accurate LM. With unsupervised LM testing we again report WDR and not WER, where the magnitude of the difference is now from the difference between the strong AM and the weak AM. 3.3. Relative measures In this paper we are ignoring the harder problem of measuring absolute accuracy. Instead we focus on relative differences between different acoustic or language models. Others have predicted absolute error measures using statistics from the training set as represented in the final acoustic models [8], without looking at testing data. But here we are interested in estimating relative performance across production data that was unseen during training. Our goal is to assess whether new models or new approaches are helping on new data, and whether the data might be changing from the distributions used during training. 4. Correlating supervised and unsupervised measures First we show that the performance on unsupervised offline tests for the AM and for the LM correlate with more traditional supervised tests. Our production data started with primarily Voice Search queries intended for google.com, but over time has included increasing amounts of general Voice Input traffic which includes a large fraction of short person-to-person messages. To start the analyses, we consider these data streams separately. For Voice Search, our traditional supervised test is built from the 200K utterance set that we sent for multiple transcriptions. For this test we exclude the 10% of the utterances where we got 4 distinct human transcriptions and sample a test set randomly from the remaining 90%. Similarly for the supervised 1686Voice Input test, we sent utterances twice and selected from the utterances with at least 80% agreement between human transcriptions. On the utterances where not all the words agreed, we randomly chose one of the human transcriptions as truth. This led to a test that excluded about 28% of the utterances. Both of these supervised tests are biased in that they only include the utterances that we could reliably transcribe. The Voice Search test has 27K utterances and 87K words. The Voice Input test has 49K utterances and 320K words. For the first unsupervised tests here, we sampled production logs for a single day of traffic. We found the median recognizer confidence for each task and then randomly selected a few hundred thousand utterances that were above median confidence for each task. For all unsupervised experiments we used the recognition results from the field as truth. Our recognition configuration for both systems is fairly standard and described in the literature. Specifically we use a PLP front-end [11] together with LDA and STC [12], and optimize our acoustic models using BMMI [13] on mostly unsupervised data mixed from both tasks. Our language models are n-grams, with Katz interpolation and entropy pruning, and the fielded Voice Input system also includes dynamic interpolation [14]. The Voice Search system used trigrams and the Voice Input system included 4-grams. 4.1. AM experiments The AM experiments use a weak LM (in this case a unigram) for each task estimated from the few hundred thousand high confidence utterances sampled for that day’s test. All the utterances in the test were also used to train the LM, so there is no OOV. This step is consistent with the matched unigram we train for discriminative acoustic model training. For Voice Search, the resulting unigram had 17K words, and for Voice Input there were 18K unique words. The acoustic models we tested here were trained using 11M (mostly unsupervised) utterances from a mix of both tasks. The parameter we vary for these experiments is the size of the acoustic models. We use the same decision tree and context state definitions for all models, but we vary the number of Gaussians assigned to each state. Each model is trained with the same number of iterations through all the data. The final model sizes range from 100K to 1M Gaussians. Decoder parameters are set in production mode, which generally means we lose around 0.5% absolute from the best possible accuracy to have faster than real-time search. # Gauss Sup VS Unsup VS Sup VI Unsup VI 100K 16.0 36.0 14.5 24.8 200K 15.3 34.4 13.6 22.8 340K 14.6 33.9 13.4 22.7 500K 14.3 33.3 13.2 22.3 1M 13.9 33.0 12.9 21.8 Table 1: WER in % on supervised (Sup) and WDR in % on unsupervised (Unsup) AM tests for Voice Search (VS) and Voice Input (VI). 4.2. LM experiments For the LM experiments we vary the number of n-grams used for the Voice Input task from around 2M to 30M by varying our final entropy pruning threshold. Unlike the production system used to generate truth for the unsupervised tests, for these tests the LM is a static n-gram. We show results with two different weak acoustic models (A/B). Condition A is a context-dependent model estimated using maximum likelihood criteria with 2 Gaussians per state for a total of 16K Gaussians. Condition B uses a similar model with a variable number of Gaussians across model states, and a total of 40K Gaussians. On supervised tests, these weak acoustic models have around two to three times the error rates of final strong production models. n-grams Sup PPL Sup WER Unsup A/B WDR 1.9M 109 15.2 38.1/25.9 3.8M 98 14.4 36.8/24.5 7.6M 92 14.1 36.0/23.8 15M 87 13.9 35.5/23.2 30M 85 13.7 35.1/22.8 Table 2: Comparing supervised (Sup) and unsupervised (Unsup) LM tests for Voice Input. WER/WDR are in %, PPL is perplexity. Unsup A and B are for different sized AMs. The relative improvement in both AM and LM experiments is consistently around 10% for a 10x increase in model size. Correlations between supervised and unsupervised tests range between 0.98 and 0.99. 5. Additional experiments Varying model size is a controlled way to generate accuracy differences. Here we include additional unsupervised measurements that show expected differences in the context of other AM and LM modeling efforts. 5.1. CMLLR To evaluate an implementation of constrained maximum likelihood linear regression [9] for adaptation, we started by testing with read speech corpora from several data collections [10] used to initialize acoustic models in a new context. With a large and regular amounts of acoustic data per speaker, we see the typical improvements of 6-10% relative, over a matched discriminative baseline. To estimate the accuracy impact of CMLLR on the production system, (where the actual distributions of amount of data per user is not imposed by the strict specifications of a data collection) we used unsupervised testing. Here we sampled all personalized users over a 30 day period, and measured the change in WDR with a weak LM and either the production AM or the production AM with CMLLR. Further we break the differences in WDR down by the amount of data available for each speaker. # Utts No Adapt Adapt 1-20 25.7 25.4 20-50 26.6 25.6 50-100 25.8 24.6 100-200 23.5 22.5 Table 3: WDR in % on adaptation tests. Input is binned by the number of utterances for a given user. From the table, it is clear that we are seeing a similar relative difference as we saw with more traditional read speech tests, and we are further able to characterize the expected satu- 1687ration of the relatively small number of parameters in CMLLR after around 20 voice input utterances. 5.2. LM update At one point we updated our language model to include a rescoring pass more explicitly matched to recent Voice Search queries. By testing this update with recent unsupervised tests we are able to show the expected win on new voice search type utterances. # Model Config Sup VS Unsup VS Original 14.6 30.0 Updated 14.6 28.6 Table 4: WER in % on supervised (Sup) and WDR in % unsupervised (Unsup) LM tests for Voice Search. One interpretation of these results is that we are updating the LM to better represent the recent query data which itself is better matched to the recent unsupervised test. It also suggests that the distribution of our data might be moving. 5.3. Estimating non-stationary distributions Finally we ran two sweeps of AM tests to estimate how stationary the acoustics for our system have been over the last 14 months. The first system is trained using the Voice Search supervised data available at the beginning of the 14 months, and the second uses only unsupervised data sampled from the last 3 months. Therefore, one model represents our initial estimate of the distribution, and the other approximates a most recent distribution. Both systems use around 350K gaussians. To evaluate the AM performance, we use a weak LM estimated from a year’s worth of production data. Figure 1: Change in WDR over time with two different AMs. Both lines show that the distribution of the data has shifted away from the original supervised data, and toward the recent unsupervised data. Additional unsupervised tests will illuminate the causes of this change in more detail. We currently suspect an increase in the fraction of voice input recognition, but it is already obvious that the distribution of the acoustics for this data is changing. The plot also suggests that with a single AM the change of WDR across conditions may also be informative. Note that since we are generalizing from the same criteria we used for AM training, and we are getting rid of some of the necessity of human transcription, we are concerned about converging away from reality. The ground is a little firmer for the LM side, since our current LM processes are in fact not yet learning from AM truth gradients the same way our unsupervised AM training learns from LM truth gradients. From the AM side, our current unsupervised tests are simply checking whether the training optimizations extend to unseen data. Pragmatically, because it is unsupervised we also have the opportunity to test that generalization with a range of weak LMs and with a range of input data, and thereby to increase our con- fidence in the generalization. Moreover, reducing the accuracy improvement provided by a strong LM seems like a safe requirement to impose on AM training. But from an experiment perspective, we have to remember what gradient we are exploiting and not cheat. In other words, augmenting the AM with features directly related to the strong LM would not lead to improvements. We also monitor coarse signals related to application use (counts of user actions in response to recognition results) to give us additional complimentary evidence of successful generalization. 6. Conclusions This paper extends unsupervised discriminative training to an unsupervised testing strategy suitable for evaluating AM and LM changes. We show strong correlations with traditional testing strategies when we change AM or LM model size. We also show expected gains on unsupervised measures with other types of AM and LM changes, and use the unsupervised measures to begin to characterize the stationarity of the input data to Google mobile. Together with unsupervised training, unsupervised testing enables development paths that no longer impose human performance as the upper bound for accuracy. 7. References [1] R. Lippmann, “Speech recognition by machines and humans,” Speech Communication, July 1997. [2] T. Kristjansson, J. Hershey, P. Olsen, S. Rennie, R. Gopinath, “Super-Human Multi-Talker Speech Recognition: The IBM 2006 Speech Separation Challenge System,” Proc. ICSLP, 2006. [3] S. Novotney, C. Callison-Burch, “Cheap, Fast and Good Enough: Automatic Speech Recognition with Non-Expert Transcription,” Proc. NAACL, 2010. [4] A. Gruenstein, I. McGraw, A. Sutherland, “A Self-Transcribing Speech Corpus: Collecting Continuous Speech with an Online Educational Game,” Proc. SLaTE, 2009. [5] J. Ma, R. Schwartz, “Unsupervised versus supervised training of acoustic models,” Proc. ICSLP, 2008. [6] L. Wang, M. Gales, P. Woodland, “Unsupervised Training for Mandarin Broadcast News Conversation Transcription,” Proc ICASSP, 2007. [7] P.C. Woodland, D. Povey, “Large scale discriminative training of hidden Markov models for speech recognition,” Comp. Speech & Lang., Jan. 2002. [8] Y. Deng, M. Mahajan, A. Acero, “Estimating Speech Recognition Error Rate without Acoustic Test Data,” Proc. Eurospeech, 2003. [9] M. J. F. Gales, “Maximum likelihood linear transformations for HMM-based speech recognition,” Comp. Speech & Lang., Vol 12.2 1998. [10] T. Hughes, K. Nakajima, L. Ha, A. Vasu, P. Moreno, M. LeBeau, “Building transcribed speech corpora quickly and cheaply for many languages,” Proc ICSLP, 2010. [11] H. Hermansky, “Perceptual linear predictive (PLP) analysis of speech,” JASA, v87.4, 1990. [12] M. Gales, “Semi-Tied Covariance Matrices for Hidden Markov Models,” Proc. IEEE Trans. SAP, May 2000. [13] D. Povey, D. Kanevsky, B. Kingsbury, B. Ramabhadran, G. Saon, K. Visweswariah, “Boosted MMI for model and feature-space discriminative training,” Proc. ICASSP, 2008. [14] B. Ballinger, C. Allauzen, A. Gruenstein, J. Schalkwyk, “OnDemand Language Model Interpolation for Mobile Speech Input,” Proc. ICSLP, 2010. 1688 Parallel Algorithms for Unsupervised Tagging Sujith Ravi Google Mountain View, CA 94043 sravi@google.com Sergei Vassilivitskii Google Mountain View, CA 94043 sergeiv@google.com Vibhor Rastogi∗ Twitter San Francisco, CA vibhor.rastogi@gmail.com Abstract We propose a new method for unsupervised tagging that finds minimal models which are then further improved by Expectation Maximization training. In contrast to previous approaches that rely on manually specified and multi-step heuristics for model minimization, our approach is a simple greedy approximation algorithm DMLC (DISTRIBUTEDMINIMUM-LABEL-COVER) that solves this objective in a single step. We extend the method and show how to ef- ficiently parallelize the algorithm on modern parallel computing platforms while preserving approximation guarantees. The new method easily scales to large data and grammar sizes, overcoming the memory bottleneck in previous approaches. We demonstrate the power of the new algorithm by evaluating on various sequence labeling tasks: Part-of-Speech tagging for multiple languages (including lowresource languages), with complete and incomplete dictionaries, and supertagging, a complex sequence labeling task, where the grammar size alone can grow to millions of entries. Our results show that for all of these settings, our method achieves state-of-the-art scalable performance that yields high quality tagging outputs. 1 Introduction Supervised sequence labeling with large labeled training datasets is considered a solved problem. For ∗The research described herein was conducted while the author was working at Google. instance, state of the art systems obtain tagging accuracies over 97% for part-of-speech (POS) tagging on the English Penn Treebank. However, learning accurate taggers without labeled data remains a challenge. The accuracies quickly drop when faced with data from a different domain, language, or when there is very little labeled information available for training (Banko and Moore, 2004). Recently, there has been an increasing amount of research tackling this problem using unsupervised methods. A popular approach is to learn from POS-tag dictionaries (Merialdo, 1994), where we are given a raw word sequence and a dictionary of legal tags for each word type. Learning from POStag dictionaries is still challenging. Complete wordtag dictionaries may not always be available for use and in every setting. When they are available, the dictionaries are often noisy, resulting in high tagging ambiguity. Furthermore, when applying taggers in new domains or different datasets, we may encounter new words that are missing from the dictionary. There have been some efforts to learn POS taggers from incomplete dictionaries by extending the dictionary to include these words using some heuristics (Toutanova and Johnson, 2008) or using other methods such as type-supervision (Garrette and Baldridge, 2012). In this work, we tackle the problem of unsupervised sequence labeling using tag dictionaries. The first reported work on this problem was on POS tagging from Merialdo (1994). The approach involved training a standard Hidden Markov Model (HMM) using the Expectation Maximization (EM) algorithm (Dempster et al., 1977), though EM does notperform well on this task (Johnson, 2007). More recent methods have yielded better performance than EM (see (Ravi and Knight, 2009) for an overview). One interesting line of research introduced by Ravi and Knight (2009) explores the idea of performing model minimization followed by EM training to learn taggers. Their idea is closely related to the classic Minimum Description Length principle for model selection (Barron et al., 1998). They (1) formulate an objective function to find the smallest model that explains the text (model minimization step), and then, (2) fit the minimized model to the data (EM step). For POS tagging, this method (Ravi and Knight, 2009) yields the best performance to date; 91.6% tagging accuracy on a standard test dataset from the English Penn Treebank. The original work from (Ravi and Knight, 2009) uses an integer linear programming (ILP) formulation to find minimal models, an approach which does not scale to large datasets. Ravi et al. (2010b) introduced a two-step greedy approximation to the original objective function (called the MIN-GREEDY algorithm) that runs much faster while maintaining the high tagging performance. Garrette and Baldridge (2012) showed how to use several heuristics to further improve this algorithm (for instance, better choice of tag bigrams when breaking ties) and stack other techniques on top, such as careful initialization of HMM emission models which results in further performance gains. Their method also works under incomplete dictionary scenarios and can be applied to certain low-resource scenarios (Garrette and Baldridge, 2013) by combining model minimization with supervised training. In this work, we propose a new scalable algorithm for performing model minimization for this task. By making an assumption on the structure of the solution, we prove that a variant of the greedy set cover algorithm always finds an approximately optimal label set. This is in contrast to previous methods that employ heuristic approaches with no guarantee on the quality of the solution. In addition, we do not have to rely on ad hoc tie-breaking procedures or careful initializations for unknown words. Finally, not only is the proposed method approximately optimal, it is also easy to distribute, allowing it to easily scale to very large datasets. We show empirically that our method, combined with an EM training step outperforms existing state of the art systems. 1.1 Our Contributions • We present a new method, DISTRIBUTED MINIMUM LABEL COVER, DMLC, for model minimization that uses a fast, greedy algorithm with formal approximation guarantees to the quality of the solution. • We show how to efficiently parallelize the algorithm while preserving approximation guarantees. In contrast, existing minimization approaches cannot match the new distributed algorithm when scaling from thousands to millions or even billions of tokens. • We show that our method easily scales to both large data and grammar sizes, and does not require the corpus or label set to fit into memory. This allows us to tackle complex tagging tasks, where the tagset consists of several thousand labels, which results in more than one million entires in the grammar. • We demonstrate the power of the new method by evaluating under several different scenarios—POS tagging for multiple languages (including low-resource languages), with complete and incomplete dictionaries, as well as a complex sequence labeling task of supertagging. Our results show that for all these settings, our method achieves state-of-the-art performance yielding high quality taggings. 2 Related Work Recently, there has been an increasing amount of research tackling this problem from multiple directions. Some efforts have focused on inducing POS tag clusters without any tags (Christodoulopoulos et al., 2010; Reichart et al., 2010; Moon et al., 2010), but evaluating such systems proves dif- ficult since it is not straightforward to map the cluster labels onto gold standard tags. A more popular approach is to learn from POS-tag dictionaries (Merialdo, 1994; Ravi and Knight, 2009), incomplete dictionaries (Hasan and Ng, 2009; Garrette and Baldridge, 2012) and human-constructed dictionaries (Goldberg et al., 2008).Another direction that has been explored in the past includes bootstrapping taggers for a new language based on information acquired from other languages (Das and Petrov, 2011) or limited annotation resources (Garrette and Baldridge, 2013). Additional work focused on building supervised taggers for noisy domains such as Twitter (Gimpel et al., 2011). While most of the relevant work in this area centers on POS tagging, there has been some work done for building taggers for more complex sequence labeling tasks such as supertagging (Ravi et al., 2010a). Other related work include alternative methods for learning sparse models via priors in Bayesian inference (Goldwater and Griffiths, 2007) and posterior regularization (Ganchev et al., 2010). But these methods only encourage sparsity and do not explicitly seek to minimize the model size, which is the objective function used in this work. Moreover, taggers learned using model minimization have been shown to produce state-of-the-art results for the problems discussed here. 3 Model Following Ravi and Knight (2009), we formulate the problem as that of label selection on the sentence graph. Formally, we are given a set of sequences, S = {S1, S2, . . . , Sn} where each Si is a sequence of words, Si = wi1, wi2, . . . , wi,|Si| . With each word wij we associate a set of possible tags Tij . We will denote by m the total number of (possibly duplicate) words (tokens) in the corpus. Additionally, we define two special words w0 and w∞ with special tags start and end, and consider the modified sequences S 0 i = w0, Si , w∞. To simplify notation, we will refer to w∞ = w|Si|+1. The sequence label problem asks us to select a valid tag tij ∈ Tij for each word wij in the input to minimize a specific objective function. We will refer to a tag pair (ti,j−1, tij ) as a label. Our aim is to minimize the number of distinct labels used to cover the full input. Formally, given a sequence S 0 i and a tag tij for each word wij in S 0 i , let the induced set of labels for sequence S 0 i be Li = |S 0 i [ | j=1 {(ti,j−1, tij )}. The total number of distinct labels used over all sequences is then φ = ∪i Li | = [ i |S [i|+1 j=1 {(ti,j−1, tij )}|. Note that the order of the tokens in the label makes a difference as {(NN, VP)} and {(VP, NN)} are two distinct labels. Now we can define the problem formally, following (Ravi and Knight, 2009). Problem 1 (Minimum Label Cover). Given a set S of sequences of words, where each word wij has a set of valid tags Tij , the problem is to find a valid tag assignment tij ∈ Tij for each word that minimizes the number of distinct labels or tag pairs over all sequences, φ = S i S|Si|+1 j=1 {(ti,j−1, tij )}| . The problem is closely related to the classical Set Cover problem and is also NP-complete. To reduce Set Cover to the label selection problem, map each element i of the Set Cover instance to a single word sentence Si = wi1, and let the valid tags Ti1 contain the names of the sets that contain element i. Consider a solution to the label selection problem; every sentence Si is covered by two labels (w0, ki) and (ki , w∞), for some ki ∈ Ti1, which corresponds to an element i being covered by set ki in the Set Cover instance. Thus any valid solution to the label selection problem leads to a feasible solution to the Set Cover problem ({k1, k2, . . .}) of exactly half the size. Finally, we will use {{. . .}} notation to denote a multiset of elements, i.e. a set where an element may appear multiple times. 4 Algorithm In this Section, we describe the DISTRIBUTEDMINIMUM-LABEL-COVER, DMLC, algorithm for approximately solving the minimum label cover problem. We describe the algorithm in a centralized setting, and defer the distributed implementation to Section 5. Before describing the algorithm, we briefly explain the relationship of the minimum label cover problem to set cover. 4.1 Modification of Set Cover As we pointed out earlier, the minimum label cover problem is at least as hard as the Set Cover prob-1: Input: A set of sequences S with each words wij having possible tags Tij . 2: Output: A tag assignment tij ∈ Tij for each word wij approximately minimizing labels. 3: Let M be the multi set of all possible labels generated by choosing each possible tag t ∈ Tij . M = [ i |S [i|+1 j=1 [ t 0∈Ti,j−1 t∈Tij {{(t 0 , t)}} (1) 4: Let L = ∅ be the set of selected labels. 5: repeat 6: Select the most frequent label not yet selected: (t 0 , t) = arg max(s 0 ,s)∈L/ |M ∩ (s 0 , s)|. 7: For each bigram (wi,j−1, wij ) where t 0 ∈ Ti,j−1 and t ∈ Tij tentatively assign t 0 to wi,j−1 and t to wij . Add (t 0 , t) to L. 8: If a word gets two assignments, select one at random with equal probability. 9: If a bigram (wij , wi,j+1) is consistent with assignments in (t, t0 ), fix the tentative assignments, and set Ti,j−1 = {t 0} and Tij = t. Recompute M, the multiset of possible labels, with the updated Ti,j−1 and Tij . 10: until there are no unassigned words Algorithm 1: MLC Algorithm 1: Input: A set of sequences S with each words wij having possible tags Tij . 2: Output: A tag assignment tij ∈ Tij for each word wij approximately minimizing labels. 3: (Graph Creation) Initialize each vertex vij with the set of possible tags Tij and its neighbors vi,j+1 and vi,j−1. 4: repeat 5: (Message Passing) Each vertex vij sends its possibly tags Tij to its forward neighbor vij+1. 6: (Counter Update) Each vertex receives the the tags Ti,j−1 and adds all possible labels {(s, s0 )|s ∈ Ti,j−1, s0 ∈ Tij} to a global counter (M). 7: (MaxLabel Selection) Each vertex queries the global counter M to find the maximum label (t, t0 ). 8: (Tentative Assignment) Each vertex vij selects a tag tentatively as follows: If one of the tags t, t0 is in the feasible set Tij , it tentatively selects the tag. 9: (Random Assignment) If both are feasible it selects one at random. The vertex communicates its assignment to its neighbors. 10: (Confirmed Assignment) Each vertex receives the tentative assignment from its neighbors. If together with its neighbors it can match the selected label, the assignment is finalized. If the assigned tag is T, then the vertex vij sets the valid tag set Tij to {t}. 11: until no unassigned vertices exist. Algorithm 2: DMLC Implementation lem. An additional challenge comes from the fact that labels are tags for a pair of words, and hence are related. For example, if we label a word pair (wi,j−1, wij ) as (NN, VP), then the label for the next word pair (wij , wi,j+1) has to be of the form (VP, *), i.e., it has to start with VP. Previous work (Ravi et al., 2010a; Ravi et al., 2010b) recognized this challenge and employed two phase heuristic approaches. Eschewing heuristics, we will show that with one natural assumption, even with this extra set of constraints, the standard greedy algorithm for this problem results in a solution with a provable approximation ratio of O(log m). In practice, however, the algorithm performs far better than the worst case ratio, and similar to the work of (Gomes et al., 2006), we find that the greedy approach selects a cover approximately 11% worse than the optimum solution. 4.2 MLC Algorithm We present in Algorithm 1 our MINIMUM LABEL COVER algorithm to approximately solve the minimum label cover problem. The algorithm is simple, efficient, and easy to distribute. The algorithm chooses labels one at a time, selecting a label that covers as many words as possible inevery iteration. For this, it generates and maintains a multi-set of all possible labels M (Step 3). The multi-set contains an occurrence of each valid label, for example, if wi,j−1 has two possible valid tags NN and VP, and wij has one possible valid tag VP, then M will contain two labels, namely (NN, VP) and (VP, VP). Since M is a multi-set it will contain duplicates, e.g. the label (NN, VP) will appear for each adjacent pair of words that have NN and VP as valid tags, respectively. In each iteration, the algorithm picks a label with the most number of occurrences in M and adds it to the set of chosen labels (Step 6). Intuitively, this is a greedy step to select a label that covers the most number of word pairs. Once the algorithm picks a label (t 0 , t), it tries to assign as many words to tags t or t 0 as possible (Step 7). A word can be assigned t 0 if t 0 is a valid tag for it, and t a valid tag for the next word in sequence. Similarly, a word can be assigned t, if t is a valid tag for it, and t 0 a valid tag for the previous word. Some words can get both assignments, in which case we choose one tentatively at random (Step 8). If a word’s tentative random tag, say t, is consistent with the choices of its adjacent words (say t 0 from the previous word), then the tentative choice is fixed as a permanent one. Whenever a tag is selected, the set of valid tags Tij for the word is reduced to a singleton {t}. Once the set of valid tags Tij changes, the multi-set M of all possible labels also changes, as seen from Eq 1. The multi-set is then recomputed (Step 9) and the iterations repeated until all of words have been tagged. We can show that under a natural assumption this simple algorithm is approximately optimal. Assumption 1 (c-feasibility). Let c ≥ 1 be any number, and k be the size of the optimal solution to the original problem. In each iteration, the MLC algorithm fixes the tags for some words. We say that the algorithm is c-feasible, if after each iteration there exists some solution to the remaining problem, consistent with the chosen tags, with size at most ck . The assumption encodes the fact that a single bad greedy choice is not going to destroy the overall structure of the solution, and a nearly optimal solution remains. We note that this assumption of cfeasibility is not only sufficient, as we will formally show, but is also necessary. Indeed, without any assumptions, once the algorithm fixes the tag for some words, an optimal label may no longer be consistent with the chosen tags, and it is not hard to find contrived examples where the size of the optimal solution doubles after each iteration of MLC. Since the underlying problem is NP-complete, it is computationally hard to give direct evidence verifying the assumption on natural language inputs. However, on small examples we are able to show that the greedy algorithm is within a small constant factor of the optimum, specifically it is within 11% of the optimum model size for the POS tagging problem using the standard 24k dataset (Ravi and Knight, 2009). Combined with the fact that the final method outperforms state of the art approaches, this leads us to conclude that the structural assumption is well justified. Lemma 1. Under the assumption of c-feasibility, the MLC algorithm achieves a O(c log m) approximation to the minimum label cover problem, where m = P i |Si | is the total number of tokens. Proof. To prove the Lemma we will define an objective function φ¯, counting the number of unlabeled word pairs, as a function of possible labels, and show that φ¯ decreases by a factor of (1−O(1/ck)) at every iteration. To define φ¯, we first define φ, the number of labeled word pairs. Consider a particular set of labels, L = {L1, L2, . . . , Lk} where each label is a pair (ti , tj ). Call {tij} a valid assignment of tokens if for each wij , we have tij ∈ Tij . Then the score of L under an assignment t, which we denote by φt , is the number of bigram labels that appear in L. Formally, φt(L) = | ∪i,j {{(ti,j−1, tij ) ∩ L}}|. Finally, we define φ(L) to be the best such assignment, φ(L) = maxt φt(L), and φ¯(L) = m − φ(L) the number of uncovered labels. Consider the label selected by the algorithm in every step. By the c-feasibility assumption, there exists some solution having ck labels. Thus, some label from that solution covers at least a 1/ck fraction of the remaining words. The selected label (t, t0 ) maximizes the intersection with the remaining feasible labels. The conflict resolution step ensures that in expectation the realized benefit is at least a half of the maximum, thereby reducing φ¯ by at least a(1 − 1/2ck) fraction. Therefore, after O(kc log m) operations all of the labels are covered. 4.3 Fitting the Model Using EM Once the greedy algorithm terminates and returns a minimized grammar of tag bigrams, we follow the approach of Ravi and Knight (2009) and fit the minimized model to the data using the alternating EM strategy. In this step, we run an alternating optimization procedure iteratively in phases. In each phase, we initialize (and prune away) parameters within the two HMM components (transition or emission model) using the output from the previous phase. We initialize this procedure by restricting the transition parameters to only those tag bigrams selected in the model minimization step. We train in conjunction with the original emission model using EM algorithm which prunes away some of the emission parameters. In the next phase, we alternate the initialization by choosing the pruned emission model along with the original transition model (with full set of tag bigrams) and retrain using EM. The alternating EM iterations are terminated when the change in the size of the observed grammar (i.e., the number of unique bigrams in the tagging output) is ≤ 5%. 1 We refer to our entire approach using greedy minimization followed by EM training as DMLC + EM. 5 Distributed Implementation The DMLC algorithm is directly suited towards parallelization across many machines. We turn to Pregel (Malewicz et al., 2010), and its open source version Giraph (Apa, 2013). In these systems the computation proceeds in rounds. In every round, every machine does some local processing and then sends arbitrary messages to other machines. Semantically, we think of the communication graph as fixed, and in each round each vertex performs some local computation and then sends messages to its neighbors. This mode of parallel programming directs the programmers to “Think like a vertex.” The specific systems like Pregel and Giraph build infrastructure that ensures that the overall system 1 For more details on the alternating EM strategy and how initialization with minimized models improve EM performance in alternating iterations, refer to (Ravi and Knight, 2009). is fault tolerant, efficient, and fast. In addition, they provide implementation of commonly used distributed data structures, such as, for example global counters. The programmer’s job is simply to specify the code that each vertex will run at every round. We implemented the DMLC algorithm in Pregel. The implementation is straightforward and given in Algorithm 2. The multi-set M of Algorithm 1 is represented as a global counter in Algorithm 2. The message passing (Step 3) and counter update (Step 4) steps update this global counter and hence perform the role of Step 3 of Algorithm 1. Step 5 selects the label with largest count, which is equivalent to the greedy label picking step 6 of Algorithm 1. Finally steps 6, 7, and 8 update the tag assignment of each vertex performing the roles of steps 7, 8, and 9, respectively, of Algorithm 1. 5.1 Speeding up the Algorithm The implementation described above directly copies the sequential algorithm. Here we describe additional steps we took to further improve the parallel running times. Singleton Sets: As the parallel algorithm proceeds, the set of feasible sets associated with a node slowly decreases. At some point there is only one tag that a node can take on, however this tag is rare, and so it takes a while for it to be selected using the greedy strategy. Nevertheless, if a node and one of its neighbors have only a single tag left, then it is safe to assign the unique label 2 . Modifying the Graph: As is often the case, the bottleneck in parallel computations is the communication. To reduce the amount of communication we reduce the graph on the fly, removing nodes and edges once they no longer play a role in the computation. This simple modification decreases the communication time in later rounds as the total size of the problem shrinks. 6 Experiments and Results In this Section, we describe the experimental setup for various tasks, settings and compare empirical performance of our method against several existing 2We must judiciously initialize the global counter to take care of this assignment, but this is easily accomplished.baselines. The performance results for all systems (on all tasks) are measured in terms of tagging accuracy, i.e. % of tokens from the test corpus that were labeled correctly by the system. 6.1 Part-of-Speech Tagging Task 6.1.1 Tagging Using a Complete Dictionary Data: We use a standard test set (consisting of 24,115 word tokens from the Penn Treebank) for the POS tagging task. The tagset consists of 45 distinct tag labels and the dictionary contains 57,388 word/tag pairs derived from the entire Penn Treebank. Per-token ambiguity for the test data is about 1.5 tags/token. In addition to the standard 24k dataset, we also train and test on larger data sets— 973k tokens from the Penn Treebank, 3M tokens from PTB+Europarl (Koehn, 2005) data. Methods: We evaluate and compare performance for POS tagging using four different methods that employ the model minimization idea combined with EM training: • EM: Training a bigram HMM model using EM algorithm (Merialdo, 1994). • ILP + EM: Minimizing grammar size using integer linear programming, followed by EM training (Ravi and Knight, 2009). • MIN-GREEDY + EM: Minimizing grammar size using the two-step greedy method (Ravi et al., 2010b). • DMLC + EM: This work. Results: Table 1 shows the results for POS tagging on English Penn Treebank data. On the smaller test datasets, all of the model minimization strategies (methods 2, 3, 4) tend to perform equally well, yielding state-of-the-art results and large improvement over standard EM. When training (and testing) on larger corpora sizes, DMLC yields the best reported performance on this task to date. A major advantage of the new method is that it can easily scale to large corpora sizes and the distributed nature of the algorithm still permits fast, efficient optimization of the global objective function. So, unlike the earlier methods (such as MIN-GREEDY) it is fast enough to run on several millions of tokens to yield additional performance gains (shown in last column). Speedups: We also observe a significant speedup when using the parallelized version of the DMLC algorithm. Performing model minimization on the 24k tokens dataset takes 55 seconds on a single machine, whereas parallelization permits model minimization to be feasible even on large datasets. Fig 1 shows the running time for DMLC when run on a cluster of 100 machines. We vary the input data size from 1M word tokens to about 8M word tokens, while holding the resources constant. Both the algorithm and its distributed implementation in DMLC are linear time operations as evident by the plot. In fact, for comparison, we also plot a straight line passing through the first two runtimes. The straight line essentially plots runtimes corresponding to a linear speedup. DMLC clearly achieves better runtimes showing even better than linear speedup. The reason for this is that distributed version has a constant overhead for initialization, independent of the data size. While the running time for rest of the implementation is linear in data size. Thus, as the data size becomes larger, the constant overhead becomes less significant, and the distributed implementation appears to complete slightly faster as data size increases. Figure 1: Runtime vs. data size (measured in # of word tokens) on 100 machines. For comparison, we also plot a straight line passing through the first two runtimes. The straight line essentially plots runtimes corresponding to a linear speedup. DMLC clearly achieves better runtimes showing a better than linear speedup. 6.1.2 Tagging Using Incomplete Dictionaries We also evaluate our approach for POS tagging under other resource-constrained scenarios. Obtain-Method Tagging accuracy (%) te=24k te=973k tr=24k tr=973k tr=3.7M 1. EM 81.7 82.3 2. ILP + EM (Ravi and Knight, 2009) 91.6 3. MIN-GREEDY + EM (Ravi et al., 2010b) 91.6 87.1 4. DMLC + EM (this work) 91.4 87.5 87.8 Table 1: Results for unsupervised part-of-speech tagging on English Penn Treebank dataset. Tagging accuracies for different methods are shown on multiple datasets. te shows the size (number of tokens) in the test data, tr represents the size of the raw text used to perform model minimization. ing a complete dictionary is often difficult, especially for new domains. To verify the utility of our method when the input dictionary is incomplete, we evaluate against standard datasets used in previous work (Garrette and Baldridge, 2012) and compare against the previous best reported performance for the same task. In all the experiments (described here and in subsequent sections), we use the following terminology—raw data refers to unlabeled text used by different methods (for model minimization or other unsupervised training procedures such as EM), dictionary consists of word/tag entries that are legal, and test refers to data over which tagging evaluation is performed. English Data: For English POS tagging with incomplete dictionary, we evaluate on the Penn Treebank (Marcus et al., 1993) data. Following (Garrette and Baldridge, 2012), we extracted a word-tag dictionary from sections 00-15 (751,059 tokens) consisting of 39,087 word types, 45,331 word/tag entries, a per-type ambiguity of 1.16 yielding a pertoken ambiguity of 2.21 on the raw corpus (treating unknown words as having all 45 possible tags). As in their setup, we then use the first 47,996 tokens of section 16 as raw data and perform final evaluation on the sections 22-24. We use the raw corpus along with the unlabeled test data to perform model minimization and EM training. Unknown words are allowed to have all possible tags in both these procedures. Italian Data: The minimization strategy presented here is a general-purpose method that does not require any specific tuning and works for other languages as well. To demonstrate this, we also perform evaluation on a different language (Italian) using the TUT corpus (Bosco et al., 2000). Following (Garrette and Baldridge, 2012), we use the same data splits as their setting. We take the first half of each of the five sections to build the word-tag dictionary, the next quarter as raw data and the last quarter as test data. The dictionary was constructed from 41,000 tokens comprised of 7,814 word types, 8,370 word/tag pairs, per-type ambiguity of 1.07 and a per-token ambiguity of 1.41 on the raw data. The raw data consisted of 18,574 tokens and the test contained 18,763 tokens. We use the unlabeled corpus from the raw and test data to perform model minimization followed by unsupervised EM training. Other Languages: In order to test the effectiveness of our method in other non-English settings, we also report the performance of our method on several other Indo-European languages using treebank data from CoNLL-X and CoNLL-2007 shared tasks on dependency parsing (Buchholz and Marsi, 2006; Nivre et al., 2007). The corpus statistics for the five languages (Danish, Greek, Italian, Portuguese and Spanish) are listed below. For each language, we construct a dictionary from the raw training data. The unlabeled corpus from the raw training and test data is used to perform model minimization followed by unsupervised EM training. As before, unknown words are allowed to have all possible tags. We report the final tagging performance on the test data and compare it to baseline EM. Garrette and Baldridge (2012) treat unknown words (words that appear in the raw text but are missing from the dictionary) in a special manner and use several heuristics to perform better initialization for such words (for example, the probability that an unknown word is associated with a particular tag isconditioned on the openness of the tag). They also use an auto-supervision technique to smooth counts learnt from EM onto new words encountered during testing. In contrast, we do not apply any such technique for unknown words and allow them to be mapped uniformly to all possible tags in the dictionary. For this particular set of experiments, the only difference from the Garrette and Baldridge (2012) setup is that we include unlabeled text from the test data (but without any dictionary tag labels or special heuristics) to our existing word tokens from raw text for performing model minimization. This is a standard practice used in unsupervised training scenarios (for example, Bayesian inference methods) and in general for scalable techniques where the goal is to perform inference on the same data for which one wishes to produce some structured prediction. Language Train Dict Test (tokens) (entries) (tokens) DANISH 94386 18797 5852 GREEK 65419 12894 4804 ITALIAN 71199 14934 5096 PORTUGUESE 206678 30053 5867 SPANISH 89334 17176 5694 Results: Table 2 (column 2) compares previously reported results against our approach for English. We observe that our method obtains a huge improvement over standard EM and gets comparable results to the previous best reported scores for the same task from (Garrette and Baldridge, 2012). It is encouraging to note that the new system achieves this performance without using any of the carefully-chosen heuristics employed by the previous method. However, we do note that some of these techniques can be easily combined with our method to produce further improvements. Table 2 (column 3) also shows results on Italian POS tagging. We observe that our method achieves significant improvements in tagging accuracy over all the baseline systems including the previous best system (+2.9%). This demonstrates that the method generalizes well to other languages and produces consistent tagging improvements over existing methods for the same task. Results for POS tagging on CoNLL data in five different languages are displayed in Figure 2. Note that the proportion of raw data in test versus train 50 60 70 80 90 DANISH GREEK ITALIAN PORTUGUESE SPANISH 79.4 66.3 84.6 80.1 83.1 77.8 65.6 82 78.5 81.3 EM DMLC+EM Figure 2: Part-of-Speech tagging accuracy for different languages on CoNLL data using incomplete dictionaries. (from the standard CoNLL shared tasks) is much smaller compared to the earlier experimental settings. In general, we observe that adding more raw data for EM training improves the tagging quality (same trend observed earlier in Table 1: column 2 versus column 3). Despite this, DMLC + EM still achieves significant improvements over the baseline EM system on multiple languages (as shown in Figure 2). An additional advantage of the new method is that it can easily scale to larger corpora and it produces a much more compact grammar that can be efficiently incorporated for EM training. 6.1.3 Tagging for Low-Resource Languages Learning part-of-speech taggers for severely lowresource languages (e.g., Malagasy) is very challenging. In addition to scarce (token-supervised) labeled resources, the tag dictionaries available for training taggers are tiny compared to other languages such as English. Garrette and Baldridge (2013) combine various supervised and semi-supervised learning algorithms into a common POS tagger training pipeline to addresses some of these challenges. They also report tagging accuracy improvements on low-resource languages when using the combined system over any single algorithm. Their system has four main parts, in order: (1) Tag dictionary expansion using label propagation algorithm, (2) Weighted model minimization, (3) Expectation maximization (EM) training of HMMs using auto-supervision, (4) MaxEnt Markov Model (MEMM) training. The entire procedure results in a trained tagger model that can then be applied to tag any raw data.3 Step 2 in this procedure involves 3 For more details, refer (Garrette and Baldridge, 2013). "We consider a model of repeated online auctions in which an ad with an uncertain click-through rate faces a random distribution of competing bids in each auction and there is discounting of payoffs. We formulate the optimal solution to this explore/exploit problem as a dynamic programming problem and show that efficiency is maximized by making a bid for each advertiser equal to the advertiser's expected value for the advertising opportunity plus a term proportional to the variance in this value divided by the number of impressions the advertiser has received thus far. We then use this result to illustrate that the value of incorporating active exploration into a machine learning system in an auction environment is exceedingly small." Accepted for publication in the Annals of Applied Statistics (in press), 09/2014 INFERRING CAUSAL IMPACT USING BAYESIAN STRUCTURAL TIME-SERIES MODELS By Kay H. Brodersen, Fabian Gallusser, Jim Koehler, Nicolas Remy, and Steven L. Scott Google, Inc. E-mail: kbrodersen@google.com Abstract An important problem in econometrics and marketing is to infer the causal impact that a designed market intervention has exerted on an outcome metric over time. This paper proposes to infer causal impact on the basis of a diffusion-regression state-space model that predicts the counterfactual market response in a synthetic control that would have occurred had no intervention taken place. In contrast to classical difference-in-differences schemes, statespace models make it possible to (i) infer the temporal evolution of attributable impact, (ii) incorporate empirical priors on the parameters in a fully Bayesian treatment, and (iii) flexibly accommodate multiple sources of variation, including local trends, seasonality, and the time-varying influence of contemporaneous covariates. Using a Markov chain Monte Carlo algorithm for posterior inference, we illustrate the statistical properties of our approach on simulated data. We then demonstrate its practical utility by estimating the causal effect of an online advertising campaign on search-related site visits. We discuss the strengths and limitations of state-space models in enabling causal attribution in those settings where a randomised experiment is unavailable. The CausalImpact R package provides an implementation of our approach. 1. Introduction. This article proposes an approach to inferring the causal impact of a market intervention, such as a new product launch or the onset of an advertising campaign. Our method generalizes the widely used ‘difference-in-differences’ approach to the time-series setting by explicitly modelling the counterfactual of a time series observed both before and after the intervention. It improves on existing methods in two respects: it provides a fully Bayesian time-series estimate for the effect; and it uses model averaging to construct the most appropriate synthetic control for modelling the counterfactual. The CausalImpact R package provides an implementation of our approach (http://google.github.io/CausalImpact/). Inferring the impact of market interventions is an important and timely Keywords and phrases: causal inference, counterfactual, synthetic control, observational, difference in differences, econometrics, advertising, market research 12 K.H. BRODERSEN ET AL. problem. Partly because of recent interest in ‘big data,’ many firms have begun to understand that a competitive advantage can be had by systematically using impact measures to inform strategic decision making. An example is the use of ‘A/B experiments’ to identify the most effective market treatments for the purpose of allocating resources (Danaher and Rust, 1996; Seggie, Cavusgil and Phelan, 2007; Leeflang et al., 2009; Stewart, 2009). Here, we focus on measuring the impact of a discrete marketing event, such as the release of a new product, the introduction of a new feature, or the beginning or end of an advertising campaign, with the aim of measuring the event’s impact on a response metric of interest (e.g., sales). The causal impact of a treatment is the difference between the observed value of the response and the (unobserved) value that would have been obtained under the alternative treatment, i.e., the effect of treatment on the treated (Rubin, 1974; Hitchcock, 2004; Morgan and Winship, 2007; Rubin, 2007; Cox and Wermuth, 2001; Heckman and Vytlacil, 2007; Antonakis et al., 2010; Kleinberg and Hripcsak, 2011; Hoover, 2012; Claveau, 2012). In the present setting the response variable is a time series, so the causal effect of interest is the difference between the observed series and the series that would have been observed had the intervention not taken place. A powerful approach to constructing the counterfactual is based on the idea of combining a set of candidate predictor variables into a single ‘synthetic control’ (Abadie and Gardeazabal, 2003; Abadie, Diamond and Hainmueller, 2010). Broadly speaking, there are three sources of information available for constructing an adequate synthetic control. The first is the time-series behaviour of the response itself, prior to the intervention. The second is the behaviour of other time series that were predictive of the target series prior to the intervention. Such control series can be based, for example, on the same product in a different region that did not receive the intervention, or on a metric that reflects activity in the industry as a whole. In practice, there are often many such series available, and the challenge is to pick the relevant subset to use as contemporaneous controls. This selection is done on the pre-treatment portion of potential controls; but their value for predicting the counterfactual lies in their post-treatment behaviour. As long as the control series received no intervention themselves, it is often reasonable to assume the relationship between the treatment and the control series that existed prior to the intervention to continue afterwards. Thus, a plausible estimate of the counterfactual time series can be computed up to the point in time where the relationship between treatment and controls can no longer be assumed to be stationary, e.g., because one of the controls received treatment itself. In a Bayesian framework, a third source ofBAYESIAN CAUSAL IMPACT ANALYSIS 3 information for inferring the counterfactual is the available prior knowledge about the model parameters, as elicited, for example, by previous studies. We combine the three preceding sources of information using a statespace time-series model, where one component of state is a linear regression on the contemporaneous predictors. The framework of our model allows us to choose from among a large set of potential controls by placing a spikeand-slab prior on the set of regression coefficients, and by allowing the model to average over the set of controls (George and McCulloch, 1997). We then compute the posterior distribution of the counterfactual time series given the value of the target series in the pre-intervention period, along with the values of the controls in the post-intervention period. Subtracting the predicted from the observed response during the post-intervention period gives a semiparametric Bayesian posterior distribution for the causal effect (Figure 1). Related work. As with other domains, causal inference in marketing requires subtlety. Marketing data are often observational and rarely follow the ideal of a randomised design. They typically exhibit a low signal-to-noise ratio. They are subject to multiple seasonal variations, and they are often confounded by the effects of unobserved variables and their interactions (for recent examples, see Seggie, Cavusgil and Phelan, 2007; Stewart, 2009; Leeflang et al., 2009; Takada and Bass, 1998; Chan et al., 2010; Lewis and Reiley, 2011; Lewis, Rao and Reiley, 2011; Vaver and Koehler, 2011, 2012). Rigorous causal inferences can be obtained through randomised experiments, which are often implemented in the form of geo experiments (Vaver and Koehler, 2011, 2012). Many market interventions, however, fail to satisfy the requirements of such approaches. For instance, advertising campaigns are frequently launched across multiple channels, online and offline, which precludes measurement of individual exposure. Campaigns are often targeted at an entire country, and one country only, which prohibits the use of geographic controls within that country. Likewise, a campaign might be launched in several countries but at different points in time. Thus, while a large control group may be available, the treatment group often consists of no more than one region, or a few regions with considerable heterogeneity among them. A standard approach to causal inference in such settings is based on a linear model of the observed outcomes in the treatment and control group before and after the intervention. One can then estimate the difference between (i) the pre-post difference in the treatment group and (ii) the pre-post difference in the control group. The assumption underlying such differencein-differences (DD) designs is that the level of the control group provides4 K.H. BRODERSEN ET AL. 20 60 100 140 (a) Y Model fit Prediction X1 X2 -40 0 20 60 (b) Point-wise impact Ground truth -2000 0 2000 6000 (c) 2013-01 2013-02 2013-03 2013-04 2013-05 2013-06 2013-07 2013-08 2013-09 2013-10 2013-11 2013-12 2014-01 2014-02 2014-03 2014-04 2014-05 2014-06 Cumulative impact Ground truth Figure 1. Inferring causal impact through counterfactual predictions. (a) Simulated trajectory of a treated market (Y ) with an intervention beginning in January 2014. Two other markets (X1, X2) were not subject to the intervention and allow us to construct a synthetic control (cf. Abadie and Gardeazabal, 2003; Abadie, Diamond and Hainmueller, 2010). Inverting the state-space model described in the main text yields a prediction of what would have happened in Y had the intervention not taken place (posterior predictive expectation of the counterfactual with pointwise 95% posterior probability intervals). (b) The difference between observed data and counterfactual predictions is the inferred causal impact of the intervention. Here, predictions accurately reflect the true (Gamma-shaped) impact. A key characteristic of the inferred impact series is the progressive widening of the posterior intervals (shaded area). This effect emerges naturally from the model structure and agrees with the intuition that predictions should become increasingly uncertain as we look further and further into the (retrospective) future. (c) Another way of visualizing posterior inferences is by means of a cumulative impact plot. It shows, for each day, the summed effect up to that day. Here, the 95% credible interval of the cumulative impact crosses the zeroline about five months after the intervention, at which point we would no longer declare a significant overall effect.BAYESIAN CAUSAL IMPACT ANALYSIS 5 an adequate proxy for the level that would have been observed in the treatment group in the absence of treatment (see Lester, 1946; Campbell, Stanley and Gage, 1963; Ashenfelter and Card, 1985; Card and Krueger, 1993; Angrist and Krueger, 1999; Athey and Imbens, 2002; Abadie, 2005; Meyer, 1995; Shadish, Cook and Campbell, 2002; Donald and Lang, 2007; Angrist and Pischke, 2008; Robinson, McNulty and Krasno, 2009; Antonakis et al., 2010). DD designs have been limited in three ways. First, DD is traditionally based on a static regression model that assumes i.i.d. data despite the fact that the design has a temporal component. When fit to serially correlated data, static models yield overoptimistic inferences with too narrow uncertainty intervals (see also Solon, 1984; Hansen, 2007a,b; Bertrand, Duflo and Mullainathan, 2002). Second, most DD analyses only consider two time points: before and after the intervention. In practice, the manner in which an effect evolves over time, especially its onset and decay structure, is often a key question. Third, when DD analyses are based on time series, previous studies have imposed restrictions on the way in which a synthetic control is constructed from a set of predictor variables, which is something we wish to avoid. For example, one strategy (Abadie and Gardeazabal, 2003; Abadie, Diamond and Hainmueller, 2010) has been to choose a convex combination (w1, . . . , wJ ), wj ≥ 0, P wj = 1 of J predictor time series in such a way that a vector of pre-treatment variables (not time series) X1 characterising the treated unit before the intervention is matched most closely by the combination of pre-treatment variables X0 of the control units w.r.t. a vector of importance weights (v1, . . . , vJ ). These weights are themselves determined in such a way that the combination of pre-treatment outcome time series of the control units most closely matches the pre-treatment outcome time series of the treated unit. Such a scheme relies on the availability of interpretable characteristics (e.g., growth predictors), and it precludes non-convex combinations of controls when constructing the weight vector W. We prefer to select a combination of control series without reference to external characteristics and purely in terms of how well they explain the pre-treatment outcome time series of the treated unit (while automatically balancing goodness of fit and model complexity through the use of regularizing priors). Another idea (Belloni et al., 2013) has been to use classical variable-selection methods (such as the Lasso) to find a sparse set of predictors. This approach, however, ignores posterior uncertainty about both which predictors to use and their coefficients. The limitations of DD schemes can be addressed by using state-space6 K.H. BRODERSEN ET AL. models, coupled with highly flexible regression components, to explain the temporal evolution of an observed outcome. State-space models distinguish between a state equation that describes the transition of a set of latent variables from one time point to the next and an observation equation that specifies how a given system state translates into measurements. This distinction makes them extremely flexible and powerful (see Leeflang et al., 2009, for a discussion in the context of marketing research). The approach described in this paper inherits three main characteristics from the state-space paradigm. First, it allows us to flexibly accommodate different kinds of assumptions about the latent state and emission processes underlying the observed data, including local trends and seasonality. Second, we use a fully Bayesian approach to inferring the temporal evolution of counterfactual activity and incremental impact. One advantage of this is the flexibility with which posterior inferences can be summarised. Third, we use a regression component that precludes a rigid commitment to a particular set of controls by integrating out our posterior uncertainty about the influence of each predictor as well as our uncertainty about which predictors to include in the first place, which avoids overfitting. The remainder of this paper is organised as follows. Section 2 describes the proposed model, its design variations, the choice of diffuse empirical priors on hyperparameters, and a stochastic algorithm for posterior inference based on Markov chain Monte Carlo (MCMC). Section 3 demonstrates important features of the model using simulated data, followed by an application in Section 4 to an advertising campaign run by one of Google’s advertisers. Section 5 puts our approach into context and discusses its scope of application. 2. Bayesian structural time-series models. Structural time-series models are state-space models for time-series data. They can be defined in terms of a pair of equations yt = Z T (2.1) t αt + t αt+1 = Ttαt + Rtηt (2.2) , where t ∼ N (0, σ2 t ) and ηt ∼ N (0, Qt) are independent of all other unknowns. Equation (2.1) is the observation equation; it links the observed data yt to a latent d-dimensional state vector αt . Equation (2.2) is the state equation; it governs the evolution of the the state vector αt through time. In the present paper, yt is a scalar observation, Zt is a d-dimensional output vector, Tt is a d × d transition matrix, Rt is a d × q control matrix, t is a scalar observation error with noise variance σt , and ηt is a q-dimensionalBAYESIAN CAUSAL IMPACT ANALYSIS 7 system error with a q × q state-diffusion matrix Qt , where q ≤ d. Writing the error structure of equation (2.2) as Rtηt allows us to incorporate state components of less than full rank; a model for seasonality will be the most important example. Structural time-series models are useful in practice because they are flexible and modular. They are flexible in the sense that a very large class of models, including all ARIMA models, can be written in the state-space form given by (2.1) and (2.2). They are modular in the sense that the latent state as well as the associated model matrices Zt , Tt , Rt , and Qt can be assembled from a library of component sub-models to capture important features of the data. There are several widely used state-component models for capturing the trend, seasonality, or effects of holidays. A common approach is to assume the errors of different state-component models to be independent (i.e., Qt is block-diagonal). The vector αt can then be formed by concatenating the individual state components, while Tt and Rt become block-diagonal matrices. The most important state component for the applications considered in this paper is a regression component that allows us to obtain counterfactual predictions by constructing a synthetic control based on a combination of markets that were not treated. Observed responses from such markets are important because they allow us to explain variance components in the treated market that are not readily captured by more generic seasonal submodels. This approach assumes that covariates are unaffected by the effects of treatment. For example, an advertising campaign run in the United States might spill over to Canada or the United Kingdom. When assuming the absence of spill-over effects, the use of such indirectly affected markets as controls would lead to pessimistic inferences; that is, the effect of the campaign would be underestimated (cf. Meyer, 1995). 2.1. Components of state. Local linear trend. The first component of our model is a local linear trend, defined by the pair of equations µt+1 = µt + δt + ηµ,t δt+1 = δt + ηδ,t (2.3) where ηµ,t ∼ N (0, σ2 µ ) and ηδ,t ∼ N (0, σ2 δ ). The µt component is the value of the trend at time t. The δt component is the expected increase in µ between times t and t + 1, so it can be thought of as the slope at time t.8 K.H. BRODERSEN ET AL. The local linear trend model is a popular choice for modelling trends because it quickly adapts to local variation, which is desirable when making short-term predictions. This degree of flexibility may not be desired when making longer-term predictions, as such predictions often come with implausibly wide uncertainty intervals. There is a generalization of the local linear trend model where the slope exhibits stationarity instead of obeying a random walk. This model can be written as µt+1 = µt + δt + ηµ,t δt+1 = D + ρ(δt − D) + ηδ,t, (2.4) where the two components of η are independent. In this model, the slope of the time trend exhibits AR(1) variation around a long-term slope of D. The parameter |ρ| < 1 represents the learning rate at which the local trend is updated. Thus, the model balances short-term information with information from the distant past. Seasonality. There are several commonly used state-component models to capture seasonality. The most frequently used model in the time domain is (2.5) γt+1 = − S X−2 s=0 γt−s + ηγ,t, where S represents the number of seasons, and γt denotes their joint contribution to the observed response yt . The state in this model consists of the S − 1 most recent seasonal effects, but the error term is a scalar, so the evolution equation for this state model is less than full rank. The mean of γt+1 is such that the total seasonal effect is zero when summed over S seasons. For example, if we set S = 4 to capture four seasons per year, the mean of the winter coefficient will be −1 × (spring + summer + autumn). The part of the transition matrix Tt representing the seasonal model is an S−1 × S−1 matrix with −1’s along the top row, 1’s along the subdiagonal, and 0’s elsewhere. The preceding seasonal model can be generalized to allow for multiple seasonal components with different periods. When modelling daily data, for example, we might wish to allow for an S = 7 day-of-week effect, as well as an S = 52 weekly annual cycle. The latter can be handled by setting Tt = IS−1, with zero variance on the error term, when t is not the start of a new week, and setting Tt to the usual seasonal transition matrix, with nonzero error variance, when t is the start of a new week.BAYESIAN CAUSAL IMPACT ANALYSIS 9 Contemporaneous covariates with static coefficients. Control time series that received no treatment are critical to our method for obtaining accurate counterfactual predictions since they account for variance components that are shared by the series, including in particular the effects of other unobserved causes otherwise unaccounted for by the model. A natural way of including control series in the model is through a linear regression. Its coefficients can be static or time-varying. A static regression can be written in state-space form by setting Zt = β Txt and αt = 1. One advantage of working in a fully Bayesian treatment is that we do not need to commit to a fixed set of covariates. The spike-andslab prior described in Section 2.2 allows us to integrate out our posterior uncertainty about which covariates to include and how strongly they should influence our predictions, which avoids overfitting. All covariates are assumed to be contemporaneous; the present model does not infer on a potential lag between treated and untreated time series. A known lag, however, can be easily incorporated by shifting the corresponding regressor in time. Contemporaneous covariates with dynamic coefficients. An alternative to the above is a regression component with dynamic regression coefficients to account for time-varying relationships (e.g., Banerjee, Kauffman and Wang, 2007; West and Harrison, 1997). Given covariates j = 1 . . . J, this introduces the dynamic regression component x T t βt = X J j=1 xj,tβj,t βj,t+1 = βj,t + ηβ,j,t, (2.6) where ηβ,j,t ∼ N (0, σ2 βj ). Here, βj,t is the coefficient for the j th control series and σβj is the standard deviation of its associated random walk. We can write the dynamic regression component in state-space form by setting Zt = xt and αt = βt and by setting the corresponding part of the transition matrix to Tt = IJ×J , with Qt = diag(σ 2 βj ). Assembling the state-space model. Structural time-series models allow us to examine the time series at hand and flexibly choose appropriate components for trend, seasonality, and either static or dynamic regression for the controls. The presence or absence of seasonality, for example, will usually be obvious by inspection. A more subtle question is whether to choose static or dynamic regression coefficients. When the relationship between controls and treated unit has been stable in the past, static coefficients are an attractive option. This is because10 K.H. BRODERSEN ET AL. a spike-and-slab prior can be implemented efficiently within a forward- filtering, backward-sampling framework. This makes it possible to quickly identify a sparse set of covariates even from tens or hundreds of potential variables (Scott and Varian, 2013). Local variability in the treated time series is captured by the dynamic local level or dynamic linear trend component. Covariate stability is typically high when the available covariates are close in nature to the treated metric. The empirical analyses presented in this paper, for example, will be based on a static regression component (Section 4). This choice provides a reasonable compromise between capturing local behaviour and accounting for regression effects. An alternative would be to use dynamic regression coefficients, as we do, for instance, in our analyses of simulated data (Section 3). Dynamic coefficients are useful when the linear relationship between treated metrics and controls is believed to change over time. There are a number of ways of reducing the computational burden of dealing with a potentially large number of dynamic coefficients. One option is to resort to dynamic latent factors, where one uses xt = But +νt with dim(ut) J and uses ut instead of xt as part of Zt in (2.1), coupled with an AR-type model for ut itself. Another option is latent thresholding regression, where one uses a dynamic version of the spike-and-slab prior as in Nakajima and West (2013). The state-component models are assembled independently, with each component providing an additive contribution to yt . Figure 2 illustrates this process assuming a local linear trend paired with a static regression component. 2.2. Prior distributions and prior elicitation. Let θ generically denote the set of all model parameters and let α = (α1, . . . , αm) denote the full state sequence. We adopt a Bayesian approach to inference by specifying a prior distribution p(θ) on the model parameters as well as a distribution p(α0|θ) on the initial state values. We may then sample from p(α, θ|y) using MCMC. Most of the models in Section 2.1 depend solely on a small set of variance parameters that govern the diffusion of the individual state components. A typical prior distribution for such a variance is (2.7) 1 σ 2 ∼ G ν 2 , s 2 , where G (a, b) is the Gamma distribution with expectation a/b. The prior parameters can be interpreted as a prior sum of squares s, so that s/ν is a prior estimate of σ 2 , and ν is the weight, in units of prior sample size, assigned to the prior estimate.BAYESIAN CAUSAL IMPACT ANALYSIS 11 𝜇1 𝑦1 𝒩 𝑦𝑡 𝜇𝑡 + 𝑥𝑡 T𝛽𝜚, 𝜎𝑦 2 𝜇𝑛 … 𝑦𝑛 𝜇𝑛+1 𝑦 𝑛+1 𝜇𝑚 𝑦 𝑚 𝑥1 𝑥𝑛 𝑥𝑛+1 𝑥𝑚 𝒩 𝜇𝑡 𝜇𝑡−1 + 𝛿𝑡−1, 𝜎𝜇 2 𝒩 𝛿𝑡 𝛿𝑡−1, 𝜎𝛿 2 … … … … … 𝜎𝜇 𝜎𝛿 𝜎𝑦 𝜇0 local trend local level observed (𝑦) and counterfactual (𝑦 ) activity controls control selection pre-intervention period post-intervention period 𝛿1 𝛿𝑛+1 𝛿𝑚 𝜚 diffusion parameters 𝒩 𝛽𝜚 𝑏𝜚, 𝜎𝜖 2 Σ𝜚 −1 −1 𝛿0 𝛿𝑛 𝜎𝜖 𝛽𝜚 regression coefficients observation noise Figure 2. Graphical model for the static-regression variant of the proposed state-space model. Observed market activity y1:n = (y1, . . . , yn) is modelled as the result of a latent state plus Gaussian observation noise with error standard deviation σy. The state αt includes a local level µt, a local linear trend δt, and a set of contemporaneous covariates xt, scaled by regression coefficients β%. State components are assumed to evolve according to independent Gaussian random walks with fixed standard deviations σµ and σδ (conditionaldependence arrows shown for the first time point only). The model includes empirical priors on these parameters and the initial states. In an alternative formulation, the regression coefficients β are themselves subject to random-walk diffusion (see main text). Of principal interest is the posterior predictive density over the unobserved counterfactual responses y˜n+1, . . . , y˜m. Subtracting these from the actual observed data yn+1, . . . , ym yields a probability density over the temporal evolution of causal impact.12 K.H. BRODERSEN ET AL. We often have a weak default prior belief that the incremental errors in the state process are small, which we can formalize by choosing small values of ν (e.g., 1) and small values of s/ν. The notion of ‘small’ means different things in different models; for the seasonal and local linear trend models our default priors are 1/σ2 ∼ G(10−2 , 10−2 s 2 y ), where s 2 y = P t (yt − y¯) 2/(n − 1) is the sample variance of the target series. Scaling by the sample variance is a minor violation of the Bayesian paradigm, but it is an effective means of choosing a reasonable scale for the prior. It is similar to the popular technique of scaling the data prior to analysis, but we prefer to do the scaling in the prior so we can model the data on its original scale. When faced with many potential controls, we prefer letting the model choose an appropriate set. This can be achieved by placing a spike-andslab prior over coefficients (George and McCulloch, 1993, 1997; Polson and Scott, 2011; Scott and Varian, 2013). A spike-and-slab prior combines point mass at zero (the ‘spike’), for an unknown subset of zero coefficients, with a weakly informative distribution on the complementary set of non-zero coefficients (the ‘slab’). Contrary to what its name might suggest, the ‘slab’ is usually not completely flat, but rather a Gaussian with a large variance. Let % = (%1, . . . , %J ), where %j = 1 if βj 6= 0 and %j = 0 otherwise. Let β% denote the non-zero elements of the vector β and let Σ−1 % denote the rows and columns of Σ−1 corresponding to non-zero entries in %. We can then factorize the spike-and-slab prior as (2.8) p(%, β, 1/σ2 ) = p(%) p(σ 2 |%) p(β%|%, σ2 ). The spike portion of (2.8) can be an arbitrary distribution over {0, 1} J in principle; the most common choice in practice is a product of independent Bernoulli distributions, (2.9) p(%) = Y J j=1 π %j j (1 − πj ) 1−%j , where πj is the prior probability of regressor j being included in the model. Values for πj can be elicited by asking about the expected model size M, and then setting all πj = M/J. An alternative is to use a more specific set of values πj . In particular, one might choose to set certain πj to either 1 or 0 to force the corresponding variables into or out of the model. Generally, framing the prior in terms of expected model size has the advantage that the model can adapt to growing numbers of predictor variables without having to switch to a hierarchical prior (Scott and Berger, 2010).BAYESIAN CAUSAL IMPACT ANALYSIS 13 For the ‘slab’ portion of the prior we use a conjugate normal-inverse Gamma distribution, β%|σ 2 ∼ N b%, σ2 (Σ−1 % ) −1 (2.10) 1 σ 2 ∼ G ν 2 , s 2 (2.11) . The vector b in equation (2.10) encodes our prior expectation about the value of each element of β. In practice, we usually set b = 0. The prior parameters in equation (2.11) can be elicited by asking about the expected R2 ∈ [0, 1] as well as the number of observations worth of weight ν the prior estimate should be given. Then s = ν(1 − R2 )s 2 y . The final prior parameter in (2.10) is Σ−1 which, up to a scaling factor, is the prior precision over β in the full model, with all variables included. The total information in the covariates is XTX, and so 1 nXTX is the average information in a single observation. Zellner’s g-prior (Zellner, 1986; Chipman et al., 2001; Liang et al., 2008) sets Σ−1 = g nXTX, so that g can be interpreted as g observations worth of information. Zellner’s prior becomes improper when XTX is not positive definite; we therefore ensure propriety by averaging XTX with its diagonal, (2.12) Σ−1 = g n n wXTX + (1 − w) diag XTX o with default values of g = 1 and w = 1/2. Overall, this prior specification provides a broadly useful default while providing considerable flexibility in those cases where more specific prior information is available. 2.3. Inference. Posterior inference in our model can be broken down into three pieces. First, we simulate draws of the model parameters θ and the state vector α given the observed data y1:n in the training period. Second, we use the posterior simulations to simulate from the posterior predictive distribution p(y˜n+1:m|y1:n) over the counterfactual time series y˜n+1:m given the observed pre-intervention activity y1:n. Third, we use the posterior predictive samples to compute the posterior distribution of the pointwise impact yt−y˜t for each t = 1, . . . , m. We use the same samples to obtain the posterior distribution of cumulative impact. Posterior simulation. We use a Gibbs sampler to simulate a sequence (θ, α) (1) ,(θ, α) (2) , . . . from a Markov chain whose stationary distribution is p(θ, α|y1:n). The sampler alternates between: a data-augmentation step that simulates from p(α|y1:n, θ); and a parameter-simulation step that simulates from p(θ|y1:n, α).14 K.H. BRODERSEN ET AL. The data-augmentation step uses the posterior simulation algorithm from Durbin and Koopman (2002), providing an improvement over the earlier forward-filtering, backward-sampling algorithms by Carter and Kohn (1994), Fr¨uhwirth-Schnatter (1994), and de Jong and Shephard (1995). In brief, because p(y1:n, α|θ) is jointly multivariate normal, the variance of p(α|y1:n, θ) does not depend on y1:n. We can therefore simulate (y ∗ 1:n , α∗ ) ∼ p(y1:n, α|θ) and subtract E(α ∗ |y ∗ 1:n , θ) to obtain zero-mean noise with the correct variance. Adding E(α|y1:n, θ) restores the correct mean, which completes the draw. The required expectations can be computed using the Kalman filter and a fast mean smoother described in detail by Durbin and Koopman (2002). The result is a direct simulation from p(α|y1:n, θ) in an algorithm that is linear in the total (pre- and post-intervention) number of time points (m), and quadratic in the dimension of the state space (d). Given the draw of the state, the parameter draw is straightforward for all state components other than the static regression coefficients β. All state components that exclusively depend on variance parameters can translate their draws back to error terms ηt , accumulate sums of squares of η, and because of conjugacy with equation (2.7) the posterior distribution will remain Gamma distributed. The draw of the static regression coefficients β proceeds as follows. For each t = 1, . . . , n in the pre-intervention period, let ˙yt denote yt with the contributions from the other state components subtracted away, and let y˙ 1:n = ( ˙y1, . . . , y˙n). The challenge is to simulate from p(%, β, σ2 |y˙ 1:n), which we can factor into p(%|y1:n)p(1/σ2 |%, y˙ 1:n)p(β|%, σ, y˙ 1:n). Because of conjugacy, we can integrate out β and 1/σ2 and be left with (2.13) %|y˙ 1:n ∼ C(y˙ 1:n) |Σ −1 % | 1 2 |V −1 % | 1 2 p(%) S N 2 −1 % , where C(y˙ 1:n) is an unknown normalizing constant. The sufficient statistics in equation (2.13) are V −1 % = XTX % + Σ−1 % β˜ % = (V −1 % ) −1 (XT % y˙ 1:n + Σ−1 % b%) N = ν + n S% = s + y˙ T 1:ny˙ 1:n + b T % Σ −1 % b% − β˜T % V −1 % β˜ %. To sample from (2.13), we use a Gibbs sampler that draws each %j given all other %−j . Each full-conditional is easy to evaluate because %j can only assume two possible values. It should be noted that the dimension of all matrices in (2.13) is P j %j , which is small if the model is truly sparse. There are many matrices to manipulate, but because each is small the overallBAYESIAN CAUSAL IMPACT ANALYSIS 15 algorithm is fast. Once the draw of % is complete, we sample directly from p(β, 1/σ2 |%, y˙ 1:n) using standard conjugate formulae. For an alternative that may be even more computationally efficient, see Ghosh and Clyde (2011). Posterior predictive simulation. While the posterior over model parameters and states p(θ, α|y1:n) can be of interest in its own right, causal impact analyses are primarily concerned with the posterior incremental effect, (2.14) p (y˜n+1:m | y1:n, x1:m). As shown by its indices, the density in equation (2.14) is defined precisely for that portion of the time series which is unobserved: the counterfactual market response ˜yn+1, . . . , y˜m that would have been observed in the treated market, after the intervention, in the absence of treatment. It is also worth emphasizing that the density is conditional on the observed data (as well as the priors) and only on these, i.e., on activity in the treatment market before the beginning of the intervention as well as activity in all control markets both before and during the intervention. The density is not conditioned on parameter estimates or the inclusion or exclusion of covariates with static regression coefficients, all of which have been integrated out. Thus, through Bayesian model averaging, we neither commit to any particular set of covariates, which helps avoid an arbitrary selection; nor to point estimates of their coefficients, which prevents overfitting. The posterior predictive density in (2.14) is defined as a coherent (joint) distribution over all counterfactual data points, rather than as a collection of pointwise univariate distributions. This ensures that we correctly propagate the serial structure determined on pre-intervention data to the trajectory of counterfactuals. This is crucial, in particular, when forming summary statistics, such as the cumulative effect of the intervention on the treatment market. Posterior inference was implemented in C++ with an R interface. Given a typically-sized dataset with m = 500 time points, J = 10 covariates, and 10,000 iterations (see Section 4 for an example), this implementation takes less than 30 seconds to complete on a standard computer, enabling nearinteractive analyses. 2.4. Evaluating impact. Samples from the posterior predictive distribution over counterfactual activity can be readily used to obtain samples from the posterior causal effect, i.e., the quantity we are typically interested in. For each draw τ and for each time point t = n + 1, . . . , m, we set φ (τ) t := yt − y˜ (τ) t (2.15) ,16 K.H. BRODERSEN ET AL. yielding samples from the approximate posterior predictive density of the effect attributed to the intervention. In addition to its pointwise impact, we often wish to understand the cumulative effect of an intervention over time. One of the main advantages of a sampling approach to posterior inference is the flexibility and ease with which such derived inferences can be obtained. Reusing the impact samples obtained in (2.15), we compute for each draw τ X t t 0=n+1 φ (τ) t (2.16) 0 ∀t = n + 1, . . . , m. The preceding cumulative sum of causal increments is a useful quantity when y represents a flow quantity, measured over an interval of time (e.g., a day), such as the number of searches, sign-ups, sales, additional installs, or new users. It becomes uninterpretable when y represents a stock quantity, usefully defined only for a point in time, such as the total number of clients, users, or subscribers. In this case we might instead choose, for each τ , to draw a sample of the posterior running average effect following the intervention, 1 t − n X t t 0=n+1 φ (τ) t (2.17) 0 ∀t = n + 1, . . . , m. Unlike the cumulative effect in (2.16), the running average is always interpretable, regardless of whether it refers to a flow or a stock. However, it is more context-dependent on the length of the post-intervention period under consideration. In particular, under the assumption of a true impact that grows quickly at first and then declines to zero, the cumulative impact approaches its true total value (in expectation) as we increase the counterfactual forecasting period, whereas the average impact will eventually approach zero (while, in contrast, the probability intervals diverge in both cases, leading to more and more uncertain inferences as the forecasting period increases). 3. Application to simulated data. To study the characteristics of our approach, we analysed simulated (i.e., computer-generated) data across a series of independent simulations. Generated time series started on 1 January 2013 and ended on 30 June 2014, with a perturbation beginning on 1 January 2014. The data were simulated using a dynamic regression component with two covariates whose coefficients evolved according to independent random walks, βt ∼ N (βt−1, 0.012 ), initialized at β0 = 1. The covariates themselves were simple sinusoids with wavelengths of 90 days and 360 days, respectively.BAYESIAN CAUSAL IMPACT ANALYSIS 17 12 14 16 18 20 22 24 (a) 2013-01 2013-02 2013-03 2013-04 2013-05 2013-06 2013-07 2013-08 2013-09 2013-10 2013-11 2013-12 2014-01 2014-02 2014-03 2014-04 2014-05 2014-06 observed predicted true (b) effect size (%) proportion of intervals excluding zero 0 0.1 1 10 25 50 0.0 0.2 0.4 0.6 0.8 1.0 50 100 150 0.0 0.2 0.4 0.6 0.8 1.0 (c) campaign duration (days) proportion of intervals containing truth Figure 3. Adequacy of posterior uncertainty. (a) Example of one of the 256 datasets created to assess estimation accuracy. Simulated observations (black) are based on two contemporaneous covariates, scaled by time-varying coefficients plus a time-varying local level (not shown). During the campaign period, where the data are lifted by an effect size of 10%, the plot shows the posterior expectation of counterfactual activity (blue), along with its pointwise central 95% credible intervals (blue shaded area), and, for comparison, the true counterfactual (green). (b) Power curve. Following repeated application of the model to simulated data, the plot shows the empirical frequency of concluding that a causal effect was present, as a function of true effect size, given a post-intervention period of 6 months. The curve represents sensitivity in those parts of the graph where the true effect size is positive, and 1 − specificity where the true effect size is zero. Error bars represent 95% credible intervals for the true sensitivity, using a uniform Beta(1, 1) prior. (c) Interval coverage. Using an effect size of 10%, the plot shows the proportion of simulations in which the pointwise central 95% credible interval contained the true impact, as a function of campaign duration. Intervals should contain ground truth in 95% of simulations, however much uncertainty its predictions may be associated with. Error bars represent 95% credible intervals. The latent state underlying the observed data was generated using a local level that evolved according to a random walk, µt ∼ N (µt−1, 0.1 2 ), initialized at µ0 = 0. Independent observation noise was sampled using t ∼ N (0, 0.1 2 ). In summary, observations yt were generated using yt = βt,1zt,1 + βt,2zt,2 + µt + t . To simulate the effect of advertising, the post-intervention portion of the preceding series was multiplied by 1 +e, where e (not to be confused with ) represented the true effect size specifying the (uniform) relative lift during the campaign period. An example is shown in Figure 3a. Sensitivity and specificity. To study the properties of our model, we began by considering under what circumstances we successfully detected a causal18 K.H. BRODERSEN ET AL. effect, i.e., the statistical power or sensitivity of our approach. A related property is the probability of not detecting an absent impact, i.e., specificity. We repeatedly generated data, as described above, under different true effect sizes. We then computed the posterior predictive distribution over the counterfactuals, and recorded whether or not we would have concluded a causal effect. For each of the effect sizes 0%, 0.1%, 1%, 10%, and 100%, a total of 2 8 = 256 simulations were run. This number was chosen simply on the grounds that it provided reasonably tight intervals around the reported summary statistics without requiring excessive amounts of computation. In each simulation, we concluded that a causal effect was present if and only if the central 95% posterior probability interval of the cumulative effect excluded zero. The model used throughout this section comprised two structural blocks. The first one was a local level component. We placed an inverse-Gamma prior on its diffusion variance with a prior estimate of s/ν = 0.1σy and a prior sample size ν = 32. The second structural block was a dynamic regression component. We placed a Gamma prior with prior expectation 0.1σy on the diffusion variance of both regression coefficients. By construction, the outcome variable did not exhibit any local trends or seasonality other than the variation conveyed through the covariates. This obviated the need to include an explicit local linear trend or seasonality component in the model. In a first analysis, we considered the empirical proportion of simulations in which a causal effect had been detected. When taking into account only those simulations where the true effect size was greater than zero, these empirical proportions provide estimates of the sensitivity of the model w.r.t. the process by which the data were generated. Conversely, those simulations where the campaign had had no effect yield an estimate of the model’s specificity. In this way, we obtained the power curve shown in Figure 3b. The curve shows that, in data such as these, a market perturbation leading to a lift no larger than 1% is missed in about 90% of cases. By contrast, a perturbation that lifts market activity by 25% is correctly detected as such in most cases. In a second analysis, we assessed the coverage properties of the posterior probability intervals obtained through our model. It is desirable to use a diffuse prior on the local level component such that central 95% intervals contain ground truth in about 95% of the simulations. This coverage frequency should hold regardless of the length of the campaign period. In other words, a longer campaign should lead to posterior intervals that are appropriately widened to retain the same coverage probability as the nar-BAYESIAN CAUSAL IMPACT ANALYSIS 19 rower intervals obtained for shorter campaigns. This was approximately the case throughout the simulated campaign (Figure 3c). Estimation accuracy. To study the accuracy of the point estimates supported by our approach, we repeated the preceding simulations with a fixed effect size of 10% while varying the length of the campaign. When given a quadratic loss function, the loss-minimizing point estimate is the posterior expectation of the predictive density over counterfactuals. Thus, for each generated dataset i, we computed the expected causal effect for each time point, φˆ i,t := hφt | y1, . . . , ym, x1, . . . , xmi ∀t = n + 1, . . . , m; i = 1, . . . , 256. (3.1) To quantify the discrepancy between estimated and true impact, we calculated the absolute percentage estimation error, ai,t := φˆ i,t − φt φt (3.2) . This yielded an empirical distribution of absolute percentage estimation errors (Figure 4a; blue), showing that impact estimates become less and less accurate as the forecasting period increases. This is because, under the local linear trend model in (2.3), the true counterfactual activity becomes more and more likely to deviate from its expected trajectory. It is worth emphasizing that all preceding results are based on the assumption that the model structure remains intact throughout the modelling period. In other words, even though the model is built around the idea of multiple (non-stationary) components (i.e., a time-varying local trend and, potentially, time-varying regression coefficients), this structure itself remains unchanged. If the model structure does change, estimation accuracy may suffer. We studied the impact of a changing model structure in a second simulation in which we repeated the procedure above in such a way that 90 days after the beginning of the campaign the standard deviation of the random walk governing the evolution of the regression coefficient was tripled (now 0.03 instead of 0.01). As a result, the observed data began to diverge much more quickly than before. Accordingly, estimations became considerably less reliable (Figure 4a, red). An example of the underlying data is shown in Figure 4b. The preceding simulations highlight the importance of a model that is sufficiently flexible to account for phenomena typically encountered in sea-20 K.H. BRODERSEN ET AL. 0 50 100 150 200 250 300 (a) absolute % error 2014-01 2014-02 2014-03 2014-04 2014-05 2014-06 15 20 25 (b) 2014-01 2014-02 2014-03 2014-04 2014-05 2014-06 Figure 4. Estimation accuracy. (a) Time series of absolute percentage discrepancy between inferred effect and true effect. The plot shows the rate (mean ± 2 s.e.m.) at which predictions become less accurate as the length of the counterfactual forecasting period increases (blue). The well-behaved decrease in estimation accuracy breaks down when the data are subject to a sudden structural change (red), as simulated for 1 April 2014. (b) Illustration of a structural break. The plot shows one example of the time series underlying the red curve in (a). On 1 April 2014, the standard deviation of the generating random walk of the local level was tripled, causing the rapid decline in estimation accuracy seen in the red curve in (a).BAYESIAN CAUSAL IMPACT ANALYSIS 21 sonal empirical data. This rules out entirely static models in particular (such as multiple linear regression). 4. Application to empirical data. To illustrate the practical utility of our approach, we analysed an advertising campaign run by one of Google’s advertisers in the United States. In particular, we inferred the campaign’s causal effect on the number of times a user was directed to the advertiser’s website from the Google search results page. We provide a brief overview of the underlying data below (see Vaver and Koehler, 2011, for additional details). The campaign analysed here was based on product-related ads to be displayed alongside Google’s search results for specific keywords. Ads went live for a period of 6 consecutive weeks and were geo-targeted to a randomised set of 95 out of 190 designated market areas (DMAs). The most salient observable characteristic of DMAs is offline sales. To produce balance in this characteristic, DMAs were first rank-ordered by sales volume. Pairs of regions were then randomly assigned to treatment/control. DMAs provide units that can be easily supplied with distinct offerings, although this finegrained split was not a requirement for the model. In fact, we carried out the analysis as if only one treatment region had been available (formed by summing all treated DMAs). This allowed us to evaluate whether our approach would yield the same results as more conventional treatment-control comparisons would have done. The outcome variable analysed here was search-related visits to the advertiser’s website, consisting of organic clicks (i.e., clicks on a search result) and paid clicks (i.e., clicks on an ad next to the search results, for which the advertiser was charged). Since paid clicks were zero before the campaign, one might wonder why we could not simply count the number of paid clicks after the campaign had started. The reason is that paid clicks tend to cannibalize some organic clicks. Since we were interested in the net effect, we worked with the total number of clicks. The first building block of the model used for the analyses in this section was a local level component. For the inverse-Gamma prior on its diffusion variance we used a prior estimate of s/ν = 0.1σy and a prior sample size ν = 32. The second structural block was a static regression component. We used a spike-and-slab prior with an expected model size of M = 3, an expected explained variance of R2 = 0.8 and 50 prior df. We deliberately kept the model as simple as this. Since the covariates came from a randomised experiment, we expected them to already account for any additional local linear trends and seasonal variation in the response variable. If one suspects22 K.H. BRODERSEN ET AL. that a more complex model might be more appropriate, one could optimize model design through Bayesian model selection. Here, we focus instead on comparing different sets of covariates, which is critical in counterfactual analyses regardless of the particular model structure used. Model estimation was carried out using 10 000 MCMC samples. Analysis 1: Effect on the treated, using a randomised control. We began by applying the above model to infer the causal effect of the campaign on the time series of clicks in the treated regions. Given that a set of unaffected regions was available in this analysis, the best possible set of controls was given by the untreated DMAs themselves (see below for a comparison with a purely observational alternative). As shown in Figure 5a, the model provided an excellent fit on the precampaign trajectory of clicks (including a spike in ‘week −2’ and a dip at the end of ‘week −1’). Following the onset of the campaign, observations quickly began to diverge from counterfactual predictions: the actual number of clicks was consistently higher than what would have been expected in the absence of the campaign. The curves did not reconvene until one week after the end of the campaign. Subtracting observed from predicted data, as we did in Figure 5b, resulted in a posterior estimate of the incremental lift caused by the campaign. It peaked after about three weeks into the campaign, and faded away after about one week after the end of the campaign. Thus, as shown in Figure 5c, the campaign led to a sustained cumulative increase in total clicks (as opposed to a mere shift of future clicks into the present, or a pure cannibalization of organic clicks by paid clicks). Specifically, the overall effect amounted to 88 400 additional clicks in the targeted regions (posterior expectation; rounded to three significant digits), i.e., an increase of 22%, with a central 95% credible interval of [13%, 30%]. To validate this estimate, we returned to the original experimental data, on which a conventional treatment-control comparison had been carried out using a two-stage linear model (Vaver and Koehler, 2011). This analysis had led to an estimated lift of 84 700 clicks, with a 95% confidence interval for the relative expected lift of [19%, 22%]. Thus, with a deviation of less than 5%, the counterfactual approach had led to almost precisely the same estimate as the randomised evaluation, except for its wider intervals. The latter is expected, given that our intervals represent prediction intervals, not confi- dence intervals. Moreover, in addition to an interval for the sum over all time points, our approach yields a full time series of pointwise intervals, which allows analysts to examine the characteristics of the temporal evolution of attributable impact. The posterior predictive intervals in Figure 5b widen more slowly than inBAYESIAN CAUSAL IMPACT ANALYSIS 23 4000 8000 12000 (a) US clicks Model fit Prediction pre-intervention intervention post-intervention -2000 0 2000 (b) Point-wise impact 0e+00 1e+05 (c) Cumulative impact week -4 week -3 week -2 week -1 week 0 week 1 week 2 week 3 week 4 week 5 week 6 week 7 Figure 5. Causal effect of online advertising on clicks in treated regions. (a) Time series of search-related visits to the advertiser’s website (including both organic and paid clicks). (b) Pointwise (daily) incremental impact of the campaign on clicks. Shaded vertical bars indicate weekends. (c) Cumulative impact of the campaign on clicks.24 K.H. BRODERSEN ET AL. the illustrative example in Figure 1. This is because the large number of controls available in this data set offers a much higher pre-campaign predictive strength than in the simulated data in Figure 1. This is not unexpected, given that controls came from a randomised experiment, and we will see that this also holds for a subsequent analysis (see below) that is based on yet another data source for predictors. A consequence of this is that there is little variation left to be captured by the random-walk component of the model. A reassuring finding is that the estimated counterfactual time series in Figure 5a eventually almost exactly rejoins the observed series, only a few days after the end of the intervention. Analysis 2: Effect on the treated, using observational controls. An important characteristic of counterfactual-forecasting approaches is that they do not require a setting in which a set of controls, selected at random, was exempt from the campaign. We therefore repeated the preceding analysis in the following way: we discarded the data from all control regions and, instead, used searches for keywords related to the advertiser’s industry, grouped into a handful of verticals, as covariates. In the absence of a dedicated set of control regions, such industry-related time series can be very powerful controls as they capture not only seasonal variations but also market-specific trends and events (though not necessarily advertiser-specific trends). A major strength of the controls chosen here is that time series on web searches are publicly available through Google Trends (http://www.google.com/trends/). This makes the approach applicable to virtually any kind of intervention. At the same time, the industry as a whole is unlikely to be moved by a single actor’s activities. This precludes a positive bias in estimating the effect of the campaign that would arise if a covariate was negatively affected by the campaign. As shown in Figure 6, we found a cumulative lift of 85 900 clicks (posterior expectation), or 21%, with a [12%, 30%] interval. In other words, the analysis replicated almost perfectly the original analysis that had access to a randomised set of controls. One feature in the response variable which this second analysis failed to account for was a spike in clicks in the second week before the campaign onset; this spike appeared both in treated and untreated regions and appears to be specific to this advertiser. In addition, the series of point-wise impact (Figure 6b) is slightly more volatile than in the original analysis (Figure 5). On the other hand, the overall point estimate of 85 900, in this case, was even closer to the randomised-design baseline (84 700; deviation ca. 1%) than in our first analysis (88 400; deviation ca. 4%). In summary, the counterfactual approach effectively obviated the need for the original randomised experiment. Using purely observationalBAYESIAN CAUSAL IMPACT ANALYSIS 25 4000 8000 12000 (a) US clicks Model fit Prediction pre-intervention intervention post-intervention -2000 0 2000 (b) Point-wise impact 0e+00 1e+05 (c) Cumulative impact week -4 week -3 week -2 week -1 week 0 week 1 week 2 week 3 week 4 week 5 week 6 week 7 Figure 6. Causal efect of online advertising on clicks, using only searches for keywords related to the advertiser’s industry as controls, discarding the original control regions as would be the case in studies where a randomised experiment was not carried out. (a) Time series of clicks on to the advertiser’s website. (b) Pointwise (daily) incremental impact of the campaign on clicks. (c) Cumulative impact of the campaign on clicks. The plots show that this analysis, which was based on observational covariates only, provided almost exactly the same inferences as the first analysis (Figure 5) that had been based on a randomised design.26 K.H. BRODERSEN ET AL. variables led to the same substantive conclusions. Analysis 3: Absence of an effect on the controls. To go one step further still, we analysed clicks in those regions that had been exempt from the advertising campaign. If the effect of the campaign was truly specific to treated regions, there should be no effect in the controls. To test this, we inferred the causal effect of the campaign on unaffected regions, which should not lead to a significant finding. In analogy with our second analysis, we discarded clicks in the treated regions and used searches for keywords related to the advertiser’s industry as controls. As summarized in Figure 7, no significant effect was found in unaffected regions, as expected. Specifically, we obtained an overall non-significant lift of 2% in clicks with a central 95% credible interval of [−6%, 10%]. In summary, the empirical data considered in this section showed: (i) a clear effect of advertising on treated regions when using randomised control regions to form the regression component, replicating previous treatmentcontrol comparisons (Figure 5); (ii) notably, an equivalent finding when discarding control regions and instead using observational searches for keywords related to the advertiser’s industry as covariates (Figure 6); (iii) reassuringly, the absence of an effect of advertising on regions that were not targeted (Figure 7). 5. Discussion. The increasing interest in evaluating the incremental impact of market interventions has been reflected by a growing literature on applied causal inference. With the present paper we are hoping to contribute to this literature by proposing a Bayesian state-space model for obtaining a counterfactual prediction of market activity. We discuss the main features of this model below. In contrast to most previous schemes, the approach described here is fully Bayesian, with regularizing or empirical priors for all hyperparameters. Posterior inference gives rise to complete-data (smoothing) predictions that are only conditioned on past data in the treatment market and both past and present data in the control markets. Thus, our model embraces a dynamic evolution of states and, optionally, coefficients (departing from classical linear regression models with a fixed number of static regressors) and enables us to flexibly summarize posterior inferences. Because closed-form posteriors for our model do not exist, we suggest a stochastic approximation to inference using MCMC. One convenient consequence of this is that we can reuse the samples from the posterior to obtain credible intervals for all summary statistics of interest. Such statistics include, for example, the average absolute and relative effect caused by theBAYESIAN CAUSAL IMPACT ANALYSIS 27 4000 8000 12000 (a) US clicks Model fit Prediction pre-intervention intervention post-intervention -2000 0 2000 (b) Point-wise impact 0e+00 1e+05 (c) Cumulative impact week -4 week -3 week -2 week -1 week 0 week 1 week 2 week 3 week 4 week 5 week 6 week 7 Figure 7. Causal effect of online advertising on clicks in non-treated regions, which should not show an effect. Searches for keywords related to the advertiser’s industry are used as controls. Plots show inferences in analogy with Figure 5. (a) Time series of clicks to the advertiser’s website. (b) Pointwise (daily) incremental impact of the campaign on clicks. (c) Cumulative impact of the campaign on clicks.28 K.H. BRODERSEN ET AL. intervention as well as its cumulative effect. Posterior inference was implemented in C++ and R and, for all empirical datasets presented in Section 4, took less than 30 seconds on a standard Linux machine. If the computational burden of sampling-based inference ever became prohibitive, one option would be to replace it by a variational Bayesian approximation (see Mathys et al., 2011; Brodersen et al., 2013, for examples). Another way of using the proposed model is for power analyses. In particular, given past time series of market activity, we can define a point in the past to represent a hypothetical intervention and apply the model in the usual fashion. As a result, we obtain a measure of uncertainty about the response in the treated market after the beginning of the hypothetical intervention. This provides an estimate of what incremental effect would have been required to be outside of the 95% central interval of what would have happened in the absence of treatment. The model presented here subsumes several simpler models which, in consequence, lack important characteristics, but which may serve as alternatives should the full model appear too complex for the data at hand. One example is classical multiple linear regression. In principle, classical regression models go beyond difference-in-differences schemes in that they account for the full counterfactual trajectory. However, they are not suited for predicting stochastic processes beyond a few steps. This is because ordinary leastsquares estimators disregard serial autocorrelation; the static model structure does not allow for temporal variation in the coefficients; and predictions ignore our posterior uncertainty about the parameters. Put differently: classical multiple linear regression is a special case of the state-space model described here in which (i) the Gaussian random walk of the local level has zero variance; (ii) there is no local linear trend; (iii) regression coefficients are static rather than time-varying; (iv) ordinary least squares estimators are used which disregard posterior uncertainty about the parameters and may easily overfit the data. Another special case of the counterfactual approach discussed in this paper is given by synthetic control estimators that are restricted to the class of convex combinations of predictor variables and do not include time-series effects such as trends and seasonality (Abadie, Diamond and Hainmueller, 2010; Abadie, 2005). Relaxing this restriction means we can utilize predictors regardless of their scale, even if they are negatively correlated with the outcome series of the treated unit. Other special cases include autoregressive (AR) and moving-average (MA) models. These models define autocorrelation among observations rather thanBAYESIAN CAUSAL IMPACT ANALYSIS 29 latent states, thus precluding the ability to distinguish between state noise and observation noise (Ataman, Mela and Van Heerde, 2008; Leeflang et al., 2009). In the scenarios we consider, advertising is a planned perturbation of the market. This generally makes it easier to obtain plausible causal inferences than in genuinely observational studies in which the experimenter had no control about treatment (see discussions in Berndt, 1991; Brady, 2002; Hitchcock, 2004; Robinson, McNulty and Krasno, 2009; Winship and Morgan, 1999; Camillo and d’Attoma, 2010; Antonakis et al., 2010; Lewis and Reiley, 2011; Lewis, Rao and Reiley, 2011; Kleinberg and Hripcsak, 2011; Vaver and Koehler, 2011). The principal problem in observational studies is endogeneity: the possibility that the observed outcome might not be the result of the treatment but of other omitted, endogenous variables. In principle, propensity scores can be used to correct for the selection bias that arises when the treatment effect is correlated with the likelihood of being treated (Rubin and Waterman, 2006; Chan et al., 2010). However, the propensityscore approach requires that exposure can be measured at the individual level, and it, too, does not guarantee valid inferences, for example in the presence of a specific type of selection bias recently termed ‘activity bias’ (Lewis, Rao and Reiley, 2011). Counterfactual modelling approaches avoid these issues when it can be assumed that the treatment market was chosen at random. Overall, we expect inferences on the causal impact of designed market interventions to play an increasingly prominent role in providing quantitative accounts of return on investment (Danaher and Rust, 1996; Seggie, Cavusgil and Phelan, 2007; Leeflang et al., 2009; Stewart, 2009). This is because marketing resources, specifically, can only be allocated to whichever campaign elements jointly provide the greatest return on ad spend (ROAS) if we understand the causal effects of spend on sales, product adoption, or user engagement. At the same time, our approach could be used for many other applications involving causal inference. Examples include problems found in economics, epidemiology, biology, or the political and social sciences. With the release of the CausalImpact R package we hope to provide a simple framework serving all of these areas. 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Google, Inc. 1600 Amphitheatre Parkway Mountain View CA 94043, U.S.A. Estimating reach curves from one data point Georg M. Goerg Google Inc. Last update: November 21, 2014 Abstract Reach curves arise in advertising and media analysis as they relate the number of content impressions to the number of people who have seen it. This is especially important for measuring the effectiveness of an ad on TV or websites (Nielsen, 2009; PricewaterhouseCoopers, 2010). For a mathematical and datadriven analysis, it would be very useful to know the entire reach curve; advertisers, however, often only know its last data point, i.e., the total number of impressions and the total reach. In this work I present a new method to estimate the entire curve using only this last data point. Furthermore, analytic derivations reveal a surprisingly simple, yet insightful relationship between marginal cost per reach, average cost per impression, and frequency. Thus, advertisers can estimate the cost of an additional reach point by just knowing their total number of impressions, reach, and cost. A comparison of the proposed one-data point method to two competing regression models on TV reach curve data, shows that the proposed methodology performs only slightly poorer than regression fits to a collection of several points along the curve. 1 Introduction Let k+ reach, rk, be the percentage of the population that is exposed to a campaign at least k times. As usual, we measure impressions in gross rating points (GRPs), which is calculated as number of impressions divided by total (target) population multiplied by 100 (measured in percent). Equipped with a functional form of the reach curve, a variety of quantities of interest can be computed, e.g., marginal cost per reach or maximum possible reach. Advertisers, however, often only have two points of the reach curve rk(g): rk(0) = 0 and rk(G) = R ∈ [0, 100], (1) where G ≥ 0 is the total GRPs and R is total reach. With this information alone one is tempted to use a linear approximation r (1) k (g) = R G g. However, reach curves are not linear and in particular, the marginal reach per GRP would equal average reach per GRP (= 1/frequency); thus (1) alone is not helpful to get a better estimate of marginal GRP (and thus cost) per reach at g = G. While the behavior of rk(g) around g = G is in general unknown, the tangent at g = 0 can be approximated quite well: starting with no exposure, adding an infinitesimally small unit of GRPs (say ) one reaches · ι % of the population, where ι = ι(k) is the reciprocal of the expected number of impressions needed for the first person to see k impressions. One can lower bound ι by 1/k. For k = 1, the bound is tight, ι = 1; getting an exact expression of ι for k > 1 is ongoing research.1 That is, for small g the reach curve can be approximated with a line through (0, 0) with slope ι: rk(g) ≈ g · ι for small g. (2) Thus, approximately, lim G→0 ∂ ∂g rk(g = G) = ι. (3) Combining (1) with (3) allows us to estimate a two-parameter model. Section 2 reviews parametric models for reach curves. Section 3 derives the parameter estimates based on 1 In practice we found that ι = (k + log2 k)−1 gives good fits for several k ≥ 1. 12.2 Conditional Logit 2 REACH CURVE MODELS the total GRP and reach. Simulations and comparisons to full least squares estimates are presented in Section 4. Finally, Section 5 summarizes the main findings and discusses future work. Details on the TV reach curve data and analytical derivations can be found in the Appendix. 2 Reach curve models Let X ≥ 0 be the number of content impressions, e.g., TV shows, websites, or commercials. For a probabilistic view of reach curves, it is useful to decompose k+ reach as P (X ≥ k,reachable) = P (X ≥ k | reachable) · P (reachable) (4) ⇔ rk = pk · ρ, (5) where ρ is the maximum possible reach, and pk is the probability of being reached k times, given that an individual is indeed reachable. This distinction allows us to model ρ and pk with separate probabilistic models. Since reach is usually denoted in percent, we also use percent for maximum possible reach ρ ∈ [0, 100], while we use proportions for pk ∈ [0, 1]. For further analytical derivations it is necessary to parametrize pk(g). Below we review two functional forms which are parsimonious (2 + 1 parameters), have excellent empirical fits, and lend themselves for simple analytical derivations. 2.1 Gamma-Mixture Jin et al. (2012) propose a Poisson distribution for the impressions g, with an exponential prior distribution with rate β on the Poisson rate λ. This yields a model of the form pk(g) = 1 − β g + β . (6) The exponential prior can be generalized to a Γ(α, β) distribution, which yields rk(g) = ρ 1 − β β + g α . (7) By construction, (6) is nested in (7), which can be tested using a hypothesis test for H0 : α = 1. 2.1.1 Marginal reach The derivative of (7) with respect to g equals2 ∂ ∂g pk(g) = α β β g + β α+1 , (8) with limg→0 ∂ ∂g rk(g) = ρα β . (9) Eq. (9) has three degrees of freedom; since only two data points are available, one parameters has to be fixed. Given the nested structure of the exponential model, it is natural to set α ≡ 1. 2.2 Conditional Logit As an alternative we propose a logistic regression logit(pk(g)) = β0 + β1 · log g, (10) where logit(p) = log p 1−p , and β0 and β1 are intercept and slope.3 Using the logit inverse expit(x) = e x 1+e x = 1 1+e−x , Eq. (10) can be rewritten as pk = expit(β0 + β1 log g) = e β0+β1 log g 1 + e β0+β1 log g (11) = 1 − 1 1 + e β0 · g β1 (12) = 1 − e −β0 e−β0 + g β1 (13) which shows similarity to (7). In fact, identifying β ≡ e −β0 , both models coincide if α = 1 and β1 = 1, respectively. Again, this can be tested using a two-sided hypothesis test for H0 : β1 = 1. The Logit conditional model can also be interpreted as the baseline Gamma mixture model with α ≡ 1, but with transformed GRPs, ˜g = g β1 , in (7). Here β1 can be interpreted as a parameter that measures the efficiency of GRPs: for β1 > 1 GRPs are more efficient than baseline; for β1 = 1 GRPs are spent according to the baseline model; and for β1 < 1 are not spent as efficiently as expected. For an empirical estimates see Section 4. 2See Section B.1 for details. 3We deliberately do not use α and β to parametrize intercept and slope, as it is prone to confusion with the (reversed) roles of α and β in (8). Google Inc. 23.1 ρ 1. (15) Thus for the logit model one has to assume β1 = 1 to use the linear approximation of R(g) at g = 0 for 1+ reach.4 3 Methodology Equipped with the two parameter model r(g; ρ, β) = ρ 1 − β β + g = ρ g β + g ∈ [0, ρ], (16) we can use the tangent approximation in (3) and total GRP and reach to estimate ρ and β. Note that β ≥ 0 is a saturation parameter and controls how efficient GRPs are: for small β reach grows quickly with GRPs, for large β it grows slowly. Its derivative equals r 0 (g; ρ, β) = ρ β (β + g) 2 , (17) which at g = 0 evaluates to r 0 (0) = ρ β . This gives a system of two equations (maximum GRP and reach & marginal reach at 0) with two unknowns, ρ ∈ [0, 100] and β > 0: ρ β = ι ⇔ ρ = β · ι, (18) ρ G β + G = R ⇔ ρ = R(G + β) G . (19) First note that for 1+ reach, ρ ≡ β since ι(k = 1) = 1. Moreover, ρ in (19) satisfies ρ ≥ 0 for all β, but it satisfies ρ ≤ 100 only for β ≤ G · 100−R R . 4For k > 1, the Logit model with β1 > 1 might become useful as the marginal k+ reach for the very first impression is 0. However, one then has to estimate three parameters again, which is not possible without any further assumptions or more than one data point. Solving for β and plugging in to ρ = ρ(β) gives ρb = min G · R G − R/ι, 100 , (20) and βb = (ρb ι = G·R/ι G−R/ι , if ρ < 100, G · 100−R R , if ρ = 100. (21) Condition ρb ≤ 100 is equivalent to G ≤ 100 ι R 100−R ; thus GRPs must be less or equal to a constant times the odds ratio of reach. Plugging them back into (16) yields expressions for reach solely as a function of R and G (details see Appendix B). According to (21) we consider the two scenarios separately. 3.1 ρ 1 − F θ pz > 1 − F. The first inequality will be close to an equality when pz and hence θ is small. For our applications 1 − F θ/pz is a reasonable approximation to the variance ratio. The second inequality reflects the fact that pooling the data cannot possibly be better than what we would get with an SSP of size n + N. From var( f ˆθI)/var(ˆθS) ≈ 1 − F θ/pz we see that using the BRP is effectively like multiplying the SSP sample size n by 1/(1−F θ/pz). Our greatest precision gains come when a high fraction of online reaches are incremental, that is, when θ/pz is largest. In our application this proportion ranges from 20% to 50% when aggregated to the campaign level. See Table 2.1 in Section 2. 3.2 Gain from the CIA alone Here we evaluate the variance reduction that would follow from the CIA. In that case, we could take advantage of the Z–Y independence, and estimate θ by ˆθC = Z¯ S(1 − Y¯ S). It is shown in the Appendix that the delta method variance of ˆθC satisfies var( f ˆθC) var(ˆθS) = 1 − py(1 − pz) 1 − θ > 1 − py, (2) when the CIA holds. This can represent a dramatic improvement, when the online reach pz and incremental reach θ are both small while the TV reach py is large. If the CIA holds, our application data suggest the variance reduction can be from 50% to 80%. The reverse setting with tiny TV reach and large online reach would not be favorable to ˆθC, but our data are not of that type. 3.3 Gain from the CIA and IDA Finally, suppose that both the CIA and IDA hold. If we apply both assumptions, we can get the estimator ˆθI,C = (fZ¯ S + FZ¯B)(1 − Y¯ S). We already gain a lot4 Example campaigns 7 from the CIA, so it is interesting to see how much more the IDA adds when the CIA holds. We show in the Appendix that under both assumptions, var( f ˆθI,C) var( f ˆθC) = f(1 − py)(1 − pz) + pypz (1 − py)(1 − pz) + pypz . If both reaches are high then we gain little, but if both reaches are small then we reduce the variance by almost a factor of f, when adding the IDA to the CIA. In our case we expect that the television reach is large but the online reach is small, fitting neither of these extremes. Consider a campaign with f = 1/3, py = 2/3 and pz = 99/100, similar to the soap campaigns. For such a campaign, var( f ˆθI,C) var( f ˆθC) = (1/9) × .99 + (2/3) × .01 (1/3) × .99 + (2/3) × .01 .= .34, so the combined assumptions then allow a nearly three-fold variance reduction compared to CIA alone. 4 Example campaigns Our data enrichment scheme is described in Section 5. Here we illustrate the results from that scheme on six marketing campaigns and discuss the differences among different algorithms. In addition to data enrichment, we also show results from tree structured models. Those split the data into groups and recursively split the groups. More about tree fitting is in Section 5. One model fits a tree to the SSP data alone and another one works with the pooled SSP and BRP data. For all three of those methods we have aggregated the predictions over the age variable, which takes six levels. In addition, we show the empirical results for age, which amount to recording the percentage of incremental reaches, that is, data with Z(1 − Y ) = 1, at each unique level of age in the SSP. There is no corresponding empirical prediction fully disaggregated by age, gender, income and education, because of the great many empty cells that would cause. We found the age related patterns of incremental reach particularly interesting. Figure 4.1 shows estimated incremental reach for all three models and the empirical counts, on all six campaigns, averaged over age groups. The beer campaign is particularly telling. The empirical data show a decreasing trend of incremental reach with age. The tree fit to SSP-only data yields a fit that is constant in age. The tree model had to explore splitting the data on all four variables without a prior focus on age. There were only 23 incremental reach events for beer in the SSP data set. With such a small number of events and four predictors, there is considerable possibility of overfitting. Cross-validation lead to a model that grouped the entire SSP into one set, that is, the tree had no splits. Both pooling and data enrichment were able to borrow strength from the BRP as well as take advantage of approximate independence of television and web exposure. They then recover the trend with age.5 Data enrichment for incremental reach 8 Age level Incremental reach (%) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1 2 3 4 5 6 ● ● ● ● ● ● Beer 2 4 6 8 10 ● ● ● ● ● ● Chrome 0.0 0.5 1.0 1.5 1 2 3 4 5 6 ● ● ● ● ● ● Salt 0.0 0.5 1.0 1.5 2.0 ● ● ● ● ● ● Soap 1 1 2 3 4 5 6 0.5 1.0 1.5 2.0 ● ● ● ● ● ● Soap 2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ● ● ● ● ● ● Soap 3 ● Emp SSP Pool DEIR Fig. 4.1: Estimated incremental reach by age, for six campaigns and three models: SSP, Pooling and DEIR as described in the text. Empirical counts are marked by Emp. The Salt campaign had a similarly small number of incremental reaches and once again the SSP only tree was constant. Fitting a tree to the SSP data always gave a flatter fit versus age than did DEIR which in turn was flatter than what we would get simply pooling the data. Section 6 gives simulations in which DEIR has greater accuracy than using pooling or SSP only. 5 Data enrichment for incremental reach For a given sample we would like to combine incremental reach estimates ˆθS, ˆθI , ˆθC and ˆθI,C whose assumptions are: none, IDA, CIA and IDA+CIA, respectively. The latter three add some value if their corresponding assumptions are nearly true, but our information about how well those assumptions hold comes from the same data we are using to form the estimates. The circumstances are similar to those in data enriched linear regression (Chen5 Data enrichment for incremental reach 9 et al., 2013). In that problem there is a regression model Yi = XT i β + εi which holds in the SSP and a biased regression model Yi = XT i (β + γ) + εi holds in the BRP. The estimates are found by minimizing S(λ) = X i∈S (Yi − XT i β) 2 + X i∈B (Yi − XT i (β + γ))2 + λ X i∈S (XT i γ) 2 , (3) over β and γ for a nonnegative penalty factor λ. The εi are independent with mean 0 and variance σ 2 S in the SSP and σ 2 B in the BRP. Taking λ = 0 amounts to fitting regressions separately in the two samples yielding an estimate βˆ that does not use the BRP at all. The limit λ → ∞ corresponds to pooling the two data sets, which would be optimal if there were no bias, i.e., if γ = 0. The specific penalty in (3) discourages the estimated γ from making large changes to the SSP; it is one of several penalties considered in that paper. Varying λ from 0 to ∞ gives a family of estimators that weight the SSP to varying degrees. The optimal λ is unknown. An oracle that knew γ and the error variance in the two data sets would be able to compute the optimal λ under a mean squared error loss. Chen et al. (2013) get a formula for the oracle’s λ and then plug estimates of γ and the variances into that formula. They show, under conditions, that the resulting plugin estimate gives better estimates of β than using the SSP only would. The conditions are that the Y values are normally distributed, and that the model have at least 5 regression parameters and 10 error degrees of freedom. The normality assumption allows a technical lemma due to Stein (1981) to be used and we believe that gains from using the BRP do not require normality. In principle we might multiply the sum of squared errors in the BRP by τ = σ 2 S /σ2 B if that ratio is known. If σ 2 BRP > σ2 SSP then we should put less weight on the BRP sample relative to the SSP sample. However the same effect is gained by increasing λ. Since the algorithm searches for optimal λ over a wide range it is less important to precisely specify τ . Chen et al. (2013) took τ = 1, simply summing all squared errors, and we will generalize that approach. For the present setting we must modify the method. First our responses are binary, not Gaussian. Second we have four estimators to combine, not two. Third, those estimators are dependent, being fit to overlapping data sets. 5.1 Modification for binary response To address the binary response there are two reasonable choices. One is to employ logistic regression. The other is to use tree-structured regression and then pool the estimators at the leaves of the tree. Regarding prediction accuracy, there is no unique best algorithm. There will be data sets for which simple logistic regression outperforms tree based classifiers and vice versa. For this paper we have adopted trees. Tree structured models have two practical advantages. First, the resulting cells that they select correspond to empirically determined market segments, which are then interpretable. Sec-5 Data enrichment for incremental reach 10 Data set Source Imputed V Assumptions D0 SSP ZS(1 − YS) none D1 BRP ZB(1 − YbSSP(XB, ZB)) IDA D2 SSP ZbSSP(XS)(1 − YbSSP(XS)) CIA D3 SSP ZbSSP+BRP(XS)(1 − YbSSP(XS)) CIA & IDA Tab. 5.1: Four incremental reach data sets and their imputed incremental reaches. The hats denote model-imputed values. For example YbSSP(XB, ZB) is a predictive model for Y based on values X and Z fit using data from SSP and evaluated at X = XB and Z = XB (from BRP). ond, within any of those cells, the model is intercept-only. Then both logistic regression and least squares reduce to a simple average. Each leaf of the regression tree defines a subset of the data that we call a cell. There are cells 1, . . . , C. The SSP has nc observations in cell c and the BRP has Nc observations there. For each cell and each set of assumptions we use a linear regression model relating an incremental reach quantity like Vei to an intercept. When there are no assumptions then Vei is the observed incremental reach for i ∈ S. Otherwise we may take advantage of the assumptions to impute values Vei using more of the data. The incremental reach values for each set of assumptions are given in Table 5.1. The predictive models shown there are all fit using rpart. For k = 0, 1, 2, 3 let Vek be vector of imputed responses under any of the assumptions from Table 5.1 and Xek their corresponding predictors. The regression framework minimizes kVe0 − Xe0βk 2 + X 3 k=1 kVek − Xek(β + γk)k 2 + X 3 k=1 λkkXe0γkk 2 . (4) over β and γk for penalties λk. In our setting each Xek is a column vector of ones of length mk. For cell c, m1k = Nc and m0k = m2k = m3k = nc. 5.2 Search for λk It is very convenient to search for suitable weights in the simplex ∆(K) = {(ω0, ω1, . . . , ωK) | ωk > 0, X K k=0 ωk = 1} because it is a bounded set, unlike the set [0, ∞] K of usable vectors λ = (λ1, . . . , λK). Chen et al. (2013) remark that it is more reasonable to use a common set of λk over all cells, stemming from unequal sample sizes. The search we use combines the advantages of both approaches.5 Data enrichment for incremental reach 11 Our search strategy for the simplex is to choose a grid of weight vectors ωg = (ωg0, ωg1, . . . , ωgK) ∈ ∆(K) , g = 1, . . . , G. For each vector ωg we find a vector λg = (λ1, . . . , λK) such that X C c=1 pcωk,c = ωgk, k = 0, 1, . . . , K, where pc is the proportion of our target population in cell c. That is, the population average weight of ωk,c matches ωgk. These weights give us the vector λg = (λg1, . . . , λgK). Using λg in the penalty criterion (4) specifies the weights we use within each cell. Our algorithm chooses the tree and the vector ω jointly using cross-validation. It is computationally expensive to make high dimensional searches. With K factors there is a K − 1 dimensional space of weights to search. Adding in the tree size gives a K’th dimension. As a result, combining all of our estimators requires us to search a 4 dimensional grid of values. We have chosen to set one of the ωk to 0 to reduce the search space from 4 dimensions to 3. We always retained the unbiased estimate ˆθS along with two others. In some computations reported in section A.4 of the Appendix we find only small differences among setting ω1 = 0, or ω2 = 0 or ω3 = 0. The best outcome was setting ω1 = 0. That has the effect of removing the estimate based on IDA only. As we saw in section 3, the IDA-only model had the least potential to improve our estimate. As a bonus, all three of the retained submodels have the same sample sizes and then common λ over cells coincides with common ω over cells. In the special case with ω1 = 0 we find after some calculus that the minimizer of (4) has βˆ c = V¯ 0c + P k∈{2,3} λk 1+λk V¯ kc 1 + P k∈{2,3} λk 1+λk ≡ X k∈{0,2,3} ωkc(λ)Vkc (5) where V¯ kc is the simple average of Vek over i ∈ S for cell c. Our default grid takes all values of ω whose coefficients are integer multiples of 10%. Populations D0, D2 and D3 all have the sample size n and of these only D0 is surely unbiased. An observation in D0 is worth at least as much as an observation in D2 or D3 and so we require ω0 > max{ω2, ω3}. Figure 5.1 shows this region and the set of 24 weight combinations that we use. 5.3 Search for tree size Here we give a brief review of regression trees in order to define our algorithm. For a full description see the monograph by Breiman et al. (1985). The version we use is the function rpart (Therneau and Atkinson, 1997) in the R programming language (R Core Team, 2012).5 Data enrichment for incremental reach 12 Weight region ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Weight points Fig. 5.1: The left panel shows the simplex of weights applied to data sets D0, D2 and D3 with the unbiased data set D0 in the lower left. The shaded region has the valid weights. The right panel shows that region with points for the 24 weights we use in our algorithm. Regression trees are built from splits of the set of subjects. A split uses one of the features in X and creates two subsets based on the values of that feature. For example it might split males from females or it might split those with the two smallest education levels from the others. Such a split defines two subpopulations of our target population and it equally defines two subsamples of our sample. A regression tree is a recursively defined set of splits. After the subjects are split into two groups based on one variable, each of those two groups may then be split again, using the same or different variables. Recursive splitting of splits yields a tree structure with subsets of subjects in the leaf nodes. Given a tree, we predict for subjects by a rule based on the leaf to which they belong. That rule uses the average within the subject’s leaf node. The tree is found by a greedy search that minimizes a measure of prediction error. In our case, the measure R(T), is the sum of squared prediction errors. By construction any tree with more splits than T has lower error and this brings a risk of overfitting. To counter overfitting, rpart adds a penalty proportional to the number |T| of leaves in tree T. The penalized criterion is R(T) + α|T| where the parameter α > 0 is chosen by M-fold cross-validation. This reduces the potentially complicated problem of choosing a tree to the simpler problem of selecting a scalar penalty parameter α. The rpart function has one option that we have changed from the default. That parameter is cp, the complexity parameter. The default is 10−2 . The cp parameter stops tree growing early if a proposed split improves R(T) by less6 Numerical investigation 13 than a factor of cp. We set cp = 10−4 . Our choice creates somewhat larger trees to get more choices to use in cross-validation. 5.4 The algorithm Here is a summary of the entire algorithm. First we make the following preprocessing steps. 1) Fit a large tree T by rpart relating observed incremental reaches Vi to predictor variables Xi in the SSP data. This tree returns a nested sequence of subtrees T0 ⊂ T1 ⊂ · · · ⊂ TL ⊂ T . Each T` corresponds to a critical value α` of the penalty. Choosing α` from this list selects the tree T`. The value L is data-dependent, and chosen by rpart. 2) Specify a grid of values ωg for g = 1, . . . , G. Here ωg = (ωg0, ωg1, . . . , ωgK) with ωgk > 0 and PG k=0 ωgk = 1. 3) Randomly partition SSP data (Xi , Yi , Zi) into M folds Sm for m = 1, . . . , M each of roughly equal size n/M. For fold m the SSP will contain ∪Sm0 for all m0 6= m. We call this S−m. The BRP for fold m is the entire BRP. We also considered using a bootstrap sample for the fold m BRP, but that was more expensive and less accurate in our numerical investigation as described in section A.4 of the Appendix. After this precomputation, our algorithm proceeds to the cross-validation shown in Figure 5.2 to make a joint selection of the tree penalty parameter α` and the simplex grid point ωg. Let the chosen values be α∗ and ω∗. We select the tree T∗ from step 1 above, corresponding to penalty parameter α∗. We treat each leaf node of T∗ as a cell c. We translate ω∗ into the corresponding λc in every cell c of tree T∗. Then we minimize (4) using this λc and the resulting βˆ c is our estimate Vbc of incremental reach in cell c. After choosing the tuning parameters ωg and α` by cross-validation, we use these parameters on the whole data set to make our final prediction. 6 Numerical investigation In order to measure the effect of data enriched estimates on incremental reach, we conducted a simulation where we knew the ground truth. Our goal is to predict for ensembles, not for individuals, so we constructed two large populations in which ground truth was known to us, simulated our process of subsampling them, and scored predictions against the ground truth incremental reach probabilities. To make our large samples realistic, we built them from our real data. We created S- and B-populations by replicating our SSP (respectively BRP) records 100 times each. Then in each simulation, we form an SSP by drawing 6000 observations at random from the S-population, and a BRP by drawing 13,000 observations at random from the B-population.6 Numerical investigation 14 for ` = 1, . . . , L do // initialize error sum of squares for g = 1, . . . , G do SSE`,g ← 0 for m = 1, . . . , M do // folds construct Table 5.1 for fold m, using S−m and B fit tree Tm for fold m by rpart prune tree Tm to T1,m, . . . , TL,m, tree T`,m uses α` for ` = 1, . . . , L do // tree sizes define cells S−m,c and Bc, c = 1, . . . , C from leaves of T`,m for g = 1, . . . , G do // simplex weights convert ωg into λg for c = 1, . . . , C do // cells compute Vek for k = 0, 2, 3 in cell c get Vbc = βˆ c from the weighted average (5) Vc ← 1 |Sm,c| X i∈Sm,c Vi // held out incr. reach pc ← fraction of true S population in cell c SSE`,g ←SSE`,g + pc(Vbc − Vc) 2 Fig. 5.2: Data enrichment for incremental reach (deir) algorithm. After precomputation described on page 13 we run this cross-validation algorithm to choose the complexity parameter α` and the weights ωg, as the joint minimizers ` ∗ and g ∗ of SSE`,g. The values pc come from a census or from the SSP if the census does not have the variables we need. We use M = 10. For each campaign, we apply deir with this sample data to estimate the incremental reach Vˆ (x). We used 10–fold cross-validation. The mean square estimation error (MSE) is P x p(x)(Vˆ (x) − V (x))2 . This sum is taken over all x values in the SSP. The simulation above was repeated 1000 times. The root mean square error was divided by the true incremental reach to get a relative RMSE. We consider two comparison methods. The first is to use the SSP only. That method computes ˆθS within the leaves of a tree. The tree is found by rpart. The second comparison is a tree fit by rpart to the pooled SSP and BRP data and using both CIA and IDA. We do not compare to the empirical fractions because many of them are from empty cells. Figure 6.1 compares the relative errors in the SSP only method to data6 Numerical investigation 15 DEIR RMSE (%) SSP RMSE (%) 1 2 3 1 2 3 ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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6 8 10 12 3 4 5 6 7 8 9 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● Chrome 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● Salt 0 1 2 3 4 5 6 0 1 2 3 4 ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Soap 1 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Soap 2 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Soap 3 Fig. 6.1: Performance comparison, SSP only versus data enrichment, predictive relative mean square errors. There is one panel for each of 6 campaigns with one point for each of 1000 replicates. The reference line is the forty-five degree line. enrichment. Data enrichment is consistently better over all 6 campaigns we simulated in the great majority of replications. It is clear that the populations are similar enough that using the larger data set improves estimation of incremental reach. Under the IDA we can pool the SSP and BRP together using rpart on the combined data to estimate Pr(Z = 1 | X). Under the CIA we can multiply this estimate by Pr(Y = 0 | X) fit by rpart to the SSP, see Table 5.1 under the assumption CIA & IDA. This method, as an implementation of statistical matching, uses two separate applications of rpart each with their own built in cross-validation. Figure 6.2 compares the relative errors of statistical matching to data enrichment. Data enrichment is consistently better over all 6 campaigns we simulated in the great majority of replications. We also investigate for each estimator, how much of the predictive error is contributed by bias. It is well known that predictive mean square error can be decomposed as the sum of variance and squared bias. These quantities are typically unknown in practice, but can be evaluated in simulation studies.6 Numerical investigation 16 DEIR RMSE (%) Pool RMSE (%) 0.6 0.8 1.0 1.2 1 2 3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Beer 5 6 7 8 9 10 3 4 5 6 7 8 9 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Chrome 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Salt 0.5 1.0 1.5 0 1 2 3 4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Soap 1 0.8 1.0 1.2 1.4 1.6 1.8 0.5 1.0 1.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Soap 2 0.4 0.6 0.8 1.0 1.2 0.0 0.5 1.0 1.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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There is one panel for each of 6 campaigns with one point for each of 1000 replicates. The reference line is the forty-five degree line. Table 6.1 reports the fractions of squared bias in predictive mean square errors for each method in all six studies. We see there that the error for statistical matching (data pooling) is dominated by bias while the error for SSP only is dominated by variance. These results are not surprising because the SSP only method has no sampling bias (only algorithmic bias) while the pooled data set has maximal sampling bias. The proportion of bias for DEIR is in between these extremes. Here we have less population bias than a typical data fusion situation because the TV and online-only panels were recruited in the same way. The bottom of Table 6.1 shows that DEIR is able to trade off bias and variance more effectively than SSP only or data pooling, because DEIR attains the smallest predictive mean squared error. Conclusions Predictions of incremental reach can be improved by making use of additional data. That improvement comes only if certain strong assumptions are true or at6 Numerical investigation 17 bias2 /mse Beer Chrome Salt Soap 1 Soap 2 Soap 3 SSP 0.35 0.42 0.26 0.12 0.28 0.12 Pool 0.88 0.82 0.88 0.88 0.88 0.93 DEIR 0.49 0.59 0.47 0.33 0.47 0.39 mse Beer Chrome Salt Soap 1 Soap 2 Soap 3 SSP 1.02 7.76 0.89 0.84 1.26 0.66 Pool 0.82 7.39 0.80 0.86 1.12 0.78 DEIR 0.61 5.42 0.48 0.52 0.68 0.42 Tab. 6.1: The upper rows show the fraction bias2 /mse of the mean squared prediction error due to bias for 3 methods to estimate incremental reach in 6 campaigns. The lower rows show the total mse, that is bias2 + var. least approximately true. Our only guide to the accuracy of those assumptions may come from the data themselves. Our data enriched incremental reach estimate uses a shrinkage strategy to pool estimates using different assumptions. Cross-validating the level of pooling gave us an algorithm that worked better than either ignoring the additional data or treating it the same as the unbiased data. Acknowledgment This project was not part of Art Owen’s Stanford responsibilities. His participation was done as a consultant at Google. The authors would like to thank Penny Chu, Tony Fagan, Yijia Feng, Jerome Friedman, Yuxue Jin, Daniel Meyer, Jeffrey Oldham and Hal Varian for support and constructive comments. References Breiman, L., Friedman, J. H. Olshen, R. A., and Stone, C. J. (1985). Classifi- cation and Regression Trees. Chapman & Hall/CRC, Baton Rouge, FL. Chen, A., Owen, A. B., and Shi, M. (2013). Data enriched linear regression. Technical report, Google. http://arxiv.org/abs/1304.1837. Collins, J. and Doe, P. (2009). Developing an integrated television, print and consumer behavior database from national media and purchasing currency data sources. In Worldwide Readership Symposium, Valencia. Doe, P. and Kudon, D. (2010). Data integration in practice: connecting currency and proprietary data to understand media use. ARF Audience Measurement 5.0.A Appendix 18 D’Orazio, M., Di Zio, M., and Scanu, M. (2006). Statistical Matching: Theory and Practice. Wiley, Chichester, UK. Gilula, Z., McCulloch, R. E., and Rossi, P. E. (2006). A direct approach to data fusion. Journal of Marketing Research, XLIII:73–83. Jin, Y., Shobowale, S., Koehler, J., and Case, H. (2012). The incremental reach and cost efficiency of online video ads over TV ads. Technical report, Google. Lehmann, E. L. and Romano, J. P. (2005). Testing statistical hypotheses. Springer, New York, Third edition. Little, R. J. A. and Rubin, D. B. (2009). Statistical Analysis with Missing Data. John Wiley & Sons Inc., Hoboken, NJ, 2nd edition. R Core Team (2012). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. R¨assler, S. (2004). Data fusion: identification problems, validity, and multiple imputation. Austrian Journal of Statistics, 33(1&2):153–171. Singh, A. C., Mantel, H., Kinack, M., and Rowe, G. (1993). Statistical matching: Use of auxiliary information as an alternative to the conditional independence assumption. Survey Methodology, 19:59–79. Stein, C. M. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In Proceedings of the Third Berkeley symposium on mathematical statistics and probability, volume 1, pages 197–206. Stein, C. M. (1981). Estimation of the mean of a multivariate normal distribution. The Annals of Statistics, 9(6):1135–1151. The Nielsen Company (2011). The cross-platform report. Quarter 2, U.S. Therneau, T. M. and Atkinson, E. J. (1997). An introduction to recursive partitioning using the RPART routines. Technical Report 61, Mayo Clinic. A Appendix A.1 Variance reduction by IDA Recall that f = n/(n + N) and F = N/(n + N) are sample size proportions of the two data sets. Under the IDA we may estimate incremental reach by ˆθI = (fZ¯ S + FZ¯B) V¯ S Z¯ S = V¯ I f + F Z¯B Z¯ S .A Appendix 19 By the delta method (Lehmann and Romano, 2005), var(ˆθI) is approximately var( f ˆθI) = var(V¯ S) ∂ ˆθI ∂V¯ S 2 + var(Z¯B) ∂ ˆθI ∂Z¯B 2 + var(Z¯ S) ∂ ˆθI ∂Z¯ S 2 + 2cov(V¯ S,Z¯ S) ∂ ˆθI ∂V¯ S ∂ ˆθI ∂Z¯ S , with partial derivatives evaluated with expectations E(V¯ S), E(Z¯ S), and E(Z¯B) replacing the corresponding random quantities. The other two covariances are zero because the S and B samples are independent. From the binomial distribution we have var(V¯ S) = θ(1 − θ)/n, var(Z¯B) = pz(1 − pz)/N and var(Z¯ S) = pz(1 − pz)/n. Also cov(V¯ S,Z¯ S) = 1 n E(ViZi) − E(Vi)E(Zi) = θ(1 − pz)/n. After some calculus, var( f ˆθI) = θ(1 − θ) n + pz(1 − pz) N θ 2F 2 p 2 z + pz(1 − pz) n θ 2F 2 p 2 z − 2 θ(1 − pz) n θF pz = var(ˆθS) + θ 2F(1 − pz) pz F N + F n − 2 n = var(ˆθS) − θ 2F(1 − pz) pz 1 n = var(ˆθS) 1 − F 1 − pz pz θ 1 − θ . A.2 Variance reduction by CIA Applying the delta method to ˆθC = Z¯ S(1 − Y¯ S), we find that var( f ˆθC) = var(Z¯ S) ∂ ˆθC ∂Z¯ S 2 + var(Y¯ S) ∂ ˆθC ∂Y¯ S 2 + 2cov(Y¯ S,Z¯ S) ∂ ˆθC ∂Y¯ S ∂ ˆθC ∂Z¯ S = var(Z¯ S)(1 − py) 2 + var(Y¯ S)p 2 z + 2cov(Y¯ S,Z¯ S)(1 − py)pz. Here var(Z¯ S) = pz(1 − pz)/n, var(Z¯ S) = pz(1 − pz)/n, and under conditional independence cov(Y¯ S,Z¯ S) = 0. Thus var( f ˆθC) = 1 n pz(1 − pz)(1 − py) 2 + py(1 − py)p 2 z = 1 n pz(1 − pz)(1 − py) 2 + py(1 − py)p 2 z = pz(1 − py) n (1 − pz)(1 − py) + pypz . When the CIA holds, θ = pz(1 − py). Note that var(ˆθS) = θ(1 − θ)/n. After some algebraic simplification we find that var( f ˆθC) var(ˆθS) = 1 − py(1 − pz) 1 − θ .A Appendix 20 A.3 Variance reduction by CIA and IDA When both assumptions hold we can estimate θ by ˆθI,C = (fZ¯ S + FZ¯B)(1 − Y¯ S). Under these assumptions, Z¯ S, Z¯B and Y¯ S are all independent, and var( f ˆθI,C) equals var(Z¯ S) ∂ ˆθI,C ∂Z¯ S 2 + var(Z¯B) ∂ ˆθI,C ∂Z¯B 2 + var(Y¯ S) ∂ ˆθI,C ∂Y¯ S 2 = pz(1 − pz) n f 2 (1 − py) 2 + pz(1 − pz) N F 2 (1 − py) 2 + py(1 − py) n p 2 z = pz(1 − py) n f(1 − py)(1 − pz) + pypz after some simplification. As a result var( f ˆθI,C) var( f ˆθC) = f(1 − py)(1 − pz) + pypz (1 − py)(1 − pz) + pypz . A.4 Alternative algorithms We faced some design choices in our algorithm. First, we had to decide which estimators to include in our algorithm. We always include the unbiased choice ˆθS as well as two others. Second, we had to decide whether to use the entire BRP or to bootstrap sample it. We ran all six choices on simulations of all six data sets where we knew the correct answer. Table A.1 shows the mean squared errors for the six possible estimators on each of the six data sets. In every case we divided the mean squared error by that for the estimator combining ˆθS, ˆθC , and ˆθI,C without the bootstrap. We only see small differences, but the evidence favors choosing λI = 0 as well as not bootstrapping. Our default method is consistently the best in this table, although only by a small amount. We saw that data enrichment is consistently better than either pooling the data or ignoring the large sample, and by much larger amounts than we see in Table A.1. As a result, any of the data enrichment methods in this table would make a big improvement over either pooling the samples or ignoring the BRP.A Appendix 21 Estimators ˆθS, ˆθI , ˆθC ˆθS, ˆθI , ˆθI,C ˆθS, ˆθC , ˆθI,C BRP All Boot All Boot All Boot Beer 1.02 1.02 1.00 1.01 1 1.01 Chrome 1.04 1.04 1.01 1.01 1 1.00 Salt 1.04 1.04 1.01 1.01 1 1.01 Soap 1 1.04 1.05 1.01 1.02 1 1.00 Soap 2 1.05 1.05 1.01 1.03 1 1.01 Soap 3 1.02 1.02 1.01 1.00 1 1.00 Tab. A.1: Relative performance of our estimators on six problems. The relative errors are mean squared prediction errors normalized to the case that uses ˆθS, ˆθC , ˆθI,C without bootstrapping. The relative error for that case is 1 by definition. Coupled and k-Sided Placements: Generalizing Generalized Assignment Madhukar Korupolu1 , Adam Meyerson1 , Rajmohan Rajaraman2 , and Brian Tagiku1 1 Google, 1600 Amphitheater Parkway, Mountain View, CA. Email: {mkar,awmeyerson,btagiku}@google.com 2 Northeastern University, Boston, MA 02115. Email: rraj@ccs.neu.edu Abstract. In modern data centers and cloud computing systems, jobs often require resources distributed across nodes providing a wide variety of services. Motivated by this, we study the Coupled Placement problem, in which we place jobs into computation and storage nodes with capacity constraints, so as to optimize some costs or profits associated with the placement. The coupled placement problem is a natural generalization of the widely-studied generalized assignment problem (GAP), which concerns the placement of jobs into single nodes providing one kind of service. We also study a further generalization, the k-Sided Placement problem, in which we place jobs into k-tuples of nodes, each node in a tuple offering one of k services. For both the coupled and k-sided placement problems, we consider minimization and maximization versions. In the minimization versions (MinCP and MinkSP), the goal is to achieve minimum placement cost, while incurring a minimum blowup in the capacity of the individual nodes. Our first main result is an algorithm for MinkSP that achieves optimal cost while increasing capacities by at most a factor of k + 1, also yielding the first constant-factor approximation for MinCP. In the maximization versions (MaxCP and MaxkSP), the goal is to maximize the total weight of the jobs that are placed under hard capacity constraints. MaxkSP can be expressed as a k-column sparse integer program, and can be approximated to within a factor of O(k) factor using randomized rounding of a linear program relaxation. We consider alternative combinatorial algorithms that are much more efficient in practice. Our second main result is a local search based approximation algorithm that yields a 15- approximation and O(k 3 )-approximation for MaxCP and MaxkSP respectively. Finally, we consider an online version of MaxkSP and present algorithms that achieve logarithmic competitive ratio under certain necessary technical assumptions. 1 Introduction The data center has become one of the most important assets of a modern business. Whether it is a private data center for exclusive use or a shared public cloud data center, the size and scale of the data center continues to rise. As a companygrows, so too must its data center to accommodate growing computational, storage and networking demand. However, the new components purchased for this expansion need not be the same as the components already in place. Over time, the data center becomes quite heterogeneous [1]. This complicates the problem of placing jobs within the data center so as to maximize performance. Jobs often require resources of more than one type: for example, compute and storage. Modern data centers typically separate computation from storage and interconnect the two using a network of switches. As such, when placing a job within a data center, we must decide which computation node and which storage node will serve the job. If we pick nodes that are far apart, then communication latency may become too prohibitive. On the other hand, nodes are capacitated, so picking nodes close together may not always be possible. Most prior work in data center resource management is focussed on placing one type of resource at a time: e.g., placing storage requirements assuming job compute location is fixed [2, 3] or placing compute requirements assuming job storage location is fixed [4, 5]. One sided placement methods cannot suitably take advantage of the proximities and heterogeneities that exist in modern data centers. For example, a database analytics application requiring high throughput between its compute and storage elements can benefit by being placed on a storage node that has a nearby available compute node. In this paper, we study Coupled Placement (CP), which is the problem of placing jobs into computation and storage nodes with capacity constraints, so as to optimize costs or profits associated with the placement. Coupled placement was first addressed in [6] in a setting where we are required to place all jobs and we wish to minimize the communication latency over all jobs. They show that this problem, which we call MinCP, is NP-hard and investigate the performance of heuristic solutions. Another natural formulation is where the goal is to maximize the total number of jobs or revenue generated by the placement, subject to capacity constraints. We refer to this problem as MaxCP. We also study a generalization of Coupled Placement, the k-Sided Placement Problem (kSP), which considers k ≥ 2 kinds of resources. 1.1 Problem definition In the coupled placement problem, we are given a bipartite graph G = (U, V, E) where U is a set of compute nodes and V is a set of storage nodes. We have capacity functions C : U → R and S : V → R for the compute and storage nodes, respectively. We are also given a set T of jobs, each of which needs to be allocated to one compute node and one storage node. Each job may prefer some compute-storage node pairs more than others, and may also consume different resources at different nodes. To capture these heterogeneities, we have for each job j a function fj : E → R, a processing requirement pj : E → R and a storage requirement sj : E → R. We note that without loss of generality, we can assume that the capacities are unit, since we can scale the processing and storage requirements of individual nodes accordingly.We consider two versions of the coupled placement problems. For the maximization version MaxCP, we view fj as a payment function. Our goal is to select a subset A ⊆ T of jobs and an assignment σ : A → E such that all capacities are observed and our total profit P j∈A fj (σ(j)) is maximized. For the minimization version MinCP, we view fj as a cost function. Our goal is to find an assignment σ : T → E such that all capacities are observed and our total cost P j∈A fj (σ(j)) is minimized. A generalization of the coupled placement problem is k-sided placement (kSP), in which we have k different sets of nodes, S1, . . . , Sk, each set of nodes providing a distinct service. For each i, we have a capacity function Ci : Si → R that gives the capacity of a node in Si to provide the ith service. We are given a set T of jobs, each of which needs each kind of service; the exact resource needs may depend on the particular k-tuple of nodes from Q i Si to which it is assigned. That is, for each job j, we have a demand function dj : Q i Si → Rk . We also have another function fj : Q i Si → R. As for coupled placement, we can assume that the capacities are unit, since we can scale the demands of individual nodes accordingly. Similar to coupled placement, we consider two versions of kSP, MinkSP and MaxkSP. 1.2 Our Results All of the variants of CP and kSP are NP-hard, so our focus is on approximation algorithms. Our first set of results consist of the first non-trivial approximation algorithms for MinCP and MinkSP. Under hard capacity constraints, it is easy to see that it is NP-hard to achieve any bounded approximation ratio to cost minimization. So we consider approximation algorithms that incur a blowup in capacity. We say that an algorithm is α-approximate for the minimization version if its cost is at most that of an optimal solution, while incurring a blowup factor of at most α in the capacity of any node. – We present a (k + 1)-approximation algorithm for MinkSP using iterative rounding, yielding a 3-approximation for MinCP. We next consider the maximization version. MaxkSP can be expressed as a k-column sparse integer packing program (k-CSP). From this, it is immediate that MaxkSP can be approximated to within an O(k) approximation factor by applying randomized rounding to a linear programming relaxation [7]. An Ω(k/ log k)-inapproximability result for k-set packing due to [16] implies the same hardness result for MaxkSP. Our second main result is a simpler approximation algorithm for MaxCP and MaxkSP based on local search. – We present a local search based 15-approximation algorithm for MaxCP. We extend it to MaxkSP and obtain an O(k 3 )-approximation. The local search result applies directly to a version where we can assign tasks fractionally but only to a single pair of machines (this is like assigning a task with lower priority and may have additional applications). We then describe asimple rounding scheme to obtain an integral version. The rounding technique involves establishing a one-to-one correspondence between fractional assignments and machines. This is much like the cycle-removing rounding for GAP; there is a crucial difference, however, since coupled and k-sided placements assign jobs to tuples of machines. Finally, we study the online version of MaxCP, in which tasks arrive online and must be irrevocably assigned or rejected immediately upon arrival. – We extend the techniques of [8] to the case where the capacity requirement for a job is arbitrarily machine-dependent. This enables us to achieve competitive ratio logarithmic in the ratio of best to worst value-per-capacity density, under necessary technical assumptions about the maximum job size. 1.3 Related Work The coupled and k-sided placement problems are natural generalizations of the Generalized Assignment Problem (GAP), which can be viewed as a 1-sided placement problem. In GAP, which was first introduced by Shmoys and Tardos [9], the goal is assign items of various sizes to bins of various capacities. A subset of items is feasible for a bin if their total size is no more than the bin’s capacity. If we are required to assign all items and minimize our cost (MinGAP), Shmoys and Tardos [9] give an algorithm for computing an assignment that achieves optimal cost while doubling the capacities of each bin. A previous result by Lenstra et al. [10] for scheduling on unrelated machines show it is NP-hard to achieve optimal cost without incurring a capacity blowup of at least 3/2. On the other hand, if we wish to maximize our profit and are allowed to leave items unassigned (MaxGAP), Chekuri and Khanna [11] observe that the (1, 2)-approximation for MinGAP implies a 2-approximation for MaxGAP. This can be improved to a ( e e−1 )-approximation using LP-based techniques [12]. It is known that MaxGAP is APX-hard [11], though no specific constant of hardness is shown. On the experimental side, most prior work in data center resource management focusses on placing one type of resource at a time: for example, placing storage requirements assuming job compute location is fixed (file allocation problem [2], [13, 14, 3]) or placing compute requirements assuming job storage location is fixed [4, 5]. These in a sense are variants of GAP. The only prior work on Coupled Placement is [6], where they show that MinCP is NP-hard and experimentally evaluate heuristics: in particular, a fast approach based on stable marriage and knapsacks is shown to do well in practice, close to the LP optimal. The MaxkSP problem is related to the recently studied hypermatching assignment problem (HAP) [15], and special cases, including k-set packing, and a uniform version of the problem. A (k + 1 + ε)-approximation is given for HAP in [15], where other variants of HAP are also studied. While the MaxkSP problem can be viewed as a variant of HAP, there are critical differences. For instance, in MaxkSP, each task is assigned at most one tuple, while in the hypermatching problem each client (or task) is assigned a subset of the hyperedges. Hence, the MaxkSP and HAP problems are not directly comparable. The k-set packing canbe captured as a special case of MaxkSP, and hence the Ω(k/ log k)-hardness due to [16] applies to MaxkSP as well. 2 The minimization version Next, we consider the minimization version of the Coupled Placement problem, MinCP. We write the following integer linear program for MinCP, where xtuv is the indicator variable for the assignment of t to pair (u, v), u ∈ U, v ∈ V . Minimize: X t,u,v xtuvft(u, v) Subject to: X u,v xtuv ≥ 1, ∀t ∈ T, X t,v pt(u, v)xtuv ≤ cu, ∀u ∈ U, X t,u st(u, v)xtuv ≤ dv, ∀v ∈ V, xtuv ∈ {0, 1}, ∀t ∈ T, u ∈ U, v ∈ V. We refer the first set of constraints as satisfaction constraints, the second and third set as capacity constraints (processing and storage). We consider the linear relaxation of this program which replaces the integrality constraints above with 0 ≤ xtuv ≤ 1, ∀t ∈ T, u ∈ U, v ∈ V . 2.1 A 3-approximation algorithm for MinCP We now present algorithm IterRound, based on iterative rounding [21], which achieves a 3-approximation for MinCP. We start with a basic algorithm that achieves a 5-approximation by identifying tight constraints with a small number of variables. Each iteration of this algorithm repeats the following round until all variables have been rounded. 1 Extreme point: Compute an extreme point solution x to the current LP. 2 Eliminate variable or constraint: Execute one of these two steps. By Lemma 3, one of these steps can always be executed if the LP is nonempty. a Remove from the LP all variables xtuv that take the value 0 or 1 in x. If xtuv is 1, then assign job t to the pair (u, v), remove the job t and its associated variables from the LP, and reduce cu by pt(u, v) and dv by st(u, v). b Remove from the LP any tight capacity constraint with at most 4 variables. Fix an iteration of the algorithm, and an extreme point x. Let nt, nc, and ns denote the number of tight task satisfaction constraints, computation constraints, and storage constraints, respectively, in x. Note that every task satisfaction constraint can be assumed to be tight, without loss of generality. Let N denote the number of variables in the LP. Since x is an extreme point, if all variables in x take values in (0, 1), then we have N = nt + nc + ns.Lemma 1. If all variables in x take values in (0, 1), then nt ≤ N/2. Proof. Since a variable only occurs once over all satisfaction constraints, if nt > N/2, there exists a satisfaction constraint that has exactly one variable. But then, this variable needs to take value 1, a contradiction. Lemma 2. If nt ≤ N/2, then there exists a tight capacity constraint that has at most 4 variables. Proof. If nt ≤ N/2, then ns + nc = N − nt ≥ N/2. Since each variable occurs in at most one computation constraint and at most one storage constraint, the total number of variable occurrences over all tight storage and computation constraints is at most 2N, which is at most 4(ns +nc). This implies that at least one of these tight capacity constraints has at most 4 variables. Using Lemmas 1 and 2, we can argue that the above algorithm yields a 5- approximation. Step 2a does not cause any increase in cost or capacity. Step 2b removes a constraint, hence cannot increase cost; since the removed constraint has at most 4 variables, the total demand allocated on the relevant node is at most the demand of four tasks plus the capacity already used in earlier iterations. Since each task demand is at most the capacity of the node, we obtain a 5- approximation with respect to capacity. Studying the proof of Lemma 2 more closely, one can separate the case nt < N/2 from the nt = N/2; in the former case, one can, in fact, show that there exists a tight capacity constraint with at most 3 variables. Together with a careful consideration of the nt = N/2 case, one can improve the approximation factor to 4. We now present an alternative selection of tight capacity constraint that leads to a 3-approximation. One interesting aspect of this step is that the constraint being selected may not have a small number of variables. We replace step 2b by the following. 2b Remove from the LP any tight capacity constraint in which the number of variables is at most two more than the sum of the values of the variables. Lemma 3. If all variables in x take values in (0, 1), then there exists a tight capacity constraint in which the number of variables is at most two more than the sum of the values of the variables. Proof. Since each variable occurs in at most two tight capacity constraints, the total number of occurrences of all variables across the tight capacity constraints is 2N − s for some nonnegative integer s. Since each satisfaction constraint is tight, each variable appears in 2 capacity constraints, and each variable takes on value less than 1, the sum of all the variables over the tight capacity constraints is at least 2nt − s. Therefore, the sum, over all tight capacity constraints, of the difference between the number of variables and their sum is at most 2(N − nt). Since there are N − nt tight capacity constraints, for at least one of these constraints, the difference between the number of variables and their sum is at most 2.Lemma 4. Let u be a node with a tight capacity constraint, in which the number of variables is at most 2 more than the sum of the variables. Then, the sum of the capacity requirements of the tasks partially assigned to u is a most the current available capacity of u plus twice the capacity of u. Proof. Let ` be the number of variables in the constraint for u, and let the associated tasks be numbered 1 through `. Let the demand of task j for the capacity of node u be dj . Then, the capacity constraint for u is P j djxj = bc(u), where bc(u) is the available capacity of u in the current LP. We know that ` − P i xi ≤ 2. Since di ≤ C(u), the capacity of u: X j dj = bc(u) +X ` j=1 (1 − xj )dj ≤ bc(u) + (` − Xm j=` xj )C(u) ≤ bc(u) + 2C(u). Theorem 1. IterRound is a polynomial-time 3-approximation algorithm for MinCP. Proof. By Lemma 3, each iteration of the algorithm removes either a variable or a constraint from the LP. Hence the algorithm is polynomial time. The elimination of a variable that takes value 0 or 1 does not change the cost. The elimination of a constraint can only decrease cost, so the final solution has cost no more than the value achieved by the original LP. Finally, when a capacity constraint is eliminated, by Lemma 4, we incur a blowup of at most 3 in capacity. 2.2 A (k + 1)-approximation algorithm for MinkSP It is straightforward to generalize the the algorithm of the preceding section to obtain a k + 1-approximation to MinkSP. We first set up the integer LP for MinkSP. For a given element e ∈ Q i Si , we use ei to denote the ith coordinate of e. Let xte be the indicator variable that t is assigned to e ∈ Q i Si . Minimize: X t,e xteft(e) Subject to: X e xte ≥ 1, ∀t ∈ T, X t,e:ei=u (dt(e))ixte ≤ Ci(u), ∀1 ≤ i ≤ k, u ∈ U, xte ∈ {0, 1}, ∀t ∈ T, e ∈ E The algorithm, which we call IterRound(k), is identical to IterRound of Section 2.1 except that step 2b is replaced by the following. 2b Remove from the LP any tight capacity constraint in which the number of variables is at most k more than the sum of the values of the variables. The claims and proofs are almost identical to the k = 2 case and are moved to Appendix A. A natural question to ask is whether a linear approximation factor of MinkSP is unavoidable for polynomial time algorithms. Unfortunately, we donot have any non-trivial results in this direction. We have been able to show that the MinkSP linear program has an integrality that grows as Ω(log k/ log log k) (see Appendix A). 3 The maximization problems We present approximation algorithms for the maximization versions of coupled placement and k-sided placement problems. We first observe, in Section 3.1, that these problems reduce to column sparse integer packing. We next present, in Section 3.2, an alternative combinatorial approach based on local search. 3.1 An LP-based approximation algorithm One can write a positive integer linear program for MaxCP. Let xtuv denote the indicator variable for the the assignment of job t to the pair (u, v), u ∈ U, v ∈ V . The goal is then to Maximize: X t,u,v xtuvft(u, v) Subject to: X u,v xtuv ≤ 1, ∀t ∈ T, X t,v pt(u, v)xtuv ≤ cu, ∀u ∈ U, X t,u st(u, v)xtuv ≤ dv, ∀v ∈ V, xtuv ∈ {0, 1}, ∀t ∈ T, u ∈ U, v ∈ V. Note that we can deal with capacities on u, v by scaling the pt(u, v) and st(u, v) values appropriately. The above LP can be easily extended to MaxkSP (see Appendix B). These linear programs are 3- and k-column sparse packing programs, respectively, and can be approximated to within a factor of 15.74 and ek + o(k), respectively using a clever randomized rounding approach. We next give a combinatorial approach based on local search which is likely to be much more efficient in practice. 3.2 Approximation algorithms based on local search Before giving the details, we start with a few helpful definitions. For any u ∈ U, Fu = Σt,vxtuvft(u, v). Similarly, for any v ∈ V , Fv = Σt,uxtuvft(u, v). We set µ = 1 n maxt,u,v ft(u, v). It follows that the optimum solution is at least nµ and at most n 2µ. The local search algorithm will maintain the following two invariants: (1) For each t, there is at most one pair (u, v) for which xtuv > 0; (2) All the linear program inequalities hold. It’s easy to set an initial state where the invariant holds (all xtuv = 0). The local search algorithm proceeds in the following steps: While ∃t, u, v : ft(u, v) > Fu pt(u,v) cu + Fv st(u,v) dv + Σu0 ,v0xtu0v 0ft(u 0 , v0 ) + µ:1. Set xtuv = 1 and set xtu0v 0 = 0 for all (u 0 , v0 ) 6= (u, v). 2. While Σt,vpt(u, v)xtuv > cu, reduce xtuv for the job with minimum cuft(u, v)/pt(u, v) such that xtuv > 0. 3. While Σu,vst(u, v)xtuv > dv, reduce xtuv for the job with minimum dvft(u, v)/st(u, v) such that xtuv > 0 Theorem 2. The local search algorithm maintains the two stated invariants. Proof. The first invariant is straightforward, because the only time we increase an xtuv value we simultaneously set all other values for the same t to zero. The only time the linear program inequalities can be violated is immediately after setting xtuv = 1. However, the two steps immediately after this operation will reduce the values of other jobs so as to satisfy the inequalities (and this is done without increasing any xtuv so no new constraint can be violated). Theorem 3. The local search algorithm produces a 3+ approximate fractional solution satisfying the invariants. Proof. When the algorithm terminates, we have for all t, u, v: ft(u, v) ≤ Fu pt(u,v) cu + Fv st(u,v) dv + Σu0 ,v0xtu0v 0ft(u 0 , v0 )µ. We sum this over t, u, v representing the optimum integer assignments: OP T ≤ ΣuFu + ΣvFv + Σt,u,vxtuvft(u, v) + OP T. Each summation simplifies to the algorithm’s objective value, giving the result. Theorem 4. The local search algorithm runs in polynomial time. Proof. Setting xtuv = 1 and setting all other xtu0v 0 = 0 adds ft(u, v)−Σu0v 0xtu0v 0ft(u 0 , v0 ) to the algorithm’s objective. The next two steps of the algorithm (making sure the LP inequalities hold) reduce the objective by at most Fu pt(u,v) cu + Fv st(u,v) dv . It follows that each iteration of the main loop increases the solution value by at least µ. By definition of µ, this can happen at most n 2/ times. Each selection of (t, u, v) can be done in polynomial time (at worst, by simply trying all tuples). Rounding Phase: When the local search algorithm terminates, we have a fractional solution with the additional guarantee from the first invariant. Note that we can extend this to the k-sided version if we increase the approximation factor to k+1+. Below, we give two different rounding schemes. The first works for general values of k and loses an O(k 2 ) factor, for an overall approximation factor of O(k 3 ). The second is specific to the k = 2 case and obtains a better approximation. 1. We randomly make each assignment with probability p times the fractional value (so pxtuv for Coupled Placement), for some p to be defined later. 2. For each assigned job t, if the other jobs t 0 6= t assigned to any one of its assigned machines violate the corresponding linear program constraint, we immediately drop job t. For Coupled Placement this means if P t 06=t,v pt 0 (u, v)xt 0uv > 1 for any t, u we set xtuv = 0.3. Note that we may still violate linear program constraints, but for any particular machine the constraint would be satisfied if we dropped any one of its assigned jobs. We divide the assigned jobs into k + 1 groups. These groups should guarantee that for any machine with at least two assigned jobs, not all its jobs are members of the same group. We then select the group with largest total objective value as our final solution. Theorem 5. For the k-sided version, the rounding scheme runs in poly-time and achieves an O(k 2 )-approximation over the fractional approximation factor (so an overall factor of O(k 3 ) using local search) for appropriate choice of p. Proof. The first two steps finish with a solution of value at least p(1 − p) k times the optimum in expectation. This is because for any job t, the probability of placing this job in step one is exactly p times its fractional value. Consider any machine m where the job is assigned; the expected total size of the other jobs t 0 6= t assigned to this machine is at most pcm and thus the probability that these other jobs exceed cm is at most p. The probability that none of the k machines where t is assigned exceed capacity from other jobs will be at most (1 − p) k . We may still violate constraints. Dividing into k + 1 groups and picking the best gives a result which is at least 1 k+1 p(1−p) k times optimum without violating constraints. Selecting p = 1 k gives the desired approximation factor. It remains to show that the division into groups can be performed in polytime. We start with all machines unmarked. For each group, we select a maximal set of jobs no two of which are assigned the same unmarked machine. We then mark all machines to which one of our current group of jobs is assigned. Note that immediately before we select group i, each remaining job is assigned to at most k−i+1 unmarked machines. For i = 1 this is obvious. Inductively, suppose that job j is assigned to more than k −i unmarked machines immediately before selecting group i + 1. Before selecting group i, job j was assigned to at most k −i+ 1 unmarked machines, and since we never “unmark” a machine it follows that job j was assigned to exactly k − i + 1 unmarked machines both before and after the selection of group i. But then none of the jobs selected in group i are assigned to any of the unmarked machines assigned to job j (else they would have become marked after selection of group i). So we can augment group i with job j without violating the constraint that no two jobs of group i are on the same unmarked machine. This contradicts the maximality of group i. We thus conclude that immediately before we select group k + 1, each remaining job is assigned only to marked machines. Thus group k + 1 selects all remaining jobs (maximality) and the jobs are divided into k+1 groups. Consider any machine m with at least two assigned jobs. Let group i be the first group to contain a job from m. Thus prior to selection of group i, we had not selected any job which was assigned to m and m was unmarked. So group i cannot include more than one job from machine m without violating the condition that no two jobs share an unmarked machine. It follows that there are at least two distinct groups which contain jobs from machine m (group i and also some later group).For MaxCP, we can improve the approximation factor. We refer the reader to Appendix B for details. Theorem 6. For MaxCP, there exists a polynomial-time algorithm based on local search that achieves a 15 + approximation for MaxCP. 4 Online MaxCP and MaxkSP We now study the online version of MaxCP, in which jobs arrive in an online fashion. When a job arrives we must irrevocably assign it or reject it. Our goal is to maximize our total value at the end of the instance. We apply the techniques of [8] to obtain a logarithmic competitive online algorithm under certain assumptions. We first note that online MaxCP differs from the model considered in [8] in that a job’s computation/storage requirements need not be the same. As demonstrated in [8] certain assumptions have to be made to achieve competitive ratios of any interest. We extend these assumptions for the MaxCP model as follows: Assumption 1 There exists F such that for all t, u, v either ft(u, v) = 0 or 1 ≤ ft(u, v) ≤ F min( pt(u,v) cu , st(u,v) dv ). Assumption 2 For = min( 1 2 , 1 ln 2F +1 ), for all t, u, v: pt(u, v) ≤ cu and st(u, v) ≤ dv. It is not hard to show that they (or some similar flavor of these assumptions) are in fact necessary to obtain any interesting competitive ratios (proof in Appendix C). Theorem 7. No deterministic online algorithm can be competitive over classes of instances where either one of the following is true: (i) job size is allowed to be arbitrarily large relative to capacities, or (ii) job values and resource requirements are completely uncorrelated. A small modification to the algorithm of [8] gives an O(log F)-competitive algorithm. Moreover, the lower bound of Ω(log F) shown in [8] applies to online MaxCP as well. (See Appendix D for proof.) Theorem 8. There exists a deterministic O(log F)-competitive algorithm for online MaxCP under Assumptions 1 and 2. For MaxkSP, this can be extended to a O(log kF)-competitive algorithm. Moreover, any online deterministic algorithm for online MaxCP has competitive ratio Ω(log F), and for online MaxkSP has competitive ratio Ω(log kF). Theorem 9. There exist a randomized O(log F)-competitive algorithm (in expectation) for online MaxCP under assumption 1 even if we weaken assumption 2 to require only that = 1 2 . No deterministic online algorithm for the problem can accomplish such a result.Acknowledgments We would like to thank Aravind Srinivasan for helpful discussions, and for pointing us to the Ω(k/ log k)-hardness result for k-set packing, in particular. We thank anonymous referees for helpful comments on an earlier version of the paper, and are especially grateful to a referee who generously offered the key insights leading to improved results for MinCP and MinkSP. References 1. Patterson, D.A.: Technical perspective: the data center is the computer. Communications of the ACM 51 (January 2008) 105–105 2. Dowdy, L.W., Foster, D.V.: Comparative models of the file assignment problem. ACM Surveys 14 (1982) 3. Anderson, E., Kallahalla, M., Spence, S., Swaminathan, R., Wang, Q.: Quickly finding near-optimal storage designs. ACM Transactions on Computer Systems 23 (2005) 337–374 4. 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In: Proceedings of the Symposium on Theory of Computing. (1995) 268–276 21. Lau, L., Ravi, R., Singh, M.: Iterative Methods in Combinatorial Optimization. Cambridge Texts in Applied Mathematics. Cambridge University Press (2011) A Proofs for MinkSP Fix an iteration of the algorithm, and an extreme point x. Let nt denote the number of tight satisfaction constraints, and ni denote the number of tight capacity constraints on the ith side. Since x is an extreme point, if all variables in x take values in (0, 1), then we have N = nt + P i ni . Lemma 5. If all variables in x take values in (0, 1), then there exists a tight capacity constraint in which the number of variables is at most k more than the sum of the variables. Proof. Since each variable occurs in at most k tight capacity constraints, the total number of occurrences of all variables across the tight capacity constraints is kN − s for some nonnegative integer s. Since each satisfaction constraint is tight, each variable appears in k capacity constraints, and each variable takes on value at most 1, the sum of all the variables over the tight capacity constraints is at least knt − s. Therefore, the sum, over all tight capacity constraints, of the difference between the number of variables and their sum is at most k(N − nt). Since the number of tight capacity constraints is N −nt, for at least one of these constraints, the difference between the number of variables and their sum is at most k. Lemma 6. Let u be a side-i node with a tight capacity constraint, in which the number of variables is at most k more than the sum of the variables. Then, the sum of the capacity requirements of the tasks partially assigned to u is at most the available capacity of u plus kCi(u). Proof. Let ` be the number of variables in the constraint for u, and let the associated tasks be numbered 1 through `. Let the demand of task j for the capacity of node u be dj . Then, the capacity constraint for u is P j djxj = bc(u).We know that m − P i xi ≤ k. We also have di ≤ Ci(u). Letting bc(u) denote the current capacity of u, we now derive X i di = bc(u) +Xm j=1 (1 − xi)di ≤ bc(u) + (m − Xm j=1 xi)Ci(u) ≤ bc(u) + kCi(u). Theorem 10. IterRound(k) is a polynomial-time k + 1-approximation algorithm for MinkSP. Proof. By Lemma 5, each iteration of the algorithm removes either a variable or a constraint from the LP. Hence the algorithm is polynomial time. The elimination of a variable that takes value 0 or 1 neither changes cost nor incurs capacity blowup. The elimination of a constraint can only decrease cost, so the final solution has cost no more than the value achieved by the original LP. Finally, by Lemma 6, we incur a blowup of at most 1 + k in capacity. We now show that the MinkSP LP has an integrality gap of Ω(log k/ log log k). We recursively construct an integrality gap instance with ` t sides, for parameters ` and t, with two nodes per side one with infinite capacity and the other with unit capacity, such that any integral solution has at least t tasks on the unit-capacity node on some side, while there is a fractional solution with load of at most t/` on the unit-capacity node of each side. Setting t = ` and k = ` ` , we obtain an instance in which the capacity used by the fractional solution is 1, while any integral solution has load ` = Θ(log k/ log log k). Each task can be placed on one tuple from a subset of tuples; for a given tuple, the demand of the task on each side of the tuple is one. We start with the construction for t = 1. We introduce a task that has ` choices, the ith choice consisting of the unit-capacity node from side i and infinite capacity nodes on all other sides. Clearly, any integral solution uses up unit capacity of one unitcapacity node, while there is a fractional solution (1/` for each choice) that uses only 1/` fraction of each unit capacity node. Given a construction for ` t sides, we show how to extend to ` t+1 sides. We take ` identical copies of the instance with ` t sides and combine the tuples for each task in such a way that for any i, any integral placement places exactly the same task on side i of each copy. Now we add task t + 1 which can be placed in one of ` tuples: unit capacity node on all sides of copy i and infinite capacity node on all other sides, for each i. Clearly, any integral solution will have to add one more task to a unit-capacity node of a side that already has load t, yielding a load t + 1, while a fractional solution can assign load of at most 1/` to the unit-capacity nodes of each side.B Proofs for MaxkSP and MaxCP We first present the linear program for MaxkSP (recall the definition in Section 1.1). Let xte denote the indicator variable for the assignment of job t to the k-tuple e. Maximize: X t,e xteft(e) Subject to: X e xte ≤ 1, ∀t ∈ T, X t,e (dt(e))ixte ≤ Ci(ei), ∀i ∈ {1, . . . , k}, xte ∈ {0, 1}, ∀t ∈ T, e ∈ Q i Si . We now present the improved approximation algorithm for MaxCP. The idea is to obtain a one-to-one correspondance between fractional assignments and machines. Essentially we view the machines as nodes of a graph where the edges are the fractional assignments (this is similar to the rounding for generalized assignment). If we have a cycle, the idea is to shift the fractions around the cycle (i.e. increase one xtuv then decrease some xt 0vw and increase some xt 00wx and so forth). Applying this directly on a single cycle may violate some constraints; while we try to increase and decrease the fractions in such a way that constraints hold, since each job has different “size” on its two endpoints we may wind up violating the constraint P t,v xtuvpt(u, v) at a single node u. This prevents us from doing a simple cycle elimination as in generalized assignment. However, if we have two adjoining (or connected) cycles the process can be made to work. The remaining case is a single cycle, where we can assign each edge to one of its endpoints. Generalized assignment rounding would now proceed to integrally assign each job to its corresponding machine; we cannot do this because each job requires two machines, and each machine thus has multiple fractional assignments (all but one of which “correspond” to some other machine). Lemma 7. Given any fractional solution which satisfies the local search invariants, we can produce an alternative fractional solution (also satisfying the local search invariants and with equal or greater value). This new fractional solution labels each job t with 0 < xtuv < 1, with either u or v, guaranteeing that each u is labeled with at most one job. Proof. Consider a graph where the nodes are machines, and we have an edge (u, v) for any fractional assignment 0 < xtuv < 1. If any node has degree zero or one, we remove that node and its assigned edge (if any), labeling the removed edge with the node that removed it. We continue this process until all remaining nodes have degree at least two. If there is a node of degree three, then there must exist two (distinct but not necessarily edge-disjoint) cycles with a path between them (possibly a path of length zero); since the graph is bipartite all cycles are even in length. We can alternately increase and decrease the fractional assignments of edges along a cycle such that the total load P t,v pt(u, v)xtuv changesonly on a single node u where the path between cycles intersects this cycle. We can do the same along the other cycle. We can then do the same thing along the path, and equalize the changes (multiplicatively) such that there is no overall change in load, but at least one edge has its fractional value changing. If this process decreases the value, we can reverse it to increase the value. This allows us to modify the fractional solution in a way that increases the number of integral assignments without decreasing the value. After applying this repeatedly (and repeating the node/edge removal process above where necessary), we are left with a graph that consists only of node-disjoint cycles. Each of the remaining edges will be labeled with one of its two endpoints (one to each). The overall effect is that we have a one-to-one labeling correspondance between fractional assignments and machines (each fractional edge to one of its two assigned machines). Note however that since each job is assigned to two machines and labeled with only one of the two, this does not imply that each machine has only one fractional assignment. Once this is done, we consider three possible solutions. One consists of all the integral assignments. The second considers only those assignments which are fractional and labeled with nodes u. For each node v, we select a subset of its fractional assignments to make integrally, so as to maximize the value without violating capacity of v. We cannot violate capacity of u because we select at most one job for each such machine. The result has at least 1 2 the value of assignments labeled with nodes u. For the third solution, we do the same but with the roles of u, v reversed. We select the best of these three solutions; our choice obtains at least 1 5 of the overall value. Proof of Theorem 6: The algorithm sketch contains most of the proof. We need to establish that we can get at least 1 2 the fractional value on a single machine integrally. This can be done by selecting jobs in decreasing order of density (ft(u, v)/pt(u, v)) until we overflow the capacity. Including the job that overflows capacity, this must be better than the fractional solution. Thus we can select either everything but the job that overflows capacity, or that job by itself. We also need to establish the 1 5 value claim. If we were to select the integral assignments with probability 1 5 and each of the other two solutions with probability 2 5 , we would get an expected 1 5 of the fractional solution. Deterministially selecting the best of the three solutions can only be better than this. ut C Proof of Theorem 7 We first show that if resource requirements are large compared to capacities, payment functions ft are exactly equal to the total amount of resources and each job requires the same amount over all resources/dimensions (but different jobs can require different amounts), then no deterministic online algorithm can be competitive. Consider a graph G with a single compute node and a single data storage node. Each node has one-dimensional compute/storage capacity of L. A jobarrives requesting 1 unit of computing and storage and will pay 2. Clearly, any competitive deterministic algorithm must accept this job, in case this is the only job. However, a second job arrives requesting L units of computing and storage and will pay 2L. In this case, the algorithm is L-competitive, and L can be arbitrarily large. Next, we show that if resource requirements are small relative to capacities, payment functions ft are arbitrary and resource requirements are identical, then no deterministic online algorithm can be competitive. This instance satisfies Assumption 2 but not Assumption 1. Consider again a graph G with a single compute node and single data storage node each with one-dimensional, unit capacities. We will use up to k + 1 jobs, each requiring 1/k units of computing and storage. The i-th job, 0 ≤ i ≤ k, will pay Mi for some large value M. Now, consider any deterministic algorithm. If it fails to accept any job j < k, then if job j is the last job, it will be Ω(M)- competitive. If the algorithm accepts jobs 0 up through k − 1 then it will not be able to accept job k and will be Ω(M)-competitive. In all cases it has competitive ratio at least Ω(M) and M and k can be arbitrarily large. Similarly, if resource requirements are small relative to capacities, payment functions ft are exactly equal to the total amount of resources requested and resource requirements are arbitrary, then no deterministic online algorithm can be competitive. Consider once more a graph G with a single compute node and single data store node with one-dimensional compute/storage capacities. However, this time the compute capacity will be 1 and the storage capacity will be some very large L. We will use up to k+1 jobs, each requiring 1/k units of computing. The i-th job, 0 ≤ i ≤ k, will require the appropriate amount of storage so that its value is Mi for very large M. Assuming L = O(kMk ), all these storage requirements are at most 1/k of L. Note that storage can accommodate all jobs, but computing can accommodate at most k jobs. Any deterministic algorithm will have competitive ratio Ω(M) and k, M and L can be suitably large. Thus, it follows that some flavor of Assumptions 1 and 2 are necessary to achieve any interesting competitive result. D Proof of Theorem 8 We adapt the framework of [8] to solve the online MaxCP problem. This framework uses an exponential cost function to place a price on remaining capacity of a node. If the value obtained from a task can cover the cost of the capacity it consumes, we admit the task. In the algorithm below, e is the base of the natural logarithm. We first show that our algorithm will not exceed capacities. Essentially, this occurs because the cost will always be sufficiently high. Lemma 8. Capacity constraints are not violated at any time during this algorithm.Algorithm 1 Online algorithm for MaxCP. 1: λu(1) ← 0, λv(1) ← 0 for all u ∈ U, v ∈ V 2: for each new task j do 3: costu(j) ← 1 2 (e λu(j) ln(2F +1) 1− − 1) 4: costv(j) ← 1 2 (e λv(j) ln(2F +1) 1− − 1) 5: For all uv let Ztuv = pj (u,v) cu costu(j) + sj (u,v) dv costv(j) 6: Let uv maximize fj (u, v) subject to Zjuv < fj (u, v) 7: if such uv exist with fj (u, v) > 0 then 8: Assign j to uv 9: λu(j + 1) ← λu(j) + pj (u,v) cu 10: λv(j + 1) ← λv(j) + sj (u,v) dv 11: For all other u 0 6= u let λu0 (j + 1) ← λu0 (j) 12: For all other v 0 6= v let λv0 (j + 1) ← λv0 (j) 13: else 14: Reject task j 15: For all u let λu(j + 1) ← λu(j) 16: For all v let λv(j + 1) ← λv(j) 17: end if 18: end for Proof. Note that λu(n + 1) will be 1 cu Σt,vpt(u, v)xtuv, since any time we assign a job j to uv we immediately increase λu(j + 1) by the appopriate amount. Thus if we can prove λu(n + 1) ≤ 1 we will not violate capacity of u. Initially we had λu(1) = 0 < 1, so suppose that the first time we exceed capacity is after the placement of job j. Thus we have λu(j) ≤ 1 < λu(j + 1). By applying assumption 2 we have λu(j) > 1 − . From this it follows that costu(j) > 1 2 (e ln(2F +1) − 1) = F, and since these costs are always non-negative we must have had Zjuv > pj (u,v) cu F ≥ fj (u, v) by applying assumption 1. But then we must have rejected job j and would have λu(j + 1) = λu(j) Identical reasoning applies to v ∈ V . Next, we bound the algorithms revenue from below using the sum of the node costs. Lemma 9. Let A(j) be the total objective value Σt,u,vxtuvft(u, v) obtained by P the algorithm immediately before job j arrives. Then (3e ln(2F + 1))A(j) ≥ u∈U costu(j) + P v∈V costv(j). Proof. The proof will be by induction on j; the base case where j = 1 is immediate since no jobs have yet arrived or been scheduled and costu(1) = costv(1) = 0 for all u and v. Consider what happens when job j arrives. If this job is rejected, neither side of the inequality changes and the induction holds. Otherwise, suppose job j is assigned to uv. We have: A(j + 1) = A(j) + fj (u, v)We can bound the new value of the righthand side by observing that since costu has derivative increasing in the value of λu, the new value will be at most the new derivative times the increase in λu. It follows that: costu(j + 1) ≤ costu(j) + (λu(j + 1) − λu(j))1 2 ( ln(2F + 1) 1 − )(e λu(j+1) ln(2F +1) 1− ) costu(j + 1) ≤ costu(j) + pj (u, v) cu ( ln(2F + 1) 1 − )(1 2 e λu(j) ln(2F +1) 1− )(e ln(2F +1) 1− ) costu(j + 1) ≤ costu(j) + pj (u, v) cu ln(2F + 1) 1 − (costu(j) + 1 2 )(e ln(2F +1)) Applying assumption 2 gives: costu(j + 1) ≤ costu(j) + (2e ln(2F + 1))(pj (u, v) cu costu(j) + 1 4 ) Identical reasoning can be applied to costv, allowing us to show that the increase in the righthand side is at most: (2e ln(2F + 1))(pj (u, v) cu costu(j) + sj (u, v) du costv(j) + 1 2 ) Since j was assigned to uv, we must have fj (u, v) > pj (u,v) cu costu(j)+sj (u,v) dv costv(j); from assumption 1 we also have fj (u, v) ≥ 1 so we can conclude that the increase in the righthand side is at most: (3e ln(2F + 1))fj (u, v) ≤ (3e ln(2F + 1))(A(j + 1) − A(j)) Now, we can bound the profit the optimum solution gets from tasks which we either fail to assign, or assign with a lower value of ft(u, v). The reason we did not assign these tasks was because the node costs were suitably high. Thus, we can bound the profit of tasks using the node costs. Lemma 10. Suppose the optimum solution assigned j to u, v, but the online algorithm either rejected j or assigned it to some u 0 , v0 with fj (u 0 , v0 ) < fj (u, v). Then pj (u,v) cu costu(n + 1) + sj (u,v) dv costv(n + 1) ≥ fj (u, v) Proof. When the algorithm considered j, it would find the u, v with maximum fj (u, v) satisfying Zjuv < fj (u, v). Since the algortihm either could not find such u, v or else selected u 0 , v0 with fj (u 0 , v0 ) < fj (u, v) it must be that Zjuv ≥ fj (u, v). The lemma then follows by inserting the definition of Zjuv and then observing that costu and costv only increase as the algorithm continues.Lemma 11. Let Q be the total value of tasks which the optimum offline algorithm assigns, but which Algorithm 1 either rejects or assigns to a uv with lower value of ft(u, v). Then Q ≤ Σu∈U costu(n + 1) + Σv∈V costv(n + 1). Proof. Consider any task q as described above. Suppose offline optimum assigns q to uq, vq. By applying lemma 10 we have: Q = Σqfq(uq, vq) ≤ Σq pq(uq, vq) cu costuq (n + 1) + sq(uq, vq) dv costq(n + 1) The lemma then follows from the fact that the offline algorithm must obey the capacity constraints. Finally, we can combine Lemmas 9 and 11 to bound our total profit. In particular, this shows that we are within a factor 3e ln(2F + 1) of the optimum offline solution, for an O(log F)-competitive algorithm. Theorem 11. Algorithm 1 never violates capacity constraints and is O(log F)- competitive. We can extend the result to k-sided placement, and can get a slight improvement in the required assumptions if we are willing to randomize. The results are given below: Theorem 12. For the k-sided placement problem, we can adapt algorithm 1 to be O(log kF)-competitive provided that assumption 2 is tightened to = min( 1 2 , 1 ln(kF +1) ). Proof. We must modify the definition of cost to: costu(j) = 1 k (e λu(j) ln(kF +1) 1− − 1) The rest of the proof will then go through. The intuition for the increase in competitive ratio is that we need to assign the first task to arrive (otherwise after this task our competitive ratio would be unbounded). This task potentially uses up space on k machines while obtaining a value of only 1. So as the value of k increases, the ratio of “best” to “worst” task effectively increases as well. Theorem 13. If we select a random power of two z ∈ [1, F] and then reject all placements with ft(u, v) < z or ft(u, v) > 2z, then we can obtain a competitive ratio of O(log F log k) while weakening assumption 2 to = min( 1 2 , 1 ln(2k+1) ). Note that in the specific case of two-sided placement this is O(log F)-competitive requiring only that no single job consumes more than a constant fraction of any machine. Proof. Once we make our random selection of z, we effectively have F = 2 and can apply the algorithm and analysis above. The selection of z causes us to lose (in expectation) all but 1 log F of the possible profit, so we have to multiply this into our competitive ratio. Collaboration in the Cloud at Google Yunting Sun, Diane Lambert, Makoto Uchida, Nicolas Remy Google Inc. January 8, 2014 Abstract Through a detailed analysis of logs of activity for all Google employees1 , this paper shows how the Google Docs suite (documents, spreadsheets and slides) enables and increases collaboration within Google. In particular, visualization and analysis of the evolution of Google’s collaboration network show that new employees2 , have started collaborating more quickly and with more people as usage of Docs has grown. Over the last two years, the percentage of new employees who collaborate on Docs per month has risen from 70% to 90% and the percentage who collaborate with more than two people has doubled from 35% to 70%. Moreover, the culture of collaboration has become more open, with public sharing within Google overtaking private sharing. 1 Introduction Google Docs is a cloud productivity suite and it is designed to make collaboration easy and natural, regardless of whether users are in the same or different locations, working at the same or different times, or working on desktops or mobile devices. Edits and comments on the document are displayed as they are made, even if many people are simultaneously writing and commenting on or viewing the document. Comments enable real-time discussion and feedback on the document, without changing the document itself. Authors are notified when a new comment is made or replied to, and authors can continue a conversation by replying to the comment, or end the discussion by resolving it, or re-start the discussion by re-opening a closed discussion stream. Because documents are stored in the cloud, users can access any document they own or that has been shared with them anywhere, any time and on any device. The question is whether this enriched model of collaboration matters? There have been a few previous qualitative analyses of the effects of Google Docs on collaboration. For example, the review of Google Docs in [1] suggested that its features should improve collaboration and productivity among college students. A technical report [2] from the University of Southern Queensland, Australia argued that Google Docs can overcome barriers to usability such as difficulty of installation and document version control and help resolve conflicts among co-authors of research papers. There has also been at least one rigorous study of the effect of Google Docs on collaboration. Blau and Caspi [3] ran a small experiment that was designed to compare collaboration on writing documents to merely sharing documents. In their experiment, 118 undergraduate students of the Open University of Israel were randomized to one of five groups in which they shared their written assignments and received feedback from other students to varying degrees, ranging from keeping texts 1Full-time Google employees, excluding interns, part-times, vendors, etc 2Full-time employees who have joined Google for less than 90 days 12 COLLABORATION VISUALIZATION private to allowing in-text suggestions or allowing in-text edits. None of the students had used Google Docs previously. The authors found that only students in the collaboration group perceived the quality of their final document to be higher after receiving feedback, and students in all groups thought that collaboration improves documents. This paper takes a different approach, and looks for the effects of collaboration on a large, diverse organization with thousands of users over a much longer period of time. The first part of the paper describes some of the contexts in which Google Docs is used for collaboration, and the second part analyzes how collaboration has evolved over the last two years. 2 Collaboration Visualization 2.1 The Data This section introduces a way to visualize the events during a collaboration and some simple statistics that summarize how widespread collaboration using Google Docs is at Google. The graphics and metrics are based on the view, edit and comment actions of all full-time employees on tens of thousands of documents created in April 2013. 2.2 A Simple Example To start, a document with three collaborators Adam (A), Bryant (B) and Catherine (C) is shown in Figure 1. The horizontal axis represents time during the collaboration. The vertical axis is broken into three regions representing viewing, editing and commenting. Each contributor is assigned a color. A box with the contributor’s color is drawn in any time interval in which the contributor was active, at a vertical position that indicates what the user was doing in that time interval. This allows us to see when contributors were active and how often they contributed to the document. Stacking the boxes allows us to show when contributors were acting at the same time. Only time intervals in which at least one contributor was active are shown, and gaps in time that are shorter than a threshold are ignored. Gray vertical bars of fixed width are used to represent periods of no activity that are longer than the threshold. In this paper, the threshold is set to be 12 hours in all examples. In Figure 1, an interval represents an hour. Adam and Bryant edited the document together during the hour of 10 AM May 4 and Bryant edited alone in the following hour. The collaboration paused for 8 days and resumed during the hour of 2 pm on May 12. Adam, Bryant and Catherine all viewed the document during that hour. Catherine commented on the document in the next hour. Altogether, the collaboration had two active sessions, with a pause of 8 days between them. Figure 1: This figure shows an example of the collaboration visualization technique. Each colored block except the gray one represents an hour and the gray one represents a period of no activity. The Y axis is the number of users for each action type. This document has three contributors, each assigned a different color. Although we have used color to represent collaborators here, we could instead use color to represent the locations of the collaborators, their organizations, or other variables. Examples with different colorings are given in Sections 2.5 and 2.6. 2 Google Inc.2 COLLABORATION VISUALIZATION 2.3 Collaboration Metrics 2.3 Collaboration Metrics To estimate the percentage of users who concurrently edit a document and the percentage of documents which had concurrent editing, we discretize the timestamps of editing actions into 15 minute intervals and consider editing actions by different contributors in the same 15 minute interval to be concurrent. Two users who edit the same document but always more than 15 minutes apart would not be considered as concurrent, although they would still be considered collaborators. Edge cases in which two collaborators edit the same document within 15 minutes of each other but in two adjacent 15 minute intervals would not be counted as concurrent events. The choice of 15 minutes is arbitrary; however, metrics based on a 15 minute discretization and a 5 minute discretization are little different. The choice of 15 minute intervals makes computation faster. A more accurate approach would be to look for sequences of editing actions by different users with gaps below 15 minutes, but that requires considerably more computing. 2.4 Collaborative Editing Collaborative editing is common at Google. 53% of the documents that were created and shared in April 2013 were edited by more than one employee, and half of those had at least one concurrent editing session in the following six months. Looking at employees instead of documents, 80% of the employees who edited any document contributed content to a document owned by others and 65% participated in at least one 15 minute concurrent editing session in April 2013. Concurrent editing is sticky, in the sense that 76% of the employees who participate in a 15 minute concurrent editing session in April will do so again the following month. There are many use cases for collaborative editing, including weekly reports, design documents, and coding interviews. The following three plots show an example of each of these use cases. Figure 2: Collaboration activity on a design document. The X axis is time in hours and the Y axis is the number of users for each action type. The document was mainly edited by 3 employees, commented on by 18 and viewed by 50+. Google Inc. 32.5 Commenting 2 COLLABORATION VISUALIZATION Figure 2 shows the life of a design document created by engineers. The X axis is time in hours and the Y axis is the number of employees working on the document for each action type. The document was mainly edited by three employees, commented on by 18 employees and viewed by more than 50 employees from three major locations. This document was completed within two weeks and viewed many times in the subsequent month. Design documents are common at Google, and they typically have many contributors. Figure 3 shows the life of a weekly report document. Each bar represents a day and the Y axis is the number of employees who edited and viewed the document in a day. This document has the following submission rules: • Wednesday, AM: Reminder for submissions • Wednesday, PM: All teams submit updates • Thursday, AM: Document is locked The activities on the document exhibit a pronounced weekly pattern that mirrors the submission rules. Weekly reports and meeting notes that are updated regularly are often used by employees to keep everyone up-to-date as projects progress. Figure 3: Collaboration on a weekly report. The X axis is time in days and the Y axis is the number of users for each action type. The activities exhibit a pronounced weekly pattern and reflect the submission rules of the document. Finally, Figure 4 shows the life of a document used in an interview. The X axis represents time in minutes. The document was prepared by a recruiter and then viewed by an engineer. At the beginning of the interview, the engineer edited the document and the candidate then wrote code in the document. The engineer was able to watch the candidate typing. At the end of the interview, the candidate’s access to the document was revoked so no further change could be made, and the document was reviewed by the engineer. Collaborative editing allows the coding interview to take place remotely, and it is an integral part of interviews for software engineers at Google. Figure 4: The activity on a phone interview document. The X axis is time in minutes and the Y axis is the number of users for each action type. The engineer was able to watch the candidate typing on the document during a remote interview. 2.5 Commenting Commenting is common at Google. 30% of the documents created in April 2013 that are shared received comments within six months of creation. 57% of the employees who used Google Docs in April commented at least once in April, and 80% of the users who commented in April commented again in the following month. 4 Google Inc.2 COLLABORATION VISUALIZATION 2.6 Collaboration Across Sites Figure 5: Commenting and editing on a design document. The X axis is time in hours and the Y axis is the number of user actions for each user location. There are four user actions, each assigned a different color. Timestamps are in Pacific time. Figure 5 shows the life of a design document. Here color represents the type of user action (create a comment, reply to a comment, resolve a comment and edit the document), and the Y axis is split into two locations. The document was written by one engineering team and reviewed by another. The review team used commenting to raise many questions, which the engineering team resolved over the next few days. Collaborators were located in London, UK and Mountain View, California, with a nine hour time zone difference, so the two teams were almost ”taking turns” working on the document (timestamps are in Pacific time). There are many similar communication patterns between engineers via commenting to ask questions, have discussions and suggest modifications. 2.6 Collaboration Across Sites Employees use the Docs suite to collaborate with colleagues across the world, as Figure 6 shows. In that figure, employees working from nine locations in eight countries across the globe contributed to a document that was written within a week. The document was either viewed or edited with gaps of less than 12 hours (the threshold for suppressing gaps in the plot) in the first seven days as people worked in their local timezones. After final changes were made to the document, it was reviewed by people in Dublin, Mountain View, and New York. Figure 7 shows one month of global collaborations for full-time employees using Google Docs. The blue dots show the locations of the employees and a line connects two locations if a document is created in one location and viewed in the other. The warmer the color of the line, moving from green to red, the more documents shared between the two locations. Google Inc. 52.6 Collaboration Across Sites 2 COLLABORATION VISUALIZATION Figure 6: Activity on a document. Each user location is assigned a different color. The X axis is time in hours and the Y axis is the number of locations for each action type. Users from nine different locations contributed to the document. Figure 7: Global collaboration on Docs. The blue dots are locations and the dots are connected if there is collaboration on Google Docs between the two locations. 6 Google Inc.3 THE EVOLUTION OF COLLABORATION 2.7 Cross Device Work 2.7 Cross Device Work The advantage of cloud-based software and storage is that a document can be accessed from any device. Figure 8 shows one employee’s visits to a document from multiple devices and locations. When the employee was in Paris, a desktop or laptop was used during working hours and a mobile device during non-working hours. Apparently, the employee traveled to Aix-En-Provence on August 18. On August 18 and the first part of August 19, the employee continued working on the same document from a mobile device while on the move. Figure 8: Visits to a document by one user working on multiple devices and from multiple locations. Not surprisingly, the pattern of working on desktops or laptops during working hours and on mobile devices out of business hours holds generally at Google, as Figure 9 shows. The day of week is shown on the X axis and hour of day in local time on the Y axis. Each pixel is colored according to the average number of employees working in Google Docs in a day of week and time of day slot, with brighter colors representing higher numbers. Pixel values are normalized within each plot separately. Desktop and laptop usage of Google Docs peaks during conventional working hours (9:00 AM to 11:00 AM and 1:00 PM to 5:00 PM), while mobile device usage peaks during conventional commuting and other out-of-office hours (7:00 AM to 9:00 AM and 6:00 PM to 8:00 PM). Figure 9: The average number of active users working in Google Docs in each day of week and time of day slot. The X axis is day of the week and the Y axis is time of the day in local time. Desktop/Laptop usage peaks during working hours while mobile usage peaks at out-of-office working hours. 3 The Evolution of Collaboration 3.1 The Data This section explores changes in the usage of Google Docs over time. Section 2 defined collaborators as users who edited or commented on the same document and used logs of employee editing, viewing and commenting actions to describe collaboration within Google. This section defines collaborators differently using metadata on documents. Metadata is much less rich than the event history logs used in Section 2, but metadata is retained for a much longer period of time. Document metadata includes the document creation time and the last time that the document Google Inc. 73.2 Collaboration for New Employees 3 THE EVOLUTION OF COLLABORATION was accessed, but no other information about its revision history. However, the metadata does include the identification numbers for employees who have subscribed to the document, where a subscriber is anyone who has permission to view, edit or comment on a document and who has viewed the document at least once. Here we use metadata on documents, slides and spreadsheets. We call two employees collaborators (or subscription collaborators to be clear) if one is a subscriber to a document owned by the other and has viewed the document at least once and the document has fewer than 20 subscribers. The owner of the document is said to have shared the document with the subscriber. The number of subscribers is capped at 20 to avoid overcounting collaborators. The more subscribers the document has, the less likely it is that all the subscribers contributed to the document. There is no timestamp for when the employee subscribed to the document in the metadata, so the exact time of the collaboration is not known. Instead, the document creation time, which is known, is taken to be the time of the collaboration. An analysis (not shown here) of the event history data discussed in Section 2 showed that most collaborators join a collaboration soon after a document is created, so taking collaboration time to be document creation time is not unreasonable. To make this assumption even more tenable, we exclude documents for which the time of the last view, comment or edit is more than six months after the document was created. This section uses metadata on documents created between January 1, 2011 and March 31, 2013. We say that two employees had a subscription collaboration in July if they collaborated on a document that was created in July. 3.2 Collaboration for New Employees Here we define the new employees for a given month to be all the employees who joined Google no more than 90 days before the beginning of the month and started using Google Docs in the given month. For example, employees called new in the month of January 2011 must have joined Google no more than 90 days before January 1, 2011 and used Google Docs in January 2011. Each month can include different employees. New employees are said to share a document if they own a document that someone else subscribed to, whether or not the person subscribed to the document is a new employee. Similarly, a new employee is counted as a subscriber, regardless of the tenure of the document creator. Figure 10 shows that collaboration among new employees has increased since 2011. Over the last two years, subscribing has risen from 55% to 85%, sharing has risen from 30% to 50%, and the fraction of users who either share or subscribe has risen from 70% to 90%. In other words, new employees are collaborating earlier in their career, so there is a faster ramp-up and easier access to collective knowledge. Figure 10: This figure shows the percentage of new employees who share, subscribe to others’ documents and either share or subscribe in each one-month period over the last two years. Not only do new employees start collaborating more often (as measured by subscription and sharing), they also collaborate with more people. Figure 11 shows the percentage of new employees with at least a given number of collaborators by month. For example, the percentage of 8 Google Inc.3 THE EVOLUTION OF COLLABORATION 3.3 Collaboration in Sales and Marketing new employees with at least three subscription collaborators was 35% in January 2011 (the bottom red curve) and 70% in March 2013 (the top blue curve), a doubling over two years. It is interesting that the curves hardly cross each other and the curves for the farthest back months lie below those for recent months, suggesting that there has been steady growth in the number of subscription collaborators per new employee over this period. Figure 11: This figure shows the proportion of new employees who have at least a given number of collaborators in each one-month period. Each period is assigned a different color. The cooler the color of the curve, moving from red to blue, the more recent the month. The legend only shows the labels for a subset of curves. The percentage of new employees who have at least three collaborators has doubled from 35% to 70%. To present the data in Figure 11 in another way, Table 1 shows percentiles of the distribution of the number of subscription collaborators per new employee using Google Docs in January 2011 and in January 2013. For example, the lowest 25% of new employees using Google Docs had no such collaborators in January 2011 and two such collaborators in January 2013. 25% 50% 75% 90% 95% January 2011 0 1 4 7 11 January 2013 2 5 10 17 22 Table 1: This table shows the percentile of number of collaborators a new employee have in January 2011 and January 2013. The entire distribution shifts to the right. 3.3 Collaboration in Sales and Marketing Section 3.2 compared new employees who joined Google in different months. This section follows current employees in Sales and Marketing who joined Google before January 1, 2011. That is, the previous section considered changes in new employee behavior over time and this section considers changes in behavior for a fixed set of employees over time. We only analyze subscription collaborations among this fixed set of employees and collaborations with employees not in this set are excluded. Figure 12: This figure shows the percentage of current employees in Sales and Marketing who have at least a given number of collaborators in each onemonth period. Figure 12 shows the percentage of current employees in Sales and Marketing who have at least Google Inc. 93.4 Collaboration Between Organizations 3 THE EVOLUTION OF COLLABORATION a given number of collaborators at several times in the past. There we see that more employees are sharing and subscribing over time because the fraction of the group with at least one subscription collaborator has increased from 80% to 95%. And the fraction of the group with at least three subscription collaborators has increased from 50% to 80%. It shows that many of the employees who used to have no or very few subscription collaborators have migrated to having multiple subscription collaborators. In other words, the distribution of number of subscription collaborators for employees who have been in Sales and Marketing since January 1, 2011 has shifted right over time, which implies that collaboration in that group of employees has increased over time. Finally, the number of documents shared by the employees who have been in Sales and Marketing at Google since January 1, 2011 has nearly doubled over the last two years. Figure 13 shows the number of shared documents normalized by the number of shared documents in January, 2011. Figure 13: This figure shows the number of shared documents created by employees in Sales and Marketing each month normalized by the number of shared documents in January 2011. The number has almost doubled over the last two years. 3.4 Collaboration Between Organizations Collaboration between organizations has increased over time. To show that, we consider hundreds of employees in nine teams within the Sales and Marketing group and the Engineering and Product Management group who joined Google before January 1, 2011, were still active in March 31, 2013 and used Google Docs in that period. Figure 14 represents the Engineering and Product Management employees as red dots and the Sales and Marketing employees as blue dots. The same dots are included in all three plots in Figure 14 because the employees included in this analysis do not change. A line connects two dots if the two employees had at least one subscription collaboration in the month shown. The denser the lines in the graph, the more collaboration, and the more lines connecting red and blue dots, the more collaboration between organizations. Clearly, subscription collaboration has increased both within and across organizations in the past two years. Moreover, the network shows more pronounced communities (groups of connected dots) over time. Although there are nine individual teams, there seems to be only three major communities in the network. Figure 14 indicates that teams can work closely with each other even though they belong to separate departments. We also sampled 187 teams within the Sales and Marketing group and the Engineering and Product Management group. Figure 15 represents teams in Engineering and Product Management as red dots and teams in Sales and Marketing as blue dots. Two dots are connected if the two teams had a least one subscription collaboration between their members in the month. Figure 15 shows that the collaboration between those teams has increased and the interaction between the two organizations has becomed stronger over the past two years. 10 Google Inc.3 THE EVOLUTION OF COLLABORATION 3.4 Collaboration Between Organizations Figure 14: An example of collaboration across organizations. Red dots represent employees in Engineering and Product Management and blue dots represent employees in Sales and Marketing Figure 15: An example of collaboration between teams. Red dots represent teams in Engineering and Product Management and blue dots represent teams in Sales and Marketing Google Inc. 113.5 Cultural Changes in Collaboration 4 CONCLUSIONS 3.5 Cultural Changes in Collaboration Google Docs allows users to specify the access level (visibility) of their documents. The default access level in Google Docs is private, which means that only the user who created the document or the current owner of the document can view it. Employees can change the access level on a document they own and allow more people to access it. For example, the document owner can specify particular employees who are allowed to access the document, or the owner can mark the document as public within Google, in which case any employee can access the document. Clearly, not all documents created in Google can be visible to everyone at Google, but the more documents are widely shared, the more open the environment is to collaboration. Figure 16: This figure shows the percentage of shared documents that are ”public within Google” created in each month. Public sharing is overtaking private sharing at Google. Figure 16 shows the percentage of shared documents in Google created each month between January 1, 2012 and March 31, 2013 that are public within Google. The red line, which is a curve fit to the data to smooth out variability, shows that the percentage has increased about 12% from 48% to 54% in the last year alone. In that sense, the culture of sharing is changing in Google from private sharing to public sharing. 4 Conclusions We have examined how Google employees collaborate with Docs and how that collaboration has evolved using logs of user activity and document metadata. To show the current usage of Docs in Google, we have developed a visualization technique for the revision history of a document and analyzed key features in Docs such as collaborative editing, commenting, access from anywhere and on any device. To show the evolution of collaboration in the cloud, we have analyzed new employees and a fixed group of employees in Sales and Marketing, and computed collaboration network statistics each month. We find that employees are engaged in using the Docs suite, and collaboration has grown rapidly over the last two years. It would also be interesting to conduct a similar analysis for other enterprises and see how long it would take them to reach the benchmark Google has set for collaboration on Docs. Not only has the collaboration on Docs changed at Google, the number of emails, comments on G+, calender meetings between people who work together has also had significant changes over the past few years. How those changes reinforce each other over time would also be an interesting topic to study. Acknowledgements We would like to thank Ariel Kern for her insights about collaboration on Google Docs, Penny Chu and Tony Fagan for their encouragement and support and many thanks to Jim Koehler for his constructive feedback. 12 Google Inc.REFERENCES REFERENCES References [1] Dan R. Herrick (2009). Google this!: using Google apps for collaboration and productivity. Proceeding of the ACM SIGUCCS fall conference (pp. 55-64). [2] Stijn Dekeyser, Richard Watson (2009). Extending Google Docs to Collaborate on Research Papers. Technical Report, The University of Southern Queensland, Australia. [3] Ina Blau, Avner Caspi (2009). What Type of Collaboration Helps? Psychological Ownership, Perceived Learning and Outcome Quality of Collaboration Using Google Docs. Learning in the technological era: Proceedings of the Chais conference on instructional technologies research (pp. 48-55). Google Inc. 13 Circulant Binary Embedding Felix X. Yu1 YUXINNAN@EE.COLUMBIA.EDU Sanjiv Kumar2 SANJIVK@GOOGLE.COM Yunchao Gong3 YUNCHAO@CS.UNC.EDU Shih-Fu Chang1 SFCHANG@EE.COLUMBIA.EDU 1Columbia University, New York, NY 10027 2Google Research, New York, NY 10011 3University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 Abstract Binary embedding of high-dimensional data requires long codes to preserve the discriminative power of the input space. Traditional binary coding methods often suffer from very high computation and storage costs in such a scenario. To address this problem, we propose Circulant Binary Embedding (CBE) which generates binary codes by projecting the data with a circulant matrix. The circulant structure enables the use of Fast Fourier Transformation to speed up the computation. Compared to methods that use unstructured matrices, the proposed method improves the time complexity from O(d 2 ) to O(d log d), and the space complexity from O(d 2 ) to O(d) where d is the input dimensionality. We also propose a novel time-frequency alternating optimization to learn data-dependent circulant projections, which alternatively minimizes the objective in original and Fourier domains. We show by extensive experiments that the proposed approach gives much better performance than the state-of-the-art approaches for fixed time, and provides much faster computation with no performance degradation for fixed number of bits. 1. Introduction Embedding input data in binary spaces is becoming popular for efficient retrieval and learning on massive data sets (Li et al., 2011; Gong et al., 2013a; Raginsky & Lazebnik, 2009; Gong et al., 2012; Liu et al., 2011). Moreover, Proceedings of the 31 st International Conference on Machine Learning, Beijing, China, 2014. JMLR: W&CP volume 32. Copyright 2014 by the author(s). in a large number of application domains such as computer vision, biology and finance, data is typically highdimensional. When representing such high dimensional data by binary codes, it has been shown that long codes are required in order to achieve good performance. In fact, the required number of bits is O(d), where d is the input dimensionality (Li et al., 2011; Gong et al., 2013a; Sanchez ´ & Perronnin, 2011). The goal of binary embedding is to well approximate the input distance as Hamming distance so that efficient learning and retrieval can happen directly in the binary space. It is important to note that another related area called hashing is a special case with slightly different goal: creating hash tables such that points that are similar fall in the same (or nearby) bucket with high probability. In fact, even in hashing, if high accuracy is desired, one typically needs to use hundreds of hash tables involving tens of thousands of bits. Most of the existing linear binary coding approaches generate the binary code by applying a projection matrix, followed by a binarization step. Formally, given a data point, x ∈ R d , the k-bit binary code, h(x) ∈ {+1, −1} k is generated simply as h(x) = sign(Rx), (1) where R ∈ R k×d , and sign(·) is a binary map which returns element-wise sign1 . Several techniques have been proposed to generate the projection matrix randomly without taking into account the input data (Charikar, 2002; Raginsky & Lazebnik, 2009). These methods are very popular due to their simplicity but often fail to give the best performance due to their inability to adapt the codes with respect to the input data. Thus, a number of data-dependent techniques have been proposed with different optimization criteria such as reconstruction error (Kulis & Darrell, 2009), data dissimilarity (Norouzi & Fleet, 2012; Weiss et al., 1A few methods transform the linear projection via a nonlinear map before taking the sign (Weiss et al., 2008; Raginsky & Lazebnik, 2009).Circulant Binary Embedding 2008), ranking loss (Norouzi et al., 2012), quantization error after PCA (Gong et al., 2013b), and pairwise misclassification (Wang et al., 2010). These methods are shown to be effective for learning compact codes for relatively lowdimensional data. However, the O(d 2 ) computational and space costs prohibit them from being applied to learning long codes for high-dimensional data. For instance, to generate O(d)-bit binary codes for data with d ∼1M, a huge projection matrix will be required needing TBs of memory, which is not practical2 . In order to overcome these computational challenges, Gong et al. (2013a) proposed a bilinear projection based coding method for high-dimensional data. It reshapes the input vector x into a matrix Z, and applies a bilinear projection to get the binary code: h(x) = sign(RT 1 ZR2). (2) When the shapes of Z, R1, R2 are chosen appropriately, the method has time and space complexity of O(d 1.5 ) and O(d), respectively. Bilinear codes make it feasible to work with datasets with very high dimensionality and have shown good results in a variety of tasks. In this work, we propose a novel Circulant Binary Embedding (CBE) technique which is even faster than the bilinear coding. It is achieved by imposing a circulant structure on the projection matrix R in (1). This special structure allows us to use Fast Fourier Transformation (FFT) based techniques, which have been extensively used in signal processing. The proposed method further reduces the time complexity to O(d log d), enabling efficient binary embedding for very high-dimensional data3 . Table 1 compares the time and space complexity for different methods. This work makes the following contributions: • We propose the circulant binary embedding method, which has space complexity O(d) and time complexity O(d log d) (Section 2, 3). • We propose to learn the data-dependent circulant projection matrix by a novel and efficient time-frequency alternating optimization, which alternatively optimizes the objective in the original and frequency domains (Section 4). • Extensive experiments show that, compared to the state-of-the-art, the proposed method improves the result dramatically for a fixed time cost, and provides much faster computation with no performance degradation for a fixed number of bits (Section 5). 2 In principle, one can generate the random entries of the matrix on-the-fly (with fixed seeds) without needing to store the matrix. But this will increase the computational time even further. 3One could in principal use other structured matrices like Hadamard matrix along with a sparse random Gaussian matrix to achieve fast projection as was done in fast Johnson-Lindenstrauss transform(Ailon & Chazelle, 2006; Dasgupta et al., 2011), but it is still slower than circulant projection and needs more space. Method Time Space Time (Learning) Full projection O(d 2 ) O(d 2 ) O(nd3 ) Bilinear proj. O(d 1.5 ) O(d) O(nd1.5 ) Circulant proj. O(d log d) O(d) O(nd log d) Table 1. Comparison of the proposed method (Circulant proj.) with other methods for generating long codes (code dimension k comparable to input dimension d). n is the number of instances used for learning data-dependent projection matrices. 2. Circulant Binary Embedding (CBE) A circulant matrix R ∈ R d×d is a matrix defined by a vector r = (r0, r2, · · · , rd−1) T (Gray, 2006)4 . R = circ(r) := r0 rd−1 . . . r2 r1 r1 r0 rd−1 r2 . . . r1 r0 . . . . . . rd−2 . . . . . . rd−1 rd−1 rd−2 . . . r1 r0 . (3) Let D be a diagonal matrix with each diagonal entry being a Bernoulli variable (±1 with probability 1/2). For x ∈ R d , its d-bit Circulant Binary Embedding (CBE) with r ∈ R d is defined as: h(x) = sign(RDx), (4) where R = circ(r). The k-bit (k < d) CBE is defined as the first k elements of h(x). The need for such a D is discussed in Section 3. Note that applying D to x is equivalent to applying random sign flipping to each dimension of x. Since sign flipping can be carried out as a preprocessing step for each input x, here onwards for simplicity we will drop explicit mention of D. Hence the binary code is given as h(x) = sign(Rx). The main advantage of circulant binary embedding is its ability to use Fast Fourier Transformation (FFT) to speed up the computation. Proposition 1. For d-dimensional data, CBE has space complexity O(d), and time complexity O(d log d). Since a circulant matrix is defined by a single column/row, clearly the storage needed is O(d). Given a data point x, the d-bit CBE can be efficiently computed as follows. Denote ~ as operator of circulant convolution. Based on the definition of circulant matrix, Rx = r ~ x. (5) The above can be computed based on Discrete Fourier Transformation (DFT), for which fast algorithm (FFT) is available. The DFT of a vector t ∈ C d is a d-dimensional vector with each element defined as 4The circulant matrix is sometimes equivalently defined by “circulating” the rows instead of the columns.Circulant Binary Embedding F(t)l = X d−1 m=0 tn · e −i2πlm/d, l = 0, · · · , d − 1. (6) The above can be expressed equivalently in a matrix form as F(t) = Fdt, (7) where Fd is the d-dimensional DFT matrix. Let F H d be the conjugate transpose of Fd. It is easy to show that F −1 d = (1/d)F H d . Similarly, for any t ∈ C d , the Inverse Discrete Fourier Transformation (IDFT) is defined as F −1 (t) = (1/d)F H d t. (8) Since the convolution of two signals in their original domain is equivalent to the hadamard product in their frequency domain (Oppenheim et al., 1999), F(Rx) = F(r) ◦ F(x). (9) Therefore, h(x) = sign F −1 (F(r) ◦ F(x)) . (10) For k-bit CBE, k < d, we only need to pick the first k bits of h(x). As DFT and IDFT can be efficiently computed in O(d log d) with FFT (Oppenheim et al., 1999), generating CBE has time complexity O(d log d). 3. Randomized Circulant Binary Embedding A simple way to obtain CBE is by generating the elements of r in (3) independently from the standard normal distribution N (0, 1). We call this method randomized CBE (CBE-rand). A desirable property of any embedding method is its ability to approximate input distances in the embedded space. Suppose Hk(x1, x2) is the normalized Hamming distance between k-bit codes of a pair of points x1, x2 ∈ R d : Hk(x1, x2)= 1 k k X−1 i=0 sign(Ri·x1)−sign(Ri·x2) /2, (11) and Ri· is the i-th row of R, R = circ(r). If r is sampled from N (0, 1), from (Charikar, 2002), Pr sign(r T x1) 6= sign(r T x2) = θ/π, (12) where θ is the angle between x1 and x2. Since all the vectors that are circulant variants of r also follow the same distribution, it is easy to see that E(Hk(x1, x2)) = θ/π. (13) For the sake of discussion, if k projections, i.e., first k rows of R, were generated independently, it is easy to show that the variance of Hk(x1, x2) will be Var(Hk(x1, x2)) = θ(π − θ)/kπ2 . (14) 0 1 2 3 4 5 6 7 8 0 0.05 0.1 0.15 0.2 0.25 log k Variance Independent Circulant (a) θ = π/12 0 1 2 3 4 5 6 7 8 0 0.05 0.1 0.15 0.2 0.25 log k Variance Independent Circulant (b) θ = π/6 0 1 2 3 4 5 6 7 8 0 0.05 0.1 0.15 0.2 0.25 log k Variance Independent Circulant (c) θ = π/3 0 1 2 3 4 5 6 7 8 0 0.05 0.1 0.15 0.2 0.25 log k Variance Independent Circulant (d) θ = π/2 Figure 1. The analytical variance of normalized hamming distance of independent bits as in (14), and the sample variance of normalized hamming distance of circulant bits, as a function of angle between points (θ) and number of bits (k). The two curves overlap. Thus, with more bits (larger k), the normalized hamming distance will be close to the expected value, with lower variance. In other words, the normalized hamming distance approximately preserves the angle5 . Unfortunately in CBE, the projections are the rows of R = circ(r), which are not independent. This makes it hard to derive the variance analytically. To better understand CBE-rand, we run simulations to compare the analytical variance of normalized hamming distance of independent projections (14), and the sample variance of normalized hamming distance of circulant projections in Figure 1. For each θ and k, we randomly generate x1, x2 ∈ R d such that their angle is θ 6 . We then generate k-dimensional code with CBE-rand, and compute the hamming distance. The variance is estimated by applying CBE-rand 1,000 times. We repeat the whole process 1,000 times, and compute the averaged variance. Surprisingly, the curves of “Independent” and “Circulant” variances are almost indistinguishable. This means that bits generated by CBE-rand are generally as good as the independent bits for angle preservation. An intuitive explanation is that for the circulant matrix, though all the rows are dependent, circulant shifting one or multiple elements will in fact result in very different projections in most cases. We will later show in experiments on real-world data that CBE-rand and Locality Sensitive Hashing (LSH)7 has almost identical performance (yet CBE-rand is significantly faster) (Section 5). 5 In this paper, we consider the case that the data points are `2 normalized. Therefore the cosine distance, i.e., 1 - cos(θ), is equivalent to the l2 distance. 6This can be achieved by extending the 2D points (1, 0), (cos θ, sin θ) to d-dimension, and performing a random orthonormal rotation, which can be formed by the Gram-Schmidt process on random vectors. 7Here, by LSH we imply the binary embedding using R such that all the rows of R are sampled iid. With slight abuse of notation, we still call it “hashing” following (Charikar, 2002).Circulant Binary Embedding Note that the distortion in input distances after circulant binary embedding comes from two sources: circulant projection, and binarization. For the circulant projection step, recent works have shown that the Johnson-Lindenstrausstype lemma holds with a slightly worse bound on the number of projections needed to preserve the input distances with high probability (Hinrichs & Vyb´ıral, 2011; Zhang & Cheng, 2013; Vyb´ıral, 2011; Krahmer & Ward, 2011). These works also show that before applying the circulant projection, an additional step of randomly flipping the signs of input dimensions is necessary8 . To show why such a step is required, let us consider the special case when x is an allone vector, 1. The circulant projection with R = circ(r) will result in a vector with all elements to be r T 1. When r is independently drawn from N (0, 1), this will be close to 0, and the norm cannot be preserved. Unfortunately the Johnson-Lindenstrauss-type results do not generalize to the distortion caused by the binarization step. One problem with the randomized CBE method is that it does not utilize the underlying data distribution while generating the matrix R. In the next section, we propose to learn R in a data-dependent fashion, to minimize the distortions due to circulant projection and binarization. 4. Learning Circulant Binary Embedding We propose data-dependent CBE (CBE-opt), by optimizing the projection matrix with a novel time-frequency alternating optimization. We consider the following objective function in learning the d-bit CBE. The extension of learning k < d bits will be shown in Section 4.2. argmin B,r ||B − XRT ||2 F + λ||RRT − I||2 F (15) s.t. R = circ(r), where X ∈ R n×d , is the data matrix containing n training points: X = [x0, · · · , xn−1] T , and B ∈ {−1, 1} n×d is the corresponding binary code matrix.9 In the above optimization, the first term minimizes distortion due to binarization. The second term tries to make the projections (rows of R, and hence the corresponding bits) as uncorrelated as possible. In other words, this helps to reduce the redundancy in the learned code. If R were to be an orthogonal matrix, the second term will vanish and the optimization would find the best rotation such that the distortion due to binarization is minimized. However, when R is a circulant matrix, R, in general, will not be orthogonal. Similar objective has been used in previous works including (Gong et al., 2013b;a) and (Wang et al., 2010). 8 For each dimension, whether the sign needs to be flipped is predetermined by a (p = 0.5) Bernoulli variable. 9 If the data is `2 normalized, we can set B ∈ {−1/ √ d, 1/ √ d} n×d to make B and XRT more comparable. This does not empirically influence the performance. 4.1. The Time-Frequency Alternating Optimization The above is a combinatorial optimization problem, for which an optimal solution is hard to find. In this section we propose a novel approach to efficiently find a local solution. The idea is to alternatively optimize the objective by fixing r, and B, respectively. For a fixed r, optimizing B can be easily performed in the input domain (“time” as opposed to “frequency”). For a fixed B, the circulant structure of R makes it difficult to optimize the objective in the input domain. Hence we propose a novel method, by optimizing r in the frequency domain based on DFT. This leads to a very efficient procedure. For a fixed r. The objective is independent on each element of B. Denote Bij as the element of the i-th row and j-th column of B. It is easy to show that B can be updated as: Bij = ( 1 if Rj·xi ≥ 0 −1 if Rj·xi < 0 , (16) i = 0, · · · , n − 1. j = 0, · · · , d − 1. For a fixed B. Define ˜r as the DFT of the circulant vector ˜r := F(r). Instead of solving r directly, we propose to solve ˜r, from which r can be recovered by IDFT. Key to our derivation is the fact that DFT projects the signal to a set of orthogonal basis. Therefore the `2 norm can be preserved. Formally, according to Parseval’s theorem , for any t ∈ C d (Oppenheim et al., 1999), ||t||2 2 = (1/d)||F(t)||2 2 . Denote diag(·) as the diagonal matrix formed by a vector. Denote <(·) and =(·) as the real and imaginary parts, respectively. We use Bi· to denote the i-th row of B. With complex arithmetic, the first term in (15) can be expressed in the frequency domain as: ||B − XRT ||2 F = 1 d nX−1 i=0 ||F(B T i· − Rxi)||2 2 (17) = 1 d nX−1 i=0 ||F(B T i·)−˜r◦F(xi)||2 2 = 1 d nX−1 i=0 ||F(B T i·)−diag(F(xi))˜r||2 2 = 1 d nX−1 i=0 F(B T i·)−diag(F(xi))˜r T F(B T i·)−diag(F(xi))˜r = 1 d h <(˜r) TM<(˜r)+=(˜r) TM=(˜r)+<(˜r) T h+=(˜r) T g i +||B||2 F , where, M=diag nX−1 i=0 <(F(xi))◦<(F(xi))+=(F(xi))◦=(F(xi)) h = −2 nX−1 i=0 <(F(xi))◦<(F(B T i·))+=(F(xi)) ◦ =(F(B T i·)) g = 2 nX−1 i=0 =(F(xi)) ◦ <(F(B T i·)) − <(F(xi)) ◦ =(F(B T i·)).Circulant Binary Embedding For the second term in (15), we note that the circulant matrix can be diagonalized by DFT matrix Fd and its conjugate transpose F H d . Formally, for R = circ(r), r ∈ R d , R = (1/d)F H d diag(F(r))Fd. (18) Let Tr(·) be the trace of a matrix. Therefore, ||RRT − I||2 F = || 1 d F H d (diag(˜r) Hdiag(˜r) − I)Fd||2 F = Tr 1 d F H d (diag(˜r) Hdiag(˜r)−I) H(diag(˜r) Hdiag(˜r)−I)Fd = Tr (diag(˜r) Hdiag(˜r) − I) H(diag(˜r) Hdiag(˜r) − I) =||˜r H ◦ ˜r − 1||2 2 = ||<(˜r) 2 + =(˜r) 2 − 1||2 2 . (19) Furthermore, as r is real-valued, additional constraints on ˜r are needed. For any u ∈ C, denote u as the complex conjugate of u. We have the following result (Oppenheim et al., 1999): For any real-valued vector t ∈ C d , F(t)0 is real-valued, and F(t)d−i = F(t)i , i = 1, · · · , bd/2c. From (17) − (19), the problem of optimizing ˜r becomes argmin ˜r <(˜r) TM<(˜r) + =(˜r) TM=(˜r) + <(˜r) T h + =(˜r) T g + λd||<(˜r) 2 + =(˜r) 2 − 1||2 2 (20) s.t. =(˜r0) = 0 <(˜ri) = <(˜rd−i), i = 1, · · · , bd/2c =(˜ri) = −=(˜rd−i), i = 1, · · · , bd/2c. The above is non-convex. Fortunately, the objective function can be decomposed, such that we can solve two variables at a time. Denote the diagonal vector of the diagonal matrix M as m. The above optimization can then be decomposed to the following sets of optimizations. argmin r˜0 m0r˜ 2 0 + h0r˜0+ λd r˜ 2 0 − 1 2 , s.t. r˜0 = r˜0. (21) argmin r˜i (mi + md−i)(<(˜ri) 2 + =(˜ri) 2 ) (22) + 2λd <(˜ri) 2 + =(˜ri) 2 − 1 2 + (hi + hd−i)<(˜ri) + (gi − gd−i)=(˜ri), i = 1, · · · , bd/2c. In (21), we need to minimize a 4 th order polynomial with one variable, with the closed form solution readily available. In (22), we need to minimize a 4 th order polynomial with two variables. Though the closed form solution is hard (requiring solution of a cubic bivariate system), we can find local minima by gradient descent, which can be considered as having constant running time for such small-scale problems. The overall objective is guaranteed to be nonincreasing in each step. In practice, we can get a good solution with just 5-10 iterations. In summary, the proposed time-frequency alternating optimization procedure has running time O(nd log d). 4.2. Learning k < d Bits In the case of learning k < d bits, we need to solve the following optimization problem: argmin B,r ||BPk−XPT k RT ||2 F +λ||RPkP T k RT −I||2 F s.t. R = circ(r), (23) in which Pk = Ik O O Od−k , Ik is a k × k identity matrix, and Od−k is a (d − k) × (d − k) all-zero matrix. In fact, the right multiplication of Pk can be understood as a “temporal cut-off”, which is equivalent to a frequency domain convolution. This makes the optimization difficult, as the objective in frequency domain can no longer be decomposed. To address this issues, we propose a simple solution in which Bij = 0, i = 0, · · · , n − 1, j = k, · · · , d − 1 in (15). Thus, the optimization procedure remains the same, and the cost is also O(nd log d). We will show in experiments that this heuristic provides good performance in practice. 5. Experiments To compare the performance of the proposed circulant binary embedding technique, we conducted experiments on three real-world high-dimensional datasets used by the current state-of-the-art method for generating long binary codes (Gong et al., 2013a). The Flickr-25600 dataset contains 100K images sampled from a noisy Internet image collection. Each image is represented by a 25, 600 dimensional vector. The ImageNet-51200 contains 100k images sampled from 100 random classes from ImageNet (Deng et al., 2009), each represented by a 51, 200 dimensional vector. The third dataset (ImageNet-25600) is another random subset of ImageNet containing 100K images in 25, 600 dimensional space. All the vectors are normalized to be of unit norm. We compared the performance of the randomized (CBErand) and learned (CBE-opt) versions of our circulant embeddings with the current state-of-the-art for highdimensional data, i.e., bilinear embeddings. We use both the randomized (bilinear-rand) and learned (bilinear-opt) versions. Bilinear embeddings have been shown to perform similar or better than another promising technique called Product Quantization (Jegou et al., 2011). Finally, we also compare against the binary codes produced by the baseline LSH method (Charikar, 2002), which is still applicable to 25,600 and 51,200 dimensional feature but with much longer running time and much more space. We also show an experiment with relatively low-dimensional data in 2048 dimensional space using Flickr data to compare against techniques that perform well for low-dimensional data but do not scale to high-dimensional scenario. Exam- C/C++ Thread Safety Analysis DeLesley Hutchins Google Inc. Email: delesley@google.com Aaron Ballman CERT/SEI Email: aballman@cert.org Dean Sutherland Email: dfsuther@cs.cmu.edu Abstract—Writing multithreaded programs is hard. Static analysis tools can help developers by allowing threading policies to be formally specified and mechanically checked. They essentially provide a static type system for threads, and can detect potential race conditions and deadlocks. This paper describes Clang Thread Safety Analysis, a tool which uses annotations to declare and enforce thread safety policies in C and C++ programs. Clang is a production-quality C++ compiler which is available on most platforms, and the analysis can be enabled for any build with a simple warning flag: −Wthread−safety. The analysis is deployed on a large scale at Google, where it has provided sufficient value in practice to drive widespread voluntary adoption. Contrary to popular belief, the need for annotations has not been a liability, and even confers some benefits with respect to software evolution and maintenance. I. INTRODUCTION Writing multithreaded programs is hard, because developers must consider the potential interactions between concurrently executing threads. Experience has shown that developers need help using concurrency correctly [1]. Many frameworks and libraries impose thread-related policies on their clients, but they often lack explicit documentation of those policies. Where such policies are clearly documented, that documentation frequently takes the form of explanatory prose rather than a checkable specification. Static analysis tools can help developers by allowing threading policies to be formally specified and mechanically checked. Examples of threading policies are: “the mutex mu should always be locked before reading or writing variable accountBalance” and “the draw() method should only be invoked from the GUI thread.” Formal specification of policies provides two main benefits. First, the compiler can issue warnings on policy violations. Finding potential bugs at compile time is much less expensive in terms of engineer time than debugging failed unit tests, or worse, having subtle threading bugs hit production. Second, specifications serve as a form of machine-checked documentation. Such documentation is especially important for software libraries and APIs, because engineers need to know the threading policy to correctly use them. Although documentation can be put in comments, our experience shows that comments quickly “rot” because they are not updated when variables are renamed or code is refactored. This paper describes thread safety analysis for Clang. The analysis was originally implemented in GCC [2], but the GCC version is no longer being maintained. Clang is a productionquality C++ compiler, which is available on most platforms, including MacOS, Linux, and Windows. The analysis is currently implemented as a compiler warning. It has been deployed on a large scale at Google; all C++ code at Google is now compiled with thread safety analysis enabled by default. II. OVERVIEW OF THE ANALYSIS Thread safety analysis works very much like a type system for multithreaded programs. It is based on theoretical work on race-free type systems [3]. In addition to declaring the type of data (int , float , etc.), the programmer may optionally declare how access to that data is controlled in a multithreaded environment. Clang thread safety analysis uses annotations to declare threading policies. The annotations can be written using either GNU-style attributes (e.g., attribute ((...))) or C++11- style attributes (e.g., [[...]] ). For portability, the attributes are typically hidden behind macros that are disabled when not compiling with Clang. Examples in this paper assume the use of macros; actual attribute names, along with a complete list of all attributes, can be found in the Clang documentation [4]. Figure 1 demonstrates the basic concepts behind the analysis, using the classic bank account example. The GUARDED BY attribute declares that a thread must lock mu before it can read or write to balance, thus ensuring that the increment and decrement operations are atomic. Similarly, REQUIRES declares that the calling thread must lock mu before calling withdrawImpl. Because the caller is assumed to have locked mu, it is safe to modify balance within the body of the method. In the example, the depositImpl() method lacks a REQUIRES clause, so the analysis issues a warning. Thread safety analysis is not interprocedural, so caller requirements must be explicitly declared. There is also a warning in transferFrom(), because it fails to lock b.mu even though it correctly locks this−>mu. The analysis understands that these are two separate mutexes in two different objects. Finally, there is a warning in the withdraw() method, because it fails to unlock mu. Every lock must have a corresponding unlock; the analysis detects both double locks and double unlocks. A function may acquire a lock without releasing it (or vice versa), but it must be annotated to specify this behavior. A. Running the Analysis To run the analysis, first download and install Clang [5]. Then, compile with the −Wthread−safety flag: clang −c −Wthread−s af et y example . cpp#include ” mutex . h ” class BankAcct { Mutex mu; i n t balance GUARDED BY(mu ) ; void d e p o s itIm p l ( i n t amount ) { / / WARNING! Must l o c k mu. balance += amount ; } void withd rawImpl ( i n t amount ) REQUIRES (mu) { / / OK. C a l l e r must have lo c ked mu. balance −= amount ; } public : void withd raw ( i n t amount ) { mu. l o c k ( ) ; / / OK. We ’ ve lo c ked mu. withd rawImpl ( amount ) ; / / WARNING! F a i l e d t o unlo c k mu. } void t r a n sf e rF r om ( BankAcct& b , i n t amount ) { mu. l o c k ( ) ; / / WARNING! Must l o c k b .mu. b . withd rawImpl ( amount ) ; / / OK. d e p o s itIm p l ( ) has no requi rement s . d e p o s itIm p l ( amount ) ; mu. unlo c k ( ) ; } } ; Fig. 1. Thread Safety Annotations Note that this example assumes the presence of a suitably annotated mutex.h [4] that declares which methods perform locking and unlocking. B. Thread Roles Thread safety analysis was originally designed to enforce locking policies such as the one previously described, but locks are not the only way to ensure safety. Another common pattern in many systems is to assign different roles to different threads, such as “worker thread” or “GUI thread” [6]. The same concepts used for mutexes and locking can also be used for thread roles, as shown in Figure 2. Here, a widget library has two threads, one to handle user input, like mouse clicks, and one to handle rendering. It also enforces a constraint: the draw() method should only be invoked only by the GUI thread. The analysis will warn if draw() is invoked directly from onClick(). The rest of this paper will focus discussion on mutexes in the interest of brevity, but there are analogous examples for thread roles. III. BASIC CONCEPTS Clang thread safety analysis is based on a calculus of capabilities [7] [8]. To read or write to a particular location in memory, a thread must have the capability, or permission, to do so. A capability can be thought of as an unforgeable key, #include ” ThreadRole . h ” ThreadRole Input Th read ; ThreadRole GUI Thread ; class Widget { public : v i r t u a l void o nC l i c k ( ) REQUIRES ( Input Th read ) ; v i r t u a l void draw ( ) REQUIRES ( GUI Thread ) ; } ; class Button : public Widget { public : void o nC l i c k ( ) o v e r r i d e { depressed = t rue ; draw ( ) ; / / WARNING! } } ; Fig. 2. Thread Roles or token, which the thread must present to perform the read or write. Capabilities can be either unique or shared. A unique capability cannot be copied, so only one thread can hold the capability at any one time. A shared capability may have multiple copies that are shared among multiple threads. Uniqueness is enforced by a linear type system [9]. The analysis enforces a single-writer/multiple-reader discipline. Writing to a guarded location requires a unique capability, and reading from a guarded location requires either a unique or a shared capability. In other words, many threads can read from a location at the same time because they can share the capability, but only one thread can write to it. Moreover, a thread cannot write to a memory location at the same time that another thread is reading from it, because a capability cannot be both shared and unique at the same time. This discipline ensures that programs are free of data races, where a data race is defined as a situation that occurs when multiple threads attempt to access the same location in memory at the same time, and at least one of the accesses is a write [10]. Because write operations require a unique capability, no other thread can access the memory location at that time. A. Background: Uniqueness and Linear Logic Linear logic is a formal theory for reasoning about resources; it can be used to express logical statements like: “You cannot have your cake and eat it too” [9]. A unique, or linear, variable must be used exactly once; it cannot be duplicated (used multiple times) or forgotten (not used). A unique object is produced at one point in the program, and then later consumed. Functions that use the object without consuming it must be written using a hand-off protocol. The caller hands the object to the function, thus relinquishing control of it; the function hands the object back to the caller when it returns. For example, if std :: stringstream were a linear type, stream programs would be written as follows:st d : : st r i n g st r e am ss ; / / produce ss auto& ss2 = ss << ” H e l l o ” ; / / consume ss auto& ss3 = ss2 << ” World . ” ; / / consume ss2 re tu rn ss3 . s t r ( ) ; / / consume ss3 Notice that each stream variable is used exactly once. A linear type system is unaware that ss and ss2 refer to the same stream; the calls to << conceptually consume one stream and produce another with a different name. Attempting to use ss a second time would be flagged as a use-after-consume error. Failure to call ss3. str () before returning would also be an error because ss3 would then be unused. B. Naming of Capabilities Passing unique capabilities explicitly, following the pattern described previously, would be needlessly tedious, because every read or write operation would introduce a new name. Instead, Clang thread safety analysis tracks capabilities as unnamed objects that are passed implicitly. The resulting type system is formally equivalent to linear logic but is easier to use in practical programming. Each capability is associated with a named C++ object, which identifies the capability and provides operations to produce and consume it. The C++ object itself is not unique. For example, if mu is a mutex, mu.lock() produces a unique, but unnamed, capability of type CapVoir également :
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Rues-Zone-Artisanale..> 30-May-2013 18:36 602KRues de France : Adresse ALEZ ALEZ AL LENN ALEZ AR GOSKER ALEZ GLAZ ALEZ GOZ KERGWENN ALEZ HIR ALEZ IZELA ALEZ IZELLA ALEZ KERBILIEZ ALEZ UHELA ABBE FRANZ STOCK ABER WRAC H ACHILLE BEL AJONCS ALAIN BOUCHART ALAIN COLAS ALBATROS ALEXANDRE DUMAS ALEXANDRE VERCHIN ALIZEES AMPERE ANATOLE FRANCE ANATOLE BRAZ ANDRE JARLAND ANEMONES ANGELA DUVAL ANGELE VANNIER ANNECY ANNE MESMEUR ANTOINE WATTEAU TRINITE ARCHIMEDE AR C HURE AR FAOU AR FOENNEG AR GALL ARGOAT ARMAND CHARPENTIER ARMAND ROBIN ARTHUR BORDERIE AUBEPINES AUGUSTE DUPOUY AUGUSTIN FRESNEL BECQUEREL BEG MENEZ BEL ABRI BELLEVUE BENIGUET BENJAMIN CONSTANT BERGERONNETTES BERLIOZ BERNACHES BERTRAND D ARGENTRE BOIS D AMOUR PINS NEUF QUENVEL ROCHE BONEZE BOULOUARN BRISE BRUYERES CALMETTE GUERIN CAMELIAS CAMILLE COROT CANAPE CAPITAINE JEANNAULT PEZENNEC CARN YAN CERISIERS CHAMONIX CHARDONNERETS GOFFIC PEGUY ROLLAND CHARMILLES CHARMILLES LANDE LOTHAN CHATAIGNERS CHATAIGNIERS CHAUMIERE FER CHENES CHENES VERTS CHEVREFEUILLES C HOAS LEORET CINQ CLAUDE DERVENN CLAUDE MONET CLAUDE PERRAULT COAT AN LEM COATANLEM COLONEL REMY COQS CORMORANS CORNOUAILLE COSMEUR COSQUER COUDRIERS COURLIS CREIS KADOR CYPRES CYTISES DA GUER D AQUITAINE BALANEC BOISEON PEN AR HOAT BREMILLIEC BROCELIANDE BUDE STRATTON DEBUSSY COAT AMOUR COAT CHAPEL COATELAN CORNOUAILLE COUBERTIN CROAS HENT GORREKEAR KERARBLEIS KER ARZEL KERAYEN KERBARS KER CERF KERDANIEL KERDEOZER KERDOUR KERGOLVEZ KERGOSTIOU KERGROACH KERHUEL KERIEL KERIGONAN KERINOU KERJEAN KERJEGU KERLAN VIAN KERLAN VRAS KERLEZANET KERLIEN KERLORET KERMARIA KERMOGUER KERMORVAN KERNENEZ KERNISY KEROUDOT KEROURVOIS KERSALE KERUSTUM KERVALLAN KERVARLAES KERVERN KERVIGNAC KERVOALIC L ABER BENOIT L ABER ILDUT L ABER WRACH CHAPELLE CHAPELLE NEUVE COUDRAIE CROIX DAME CARDE FEE VIVIANE FONTAINE FONTAINE AUX ANGLAIS GUADELOUPE GUYANE L ALSACE LANADAN LANNEVEL LANNIRON PENFELD PLEIADE LARC HANTEL REUNION L ARGOAT ROCHE PERCEE LAVALOT VOIE ROMAINE L ELORN LENHESQ L ETANG L HERMITAGE L IRLANDE L IROISE L ISOLE LOCAL HILAIRE L ODET LOSSULIEN MANAR LAK MENEZ KERVEADY MEZ YAN DEMOISELLES MOLENE NAVALHARS DENIS PAPIN DENTELLIERE PARK AN TI PENANCREACH PEN AR C HOAT PEN AR GUER PEN AR MENEZ PEN AR PEN AVEL PEN ERGUE PENFOULIC PENHOAT PENNERVAN PEN RUIC PRAT AR ROUZ PRAT AR ROZ PRAT GUEN PRAT HIR PROVENCE ROZARGUER ROZ AVEL 4 VENTS ABERS ACACIAS AIGRETTES EUGENE NEVEZ AJONCS AJONCS D OR ALIZES ALOUETTES AUBEPINES AVOCETTES BERGERONNETTES BERGERS BOULISTES BOUVREUILS BRUYERES CAMELIAS CEDRES CELTES CERISES CERISIERS CHAPERONS CHARMILLES CHATAIGNIERS CHENES CIGOGNES COLOMBES COQUILLAGES CORMIERS CORMORANS COURLIS CYPRES DAHLIAS DEPORTES ECUREUILS ECUYERS EGLANTINES EIDERS ENCLOS ERABLES ETANGS KERUZAS FAUVETTES FILETS BLEUS FLEURS D AJONCS FONTAINES FRAISES FRENES FRERES LUMIERE FUCHSIAS FUSAINS GENETS GENETS D OR GERANIUMS GLENANS GLYCINES GOELANDS GRANDS SABLES GROSEILLES GUILLEMOTS HETRES HIRONDELLES HORTENSIAS HOUX IRIS JARDINS JASMINS JONQUILLES LAURIERS LILAS LYS MACAREUX MAGNOLIAS MANDARINS MARAICHERS MARRONNIERS MESANGES MIMOSAS MOUETTES MUGUETS MURES MYRTILLES NARCISSES NEFLES NOISETIERS ORCHIDEES ORMES PECHEURS PECHEUSES PENSEES PERVENCHES PETITS CHENES PETUNIAS PEUPLIERS PINS PINSONS PIVERTS PLATANES POMMIERS PRIMEVERES PRUNELLES PRUNIERS RAINETTES ROCHES BLANCHES ROITELETS ROMARINS ROSES SAPINS SAUGES SOEURS SIBIRIL SOUPIRS SOURCES STERNES SYCOMORES TAMARIS STANG ROZ STANG YOUENN STANG ZU STELLE THUYAS TILLEULS TROENES TROIS TULIPES VERDIERS VIOLETTES TREGOUZEL TREMARIA TREOUZON VIHAN TREQUEFFELEC DEUX PLAGES VERLEDAN ARISTIDE PILVEN LUCAS ROYER DOUR 19 MARS 1962 DU BANELLOU BELIER BOIS QUENVEL BOT BOURDONNEL CANDY CANIK AR HARO FER CHEVREFEUILLE DUC HOEL GUILLY LARGE MAINE MESTO MUGUET DUNES PARC BRAZ PARC HUELLA PEN DUICK PERIGORD PETIT KERVAO MANOIR PETIT PARIS PHARE POAN BEN PONTOIS PRADIC PRESIDENT ROOSEVELT PUITS ROUZ CREIS RUISSEAU STADE STANCOU STANG STANGALA STIFF THYM TROMEUR VALLON MINOU VERGER EDGAR DEGAS EGLANTINE ELORN EMBRUNS ERISPOE ERNEST L ECLUSE ERNEST PSICHARI ERNEST RENAN FEUNTEUN SANE FILANDIERES FILOMENA CADORET FLEURS FONTAINE FORGES ARAGO BUZOT COPPEE DUINE KERBOURCH MAURIAC MENEZ FREDERIC JOLIOT FREDERIC GUYADER FREGATES FRERES CLEMENT FROMVEUR GALLIENI GASTON ESNAULT G BRIOT MALLERIE GENETS GENETS LOCQUERAN MACE GOAREM BIHAN GOAREM PIN GOELANDS GOUNOD GOZ G POITOU DUPLESSY GRAINVILLE GUESTEN GUIAUTEC APOLLINAIRE BUYS GUSTAVE COURBET GWELL KAER TREBEUZEC H BOURDE ROGERIE HELENE BOUCHER HENRI WACQUET HENRY CHATELIER HERONS HIRONDELLES HORTENSIAS IRIS IROISE IZELLA JACQUES PREVERT JAKEZ RIOU JARDINS JARL PRIEL BAPTISTE CHARCOT BAPTISTE LOUVET BOUIN DONNARD MILLET JULIEN LEMORDANT LAGADIC CORRE LOUIS CHUTO DEGUINET MESCHINOT MORAND JEANNE FLAMME JEANNE VALMIER QUEIGNEC YVES LEVENES JEFF PENVEN J L G NAOUR J L GREVELLEC JONQUILLES JOSE MARTINACHE JOSEPH GAY LUSSAC KERADEN KERALLAIN KERANGUDEN KERAUDREN KERAUTRET KERAVILIN BIAN KERBLEUNIOU KER BREACH KER EOL KERGALY KERGONDA KERGOZ KERINER KERIVOT KERJEAN KERJEROME KERLIDOU KERMARC KERMOOR KERSALE KERSALE D EN HAUT KERVILLORE KREIZKER L BOUGAINVILLE LAE LAGADENOU LAM AR ZANT LAND LANN ROCHEFOUCAULT LAURIERS LAVANDIERES LAVIGERIE LAVOISIER LEA LEACH LEAC LILIA LEACH LILIA LEIGN LEN LENN LEO LEON LERN LESTONAN VRAZ LEUN LEUR LEURE LEURIOU LEURVEAN LIA LIJOU LILAS LIORZHOU LIORZIOU LIORZOU LIOU LOAR RUZ KERMABILOU LOCH LOEIZ AR FLOCH LOUARN LOUC H LOUIS BOUGUENNEC LOUIS FEUNTEUN LOUIS GUILLOUX LOUIS HEMON LOUIS OGES LOUIS PASTEUR LUCIEN SIMON LUDU MADAME SEVIGNE MADAME NESTOUR MANOIR MARAICHERS MARCEL PAGNOL MARGUERITES PAULE SALONNE MARONNIERS MARRONNIERS MARTILIN AN DALL MARYSE BASTIE MAURICE BON MAX JACOB MECHOU GOAREM MEILH GLAZ MEILH STANG VIHAN MENEZ MENEZ GOUERON MENEZ KADOR MENEZ TRAON MEN FOUES MENGLEUZ ROUZ MESANGES MESCANTON M FRIAND MEUNIER MICHAEL FARADAY MICHEL JAOUEN MICHEL JULIEN MIMOSAS MOLENE MORVAN LEBESQUE MOUETTES MOZART MUGUET NATHALIE LEMEL NAVALHARS NAVIGATEURS NEIZ KAOUEN NEUCHATEL NEUVE NOEL ROQUEVERT NOISETIERS NOROIT OISEAUX OLIVIER SOUVESTRE ONDINES OSISMES PARC PARC AR GROAS PARC AR VILIN PARK AN ABER PARK BIHAN PARK BRAZ PARK LANN PARK LENDU CLAUDEL GAUGUIN LANGEVIN VERLAINE PECHEURS PELICANS PEN AN TREZ PEN AR GUER PEN AR HAN PEN AR MENEZ PEN AR STREJOU PEN AR VALLY PEN AR VALY PEN AR VERN PENLAN PENNANEAC H PENNKER PER JAKEZ HELIAS PERVENCHES PETRELS PEUPLIERS PHARE BLAYAU CURIE DAMALIX COUBERTIN LEPINE LOUET PINS PLUVIERS POMMIERS PONTICOU PONTIGOU ROMAIN PORSMEUR POTAGERS POULDIGUY PRAT PRAT AR FEUNTEUN PRAT AR ZARP PRAT COULM PRESIDENT WILSON PRIMEVERES PROSPER PROUX PRUNELLES QUATRE VENTS QUINQUIS RAYMOND CANVEL RENE MEN RENE GUY CADOU R GEN PENFENTENYO ROLAND DORE ROLAND DORGELES RONSARD ROSA FLOCH ROSCOGOZ ROSEAUX ROSEAUX TREZ HIR ROSERAIE ROSIERS ROSMEUR ROSSINI ROUZ ROZ AN AODIG ROZ AVEL GENERAL PENFENTENYO KREIS RUPOEZ ARMEL CLET CLOUD SAINTE ANNE SAINTE CROIX SAINTE THUMETTE GILDAS GUENOLE GURLOES SAINT LUC MALO POL ROUX PRIMEL ROCH ROCK RONAN TUGDUAL URSULE VALENTIN SAMUEL CHAMPLAIN SAULES STADE STANG ZU STEIR STIFF SUFFREN SULLY SURCOUF SUROIT TAMARIS THEODORE BOTREL THEODORE VILLEMARQUE THORENS TI LIPIG TILLEULS TORRENS TOULIFO TOULIGUIN TOULOUSE LAUTREC TOURIGOU HUELLA TOURNE PIERRES TOURTERELLES TREBEHORET TREGOR TRIELEN TRISTAN CORBIERE TROENES TROUIDY TULIPES TY BOUT TY NEVEZ U JOSEPH COUCHOUREN VANNETAIS VENELLE VERTE VIAN KERDEACH VIBERT 1 VIBERT VICTOR SALEZ VICTOR SEGALEN VILLIERS L ISLE ADAM VILLOURY VINCENT VAN GOGH VIVALDI VOLTAIRE VRAS VRAS LAMBELL WATTEAU W CATHERINE BOOTH XAVIER GRALL YANN SOHIER YOUEN DREZEN YVES ELLEOUET YVES AUDREIN ALLEGOT ALLEGUENNOU ALLEGUENOU AMARINE KERSALIOU AMIRAL TRAP AN ALE BIHAN AN ALE VRAS AN ALLE VRAS AN ARAGON AN ATANT GOZ TOULIGUIN ANCIENNE ANCIENNE GUILLY ANCIENNE PRIVE KERGROES ANCIEN PRESBYTERE ANCIEN PRESBYTERE BOURG ANCIEN PRESBYTERE MENEZ STEUD AN DACHENNIG AN DAOU BARZ AN DIRZO HENT AN DISKUIZ AN DISTRO RHUN AN DIVISION AN DOURDU AN DOURIG AN DREINDED AN DREUZEC KERBEURNES AN DURZHUNELL TI GARDE TREVOURDA AN ENEZ ANGLE AN HENT COZ AN ILISVEN AN ILIZ VENN AN OALEJOU AN ODE BRI AN ODE WENN ANSE KERJEGU ANSE KERAMBACCON ANSE TY MARK COSQUER ANSE ROHOU ANSE ROSPICO ANSE ROZ ANSE GILDAS ANSE LAURENT ANTER AR VALY ANTEREN ANTER HENT AN TI GWER RUVEIC AN TRAON AN TREIZ AR BALAN AOUR AR BARADOS STANG AR WENNEG AR BARADOZ AR BAR HIR ARBRE CHAPON AR C AR CHERIGOU BIHAN AR C HEUN ARC HOAT AR COSQUER AR CREAC H ARDOISIERE KERMANACH AR DREO AR FELL AR FOTOU BIHAN AR FOTOU VRAS AR GEBOG ARGENTON 10 STREAT TOUL AR LIN ARGENTON 11 LANHALLES ARGENTON 11 STREAT TOUL AR LIN ARGENTON 12 LANHALLES ARGENTON 12 STREAT AR ARGENTON 12 STREAT TOUL AR LIN ARGENTON 13 LANHALLES ARGENTON 14 VERLEN ARGENTON 15 GORREQUEAR ARGENTON 15 LANHALLES ARGENTON 16 HENT SANT GONVEL ARGENTON 17 HENT AOD PENFOUL ARGENTON 17 HENT AOD VERLEN ARGENTON 18 HENT AOD VERLEN ARGENTON 18 VERLEN ARGENTON 1 GORREQUEAR ARGENTON 1 HENT AOD GWEN TREZ ARGENTON 23 HENT AOD VERLEN ARGENTON 24 GONVELD ARGENTON 25 ARGENTON 26 LANHALLES ARGENTON 27 ARGENTON 27 KERRIOU ARGENTON ARGENTON 2 GORREQUEAR ARGENTON 2 HENT SANT GONVEL ARGENTON 2 KERLEGUER ARGENTON 2 LANHALLES ARGENTON 36 KERRIOU ARGENTON 38 LANHALLES ARGENTON 4 STREAT PRAT AR C ARGENTON 4 STREAT TOUL AR RAN ARGENTON 5 HENT SANT GONVEL ARGENTON 5 KERRIOU ARGENTON 6 STREAT AR ARGENTON 7 HENT AOD VERLEN ARGENTON 7 STREAT TOUL AR LIN ARGENTON 7 STREAT TOUL AR RAN ARGENTON 8 VIVIER ARGENTON 8 STREAT TOUL AR RAN ARGENTON 9 STREAT TOUL AR LIN ARGENTON VIVIER ARGENTON GONVELD ARGENTON VERLEN AR GLOANEG AR GOAREM GOUR AR GOAREN KEROZAL AR GOUDORIC AR GOUENT AR GOZKER AR GUILI AR GUILI HERVE AR GUILY AR HOAD ARHOAT AR HOAT AR MAJENN AR MEAN AR MENEZ AR MENEZIG KEROUANT AR MENNONT AR MEN TOUL KERDOUALEN AR MERDI AR MILIN COZ KERGLIEN AR MOR RUST AR PALUD AR PONTIG AR POULL AR REUNGOL AR ROUDOUR AR ROUZ AR ROZ COZ AR ROZ NEVEZ AR RUGELL ARSENAL AR STIVELL AR STOUNK AR STYVEL AR SULIAO AR VANEL AR VARREC AR VEIN GLAS AR VENEC AR VENGLEUZ AR VERN AR VERNIG AR VEROURI AR VEROURI NEVEZ AR VEROURY NEVEZ AR VEVID AR VILLASE AR VOAREM AR VOURC H AR VRENNEC AR VRENNIG AR VUJID VRAS AR WOAREM AR YUN AR ZAL AR ZALOU ASKEL ATELIER KERAVAL T KERNEVEN ATTILOU AUBERGE NEUVE AULNAYS AUX QUATRE AUX QUATRE VENTS AVEL AR MENEZ AVEL CORNOG GOAREM KERMADORET AVEL DRO AVEL HUELLA KERHUELLA AVEL MAD AVEL MENEZ AVEL MOOR CRINQUELLIC VIAN AVEL MOR AVEL MOR KERSCOUARNEC AVEL VOR AVEL VOR KERHORNOU 1 ERE DFL 8 MAI 1945 GANTIER ALAIN LAY ALEXIS ROCHON AMIRAL REVEILLERE ANNE BRETAGNE ANSE PENFOUL ARISTIDE BRIAND ARMAND PEUGEOT AUGUSTE GANTIER BEL AIR D ISIS BOUILLOUX LAFONT BUSUM CALLINGTON CAMILLE DESMOULINS CHAPELLE FOUCAULD CHARLES TILLON COATMEZ BANTRY BECHARLES BIELEFELD SENNE BRETAGNE COAT KAER COATMEUR COAT MEZ CORAY GARSALEC KERADENNEC KERARTHUR KERBONNE KERDREZEC KERGOAT AL LEZ KERHUEL KERNEGUES KERRIEN KERVEGUEN KERVOALIC CASCADE FRANCE LIBRE LIBERATION MER PLAGE PLAGE GUEUX POINTE REPUBLIQUE RESISTANCE L NAVALE LIMERICK L OCEAN MENEZ BIHAN NORMANDIE PEN CARN LESTONAN PROVENCE QUIMPER REMSCHEID BOUVREUILS CARMES CASTORS CHASSEURS SCHLEIDEN COLS VERTS CORMORANS DUNES ECUREUILS GIRONDINS GLENAN GOELANDS SKIBBEREEN OISEAUX PRES SAULES SPORTS TOURTERELLES TALLINN TARENTE TI DOUAR TI TREBEHORET TRURO TY BOS WALDKAPPEL DIGUE BARON LACROSSE D ISIS BRADEN CABELLOU DUCHESSE ANNE CLAIR LOGIS CORNIGUEL DORLETT DOURIC GUERDY GUIRIC LEZARDEAU LYCEE MARECHAL FOCH COQ POLYGONE POULDUIC PRESIDENT ALLENDE PRESIDENT ROBERT SCHUMAN ROUILLEN SAULE SILLON TECHNOPOLE TEVEN TOUROUS ERIC TABARLY FOUESNANT FRANCAIS LIBRES GENERAL BAIL BRASSENS CLEMENCEAU BAIL POMPIDOU GHILINO G BAIL HENT GLAZ JACQUES VIOL BERNARD DESSAUX JAURES DANIEL JEANNE D ARC JOHN KENNEDY JULES FERRY KERCREVEN KERDANET KERDIVICHEN KERINCUFF KERISTIN KER IZELLA KERLECH KERLECK TOUR D AUVERGNE LEON BLUM LESTONAN LOUIS LEZ LOUISON BOBET MANU BRUSQ MARECHAL FOCH MARECHAL MARGUERITE FAYOU MATHIEU DONNART MER MIOSSEC OCEAN ODET PABLO NERUDA GUEGUIN JACKEZ HELIAS MENDES FRANCE PLAGE PORT PORTSALL PREMIER MAITRE L HER QUELEREN RENE COADOU REPUBLIQUE ROBERT JAN ROBERT SCHUMANN DENIS JOSEPH SALVADOR ALLENDE SCHUMANN TALADERCH TEVEN TOUL AN TOUR D AUVERGNE VICTOR HUGO VICTOR GORGEU WALTENHOFENN WALTENHOFFEN WURSELEN YVES THEPOT BACON BAIE TREPASSES BAIE TREPASSES ROZ VEUR BAIE PENFOUL BAILLAOU BALAENNOU BALAENOU BALAN AR GOFF BALANDREO BALANEC BALANEC HUELLA BALANEG BALANEIR BALANOU BALAREN BALIALEC BALY PLUFERN BALY VERN BANALEC AR LOUET BANALOU BAND BANDELLOU BANELL AL LENN BANELL AR GROAZ BANELL AR MERC HED BANELL DALL BANIGUEL BANINE BANTHOU BARADOS BARADOZ BARADOZIC BARADOZIG BARBARY BARBOA BARBU BARENNOU BARGUET BARHIR BARNAO BARNAOU BARNENEZ BARONNOU BAROUZ BARRACHOU BARR AVEL BARRE BARRE NEVEZ BARRIERE CROIX BARRIERE CROIX BARRIERE ROUGE BARVEDEL BASCAM BAS BAS BOURG BASINIC BASSIN 9 MARINE LANINON BASSINIC DANIEL KERLEGAN KERLEGAN KERSEGALOU COAT GRANGE ROCHE D ELLIANT PINS PLEUVEN DUC DUC RELECQ GARIN JAFFRAY KERDALEM KERGOAT DUC NEUF PLEUVEN GERMAIN BOISSIERE BOLAST BOLAZEC VIAN BOLE BOLEDER BOLORE BON COIN BONNE CHANCE BONNE NOUVELLE BONNE RENCONTRE BONNIOUL BON PLAISIR BON REPOS BONTUL BORCH LESTREQUEZ BORDENEN BORHOU BIHAN BORODOU BOSCADEC BOSCAO BOSSAVARN BOSSULAN BOTANIEC BOTAVAL BOT AVAL BOT BALAN BOTBALAN BOT BALAN KER ANNA BOTBEGUEN BOTBERN BOTBIAN BOTBIHAN BOTBODERN BOTCABEUR BOTCADOR BOT CARREC BOT CARREC IZELLA BOTCOAT BOTDOA BOTDREIN BOT DREIN BOTDUEL BOTEDEN BOTFAO BOT FAO BOTFAVEN BOTFORN BOTFRANC BOT GUEZ BOTHALEC BOTHENEZ BOTHUAN BOTIGNERY BOTLAN BOTLANN BOTLAN MATHIEU BOTLAVAN BOTLENAT BOTLOVAN BOTMEUR BOTMEZER BOT ONN BOT PIN BOT QUELEN BOTQUELEN BOTQUENAL BOTQUEST BOTQUIGNAN BOT RADEN BOT REPOS BOTREVY BOTSAND BOTSPERN BOTTREIN BOTVELLEC BOUCHEOZEN BOUDIC BOUDIGOU PEAR PEN AR MENEZ BOUDIGUEN BOUDOU BOUDOUGUEN BOUDOULAN BOUDOULAND BOUDOUREC BOUDRACH BOUDUEL BOUED MANOIR KERSKLOEDENN BOUERES BOUGES BOUGOUGNES BOUGOUROUAN BOUILLARD BOUILLEN BOUILLEN HOZ BOUILLEN AR HOZ BOUILLEN BRAS BOUILLENNOU BOUILLENNOU TREGONDERN BOUILLEN VIAN BOUIS BOULACH BOULAIE BOULAOUIC BOULARD GUILLOU GUILLOU AMIRAL KERGUELEN BOUGAINVILLE CAMILLE REAUD CHATEAUBRIAND CLEMENCEAU COMMANDANT MOUCHOTTE CORNEILLE CORNICHE CORNICHE TREZ HIR CORNICHE TREZ HIR BRETAGNE COATAUDON CREAC GWEN KERNEUZEC KERVEGUEN FRANCE LIBRE LAITA PLAGE REPUBLIQUE L EUROPE L OCEAN PLYMOUTH PROVENCE ACACIAS FRANCAIS LIBRES FRERES MAILLET SLIGO MYOSOTIS POILUS D ESTIENNE D ORVES DUPLEIX PONANT EUROPE GAMBETTA GENERAL ISIDORE MARFILLE RICHEPIN KATERINE WYLIE FONTAINE LAITA LEON BLUM LEOPOLD MAISSIN MARECHAL MARINE MAR MENDES FRANCE MER MER TREZ HIR MICHEL BRIANT MOLIERE MONTAIGNE OCEAN PLAGE PLAGES ROBERT SURCOUF SAINTE BARBE MARTIN TANGUY PRIGENT THIERRY D ARGENLIEU VICTOR HUGO VIOLLARD YVES NORMAND YVES NORMANT BOULFRET BOULLAC H BOULLACH BOULLAOUIC BOULLEN BOULOUZARD BOULVA BOULVAS BOULVERN BOULWENN BOUQUIDIC BOURAPA BOURAPPA BOURBONNAISE TY NEVEZ KERVENAL BOURDEL BOURDIDEL BOURETTE BOURG 11 LEUR SANT MERYNN BOURG BOURGADEN BOURG BERRIEN BOURG RUMENGOL BOURG GROUANEC BOURG LAMBER BOURG LAMPAUL BOURG LILIA BOURG LOGONNA BOURG NEUF BOURGNEUF BOURG NEVEZ IZELLA BOURG ROUTE FOUESNANT BOURG ROUTE POULDREUZIC BOURG ROUTE QUIMPER BOURG CADOU BOURG TI BRUG BOURG TREZIEN BOURLACH BOURLAND BOURNAZOU BOUROUGUEL BOUROUILLES BOURRET BOUT BOUTEFELEC BOUTIQUE BOUTOIGNON BOUTREC BOUTROUILLE BOUTROUILLES BOUYOUNOU BOZENOC CRUGUEL BRAMOULLE BRANCOU BRAS BRANCOU BRAZ BRANDERIEN BRANILIEC BREC HLEUZ BREC HOUEL BREGALOR BREGOULOU BREHARADEC BREHARN BREHELIEC BREHELLEN BREHEUNIEN BIAN BREHEUNIEN BRAS BREHICHEN BREHOAT BREHOAT KERDEZPET BREHONNET BREHORAY BREHOSTOU BREHOUNIC BREIGNOU BREIGNOU COZ BRELEIS BRELEVENEZ BREMEL BREMELLEC BREMEUR BREMILLEC BREMILLIEC BREMOGUER BREMPHUEZ BRENANVEC BRENANVEC NEVEZ BRENAVALAN BRENAVELEC BRENDAOUEZ BRENDEGUE BRENDEGUE BIHAN BRENDEGUE BRAS BRENELEC BRENELIO BRENENTEC BRENEOL BRENESQUEN BRENGOEL BRENGOULOU BRENGURUST BRENINGANT HUELLA BRENIZENEC BRENIZENNEC BRENN BRENNANVEC BRENNANVEC NEVEZ BRENNTENOES BRENOT BRENOT VIAN BRENOT VIAN TY BIAN BRENTERC H BRENUMERE BRENVARCH BRESLAU BRETIEZ BRETIN BRETOUARE BRETTIN BREUGNOU BREUGUNTUN BREUIL BREUIL BRAS BREUNEN BREUNTEUNVEZ BREVENTEC BREZAL BREZECHEN BREZEHAN BREZEHANT BREZEHEN BREZEUHEN BREZOULOUS BRIAC VIAN BRIAC VRAS BRIANTEL BRIDEN BRIEC BIHAN BRIEC BRAS BRIELLEC BRIEMEN BRIGNEAU KERABAS BRIGNEAU KERNON ARMOR BRIGNEAU KERVETOT BRIGNEAU KERZIOU BRIGNEAU MALACHAPPE BRIGNEAU TEMPLE BRIGNEAU TRELAZEC BRIGNEOCH BRIGNEUN BRIGNIOU BRIGNON BRIGNOU BRIGOULAER BRINGALL BRINGALL HUELLA BRINGALL IZELLA BRIOU BRIOU HUELLA BRISCOUL BRISCOUL HUELLA BROAL BROC HET BROENNIC BROENNOU BROGADEON BROGARONNEC BROGODONOU BROGORONEC BROHEON BROMEUR BROMINOU VRAS BRONDUSVAL BRONE BRONENOU BRONNOLO BRONNUEL BRONOLO BRONUA BROUSTOU BRUC BRUCOU BRUG AN IZEL BRUG BRUGOU BRUGUET BRUKIGER 54 HENT KERMINALOU BRULERIE BRULUEC BRULY BRUMPHUEZ BRUMPHUEZ HUELLA BRUMPHUEZ IZELLA BRUNEC BRUNGUEN BRUNGUEN COZ BRUNGUENNEC BRUNOC BRUYERE BRUYERES BUDOU BUELHARS BUGNET 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CANN CARBON CARO CERF CLERCH BURTULL CLEUN CLOASTRE CLOS HERRY COMTE COSQUER CRANN CREAC H CREACH CROISSANT CROIZIOU CUN DORGUEN DOURIC DREFF DRENNEC FESSIOU FOZOU FRESQ FRET FUME GARO GARZON GOLLEN GUELLEC GUENDON GUERN LAUNAY GUILLOC GUIP HAFFOND HELLES LEIGNOU LEIGN ROUX JOA LANNOU LAUNAY LEING LEURE LICHERN LOURCH MANOIR MENDY MENHIR MERDY MESTO MEZOU L ENCLOS NEC HOAT PALAIS PARADIS PARCOU PEDEL PEULIOU PILLION PLESSIS NEUF PONTOIS POTEAU VERT QUEAU QUINQUIS RADEN RELAIS RELUET RESTAURANT REUN RHUN ROHOU ROSIER ROUDOUR ROUDOUS LESCAOUIDIC LESCOAT EOZEN SCRAIGN SEQUER NEVEZ FRIANTIS GENETS HARAS HAUTS PESMARCH HAUTS DOURDUFF LESIVY BRAS LESMEL LESMOUALCH BRAS LESNON SPINS PLANTS LESPLOMEUR ROCAILLES SALLES LESSOUGAR STANG LESTARIDEC LESTEMBERT LESTEVEN LESTIALA LESTRAOUEN STREAT STREJOU LESTREMINOU LESTREVIGNON LESTRIOUAL LESVINGANT TEMPLE TRAON TREFF TREILLIC TY ZONT LEUR AR MARCHE LEUR AR MORIS LEURIOU VAL ROUGE VAL VERT VERGEZ VERN VEZEC VIBEN VIEUX HAFFOND VIZIER VRENNIT 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MESNOTER MESNUS MESONAN MESPAOL MESPAUL MESPAULIOU MESPELER MESPERENES MESPERLEUC MESPILAT MESPIRIOU MESPIRIT MESPONT MESPOT MESPOUL BIAN MESPRIGENT MESQUEAU MESQUEFFURUS MESQUEO MESQUEON MESQUERNIC MESQUILLY MESROHIC MESSELLOU MESSILIO MESSINGANT MESSINOU MESSIOU MESSIOUAL MESSIOUMEUR MESTALLIC MESTANEN NEVEZ MESTELER MESTELHOEN MESTELLAN MESTEURNEL MESTINIOU MESTINOU MESTIRIEN MESTUAL MESVENNOU MESVOACET METAIRIE METAIRIE BOISEON METAIRIE CHEFFONTAINES METAIRIE TRESSEOL METAIRIE NEUVE METAIRIE QUEBLEIN METAIRIE STANG METAIRIE TRESSEOL MEZAHERE MEZALER MEZALIA MEZALLAP MEZAMBELEG MEZAMER MEZ AN DIB MEZANGAZEC MEZ AN HOSTIS MEZANLAP MEZANLEZ MEZANRUN MEZANSTOURN MEZ ANTELLOU MEZ AN VRAC H MEZAOLET MEZ AR MEZ AR BEZ MEZ AR C HANT MEZ AR C HEFF MEZARES MEZAREUN MEZ AR FEUNTEUN MEZARGOFF MEZARGROAS MEZ AR GROAS MEZ AR REUN MEZ AR ROZEC MEZ AR SEILER MEZ AR SELER MEZ AR SELLER MEZARUN MEZARVERN MEZ AR VILIN MEZ AR VOAZ MEZASCOUR MEZASTEN MEZAUDREN MEZAVEL MEZAVEN BIHAN MEZAVERN MEZCOUEZEL MEZ DOUN NIOU MEZEDERN MEZELES MEZENROCH MEZEOZEN MEZER BRAS MEZERMIT MEZGLAZ MEZ GUEN LILIA MEZ GUILDEC MEZHIR MEZ HIR MEZHUEL MEZ HUEL MEZ HUELEN MEZ HUELLA MEZ HUELLEN MEZILE TY NEVEZ MEZIRVIN MEZIVINOU MEZ KEREVER MEZKWIREG MEZ LANN MEZLIDEC MEZ MEAN MEZ MEUR TAL AR GROAS MEZNEN MEZNOALET MEZONAN MEZORMEL MEZOU MEZOUARCLOZ MEZOU BAR HIR MEZOU BRAS MEZOU CRANOC MEZOU DOLVEN MEZOU GRANOC MEZOUGUEN MEZOU KERVIDRE MEZOU KERYEVEL MEZOU KERZEVEN MEZOU KOPARKOU MEZOU MANACH MEZOUMEUR MEZOU MILIN AVEL MEZOUT BIAN MEZOUT BRAS MEZOU VILIN MEZPLEUN KERGONAN ROSTUDEL MEZ TALEDEC MEZ VIHAN MEZ YAR MIGOURON MIKAEL MIL AR ROCHOU MILAUDREN MILBARON MILH AR PALUD MILIN AN AOD MILIN AN ESKOP MILIN AN OUARNER MILIN AN TRAON MILIN AR MILIN AR C HASTEL MILIN AR CRANKET MILIN AR ROCH MILIN AVEL MILIN GOUENNOU MILIN GOZ MILIN HUS MILIN KERLOC H MILINN AVEL QUATRE MILIN NEVEZ MILINN NEVEZ MILINOU MILIN RIOU MILIN ROCH MILIN TOARE MILIN TRAON KERVAN MILIN VARC H MILIN VOR MIL JANDRIG MILLE MOTTES MILLMEICK MINE MINE BRENELIO MINE DERO MINE GLAZ MINE GROAS NE MINE GUERVEUR MINE PORS AR MANER MINE RULAN MINE DAVID MINE MINE SEACH MINE TREOUZAL MINE YEUN MINEZ MINEZ BRENELIO MINEZ GWE FAO MINEZ KERGUEN MINEZ KERSCLIPPON MINEZ MEUR MINEZ POAN MINEZ MINEZ TREOUZAL MINGAM MINGAN HUELLA TRAON HIR MINGANT MINIHY MINIOC MINIOU MINIOU BRAS MINIOU ROUZ MINODIC MINOU MIN PEN AR ROZ MINTERNOC MINUELLO MINVEN MINVEN KEREST MIRINIT MI ROUTE MISECLUN MISELCUN MISSIOUARN MLINN NEVEZ MOABREN MOC GOUEZ KERABUS MOELLIEN MOENNEC MOGERAN IZELLA MOGUEL BRAS MOGUER MOGUERAN MOGUERAN CREAC AN AVEL MOGUERAN HUELLA MOGUERAN IZELLA MOGUERANT MOGUERGREAN MOGUER HALVEZ MOGUERIC MOGUERIEC MOGUERIEC AR MANER MOGUERIEC CHANEL MOGUERIEC KERAVAL MOGUERIEC POUL AR STREAT MOGUERIEC POULMORVAN MOGUERIGOU MOGUEROU MOITIE MOLETTE MONCOUAR MON DESIR MONDRAGON MONGARDI MONGARIOU MONIVEN MONIVEN IZELLA MON REPOS KERBREZILLIC MONTAGNE MONT NEVET MONTORET MONTORET VIHAN MONTOURGARD MONTSERRAT MORBIC MORBOULLOU MORDUC MORGANT MORGAT BEG AR GADOR MORGAT BOUIS MORGAT KERABARS MORGAT KERAVEL MORGAT KERBASGUEN MORGAT KERDROEN MORGAT KERELLOT TREFLEZ MORGAT KERGLINTIN MORGAT KERGOLEZEC MORGAT KERIGOU MORGAT PALUE MORGAT LOSTMARCH MORGAT PEN AR GUER MORGAT RHEUN DAOULIN MORGOAT MORGUIGNEN MOR GWEN 20 HENT BEG CROASSEN MORIOU MORIZUR MORVANNED COZ MORVANNED NEVEZ MORVE MOTTE MOUDENNOU MOUEZ AR MOR EGAREC MOUEZ AVEL COAT MELEN MOUGAU BIHAN MOUGAU BRAS MOUGOULOU AN TRAON PAPIER AU DUC VENT AVOINE BAILLEGUEN BAND BANT BARBARY BARON BEL AIR BENVERN BERLUHEC BERNAL BEUZIDOU BEUZIT BLANC BOIS ROCHE GARIN BOISSIERE BONDIVY BONIS BONNESCAT BOUROUGUEL BRENIZENEC BREZAL BRODEZAL BROTEL BRUNE BURUNOU CALLAC CARMAN CARN CARPONT CASCADEC CASTEL CHANEU CHANTRE CHATEL CHEF COADIC COATBLOHOU COAT FORN COATIGRACH COAT JUNVAL COAT LOSQUET COAT MERET COAT QUEAU COAT SAO COING COMMOU COMMUNAL CONAN COSQUER COTE GOALARN COZ CREACH CREACH GWEN CREACH PENVIDIC CREACH QUETTA CREMENEC CUZULLIEC DAMANY D ARGENT D ASCOET D AULNE BAND BERNAL BRONOLO CABORNES CASTELLIEN CASTELLIN CHEF CHOUENNER COAT CAER COAT CANTON COATSQUIRIOU DOULVEN FONTEYOU GLUJEAU GOASMOAL GUENGAT GUERVENNEC GUINIGOU KERANGOC KERANROCH KERBIRIOU KERDU KEREFFREN KERFRES KERGAERIC KERGONAN KERGONVAN KERGUELEN KERHUON KERLEVENE KERLOAS KERLOSTREC KERMARGUIN KERMEUR KERMORVAN KERNAULT KERNEVEZ KERNONGAL KEROLVEN KEROUALLON KERROCH KERSTRAD KERVELLA KERVERNIOU KERVOUSTER L ABBE BOISSIERE BOULAIE CHAUSSEE HAIE MER LESTREVET MOTHE MOTTE DELANGUIONAS LANHOULOU LANNIDY LANNIEC PALUE ROCHE TOUR LAUNAY VILLENEUVE LESCUZ LESVOYEN LEZAFF LEZERAZIEN L HERMITAGE LIVINOT LOSSULIEN LOSTEVIR D ENEZ D EN HAUT PENLANN PENQUELEN AR MARC HAT PORSMOGUER QUINQUIS QUISTINIC ROGNAN ROUDOUMEUR ROZVEUR ALOUARN CLOUD MOULIN PRES TREMADUR TRESSEOL TREVIEN TROAGUILLY TROANEZ TROYON TY VARLEN DEUFFIC DONAN DONANT DOURGUEN DREZEC BUIS BEG AEL DUC CAM CANAL CHANTRE CHAT CRANN CRECQ DUC FAOU FOLGOET GUERN HILGUY HILGUY TY CHASS JET JUCH LAY LAZ PENHOAT PLESSIX MOULIN PRE PRIEURE QUINQUIS ROI ROY ROZ ROZOS RUFFA SEQUER STALE STANG TEMPLE VARCH ENFER ETANG FOLGOET FONTAINE BLANCHE FOURDEN GARDES GARDEZ GARO GAUCHE GILLETTA GOASSELEN GOEL GOFF GOUENNOU GOUEZ GOUZIOU PALUD GUILER HAIE HASCOET HAUT HAUT HELOU HENRY HENVEZ HILBARS HILVERN JET JEUNE JOANET JOANNET KASTELL BOC H KERALLE KERAMBELLEC KERANFLEC H KERANGOC KERANGOMAR KERANY KERBEOC H KERBERN KERBIZIEN KERDREIN KERDUNIC KERELLEC KERELLOU KERENIEL KEREON KEREUZEN KERFEUTEUNIOU KERGAER KERGANAPE KERGARAOC KERGARAOC NORD KERGOAT KERGOFF KERGONADEACH KERGONIOU KERGONVAN KERGOULOUARN KERGUESTEN KERGUNUS KERGUS KERHALS KERHAO PENHARN KERHAS KERHINE KERIAR KERIBER KERIBERT KERINGARD KERIVARCH KERLEAU KERLEO KERLIGONAN KERLOCH KERNABAT KERNAVENO KERNO KEROCUN KEROMNES KEROUE KERRIEGARS KERROT KERROUANT KERSABIEC KERSANDY KERSCOULET KERSTRAD KERSTRAT KERVAO KERVASTAL KERVEN KERVIZIN KERZUOT KOZ KASTEL L ABBE LABBE LANNIEC LANNOGAT LANNOUEDIC LANORGARD LANSOLOT LANVERN LAUNAY LESCOAT LESCOP LESPOUL LIDIEN MANACH MARANT MARCHE MAREGUEZ MER MEUR MOALHIC MOELLIEN MOGUEL MOGUEROU MOTTE NARCH NEUF NEUF QUILLIEN NEUF MATHIEU NEZET NOHENNEC PAPIER PARLAVANT PELL PEN AR GUER PENARUN PEN AR VERN PEN ENEZ PENFROUT PENGUERN PENHOAT PENLAN PENMARC H PENNAYEUN PERAN PLOMB PLUFERN POAS PONCHOU MOULIN PONTALENNEC CARADEC COUALCH CROACH GARVAN HERE LEY MORVAN OURS QUEAU RHEUN PORS MILIN VALLY PORSMOGUER PORS MORO POUL BLEIS POULFANC PRE PROVOST PRY QUELENNEC QUENIC BEUZEC QUENICTUEC QUESTEL QUIDAN QUILLIEN QUILLIOU QUINIOU QUISTILLIC RABAT RAGOT RICHET RIVE RIVOALEN ROCHE ROGUAN RONVARCH ROSQUIJEAN ROSQUIJEAU ROSSIOU ROSTIC ROUAL ROUDOU GLAZ ROUDOUGOALEN ROUDOUS ROUX ROY ROZOS CLOUD CLOUE ELOI MOULIN NORGARD OUARNEAU SINAN SOUL SQUIRRIOU STANG STANG ZU SUGUENSOU MOULINS VERTS TALARN PENFELD TANNE TARIEC TERRE TOULGOAT TRAON TRAOUNEVEZEC TREANNA TREAS TREDIEC TREGALET TREGONGUEN TREGUIER TREOURON TREUSCOAT TREVAN TREVILIS TROGOFF TROMELIN TY MEN TY VARLEN VERGRAON VERN VERT VEYER VIEUX MOUSTANGUERN MOUSTARGOLL MOUST AR GUERN MOUSTER MOUSTERLANN MOUSTERLIN 13 METAIRIE MOUSTERLIN 17 GARN GLAZ MOUSTERLIN 19 LEN C HANKED MOUSTERLIN 27 KARN STER MOUSTERLIN 4 HENT KROAS KERNEING MOUSTERLIN 9 METAIRIE MOUSTERLIN HENT CLEUT ROUZ MOUSTERLIN KARN MENEZ KERBADER MOUSTEROU MOUSTOIR MOUSTOIR MENEC MOUSTOIR CADO MOUSTOIR VIHAN MOUSTOULGOAT MOUSTREN MOUSTRENN MUNUGUIC MUNUT MUR MUR HUELLA MURIOU MUR MESGOUEZ MUR NEUF NAMOUHIC NANCH NANCLIC NANGELOU NAONT NAOT HIR NAOUGOAD NAOUNT NAOUT NARC GWEN NARGOAT NARRET NARTOUS NATELLIOU NATOR NAVALENNOU NAVALHARS NAVALIC NAVAN NAVANT NAVARO KERASCOET NEC HOAT NEDER NEHELEN NEILLOU NEISCAOUEN NEIS VRAN NEIZ VRAN NELEHEN NELLACH NELOHAN NEMBLACH NENCHES NENEZ NENEZ NEVEZ NENEZ TREFLADUS NENGOAT NENVEZ NEOLENNOU NEONEC NERGOAT NERGOZ NERIZELEC NERODINE NESLARCH HUELLA NESLARCH IZELLA NESTAVEL BIAN NESTAVEL BRAS NESTOU NEUVE NEUZIOU NEVARS NEVARS BIHAN NEVARS HUELLA NEVARS IZELLA NEVEIC NEVEIT NEVEN NEVEZ CREACH NEZ AR C HANT NEZARD NEZERDY NEZERS NEZERT NIADEC QUILLOUARN NIGOLOU NILIZIC NILVIT NIOU NIOU HUELLA NIOU IZELLA NIQUELVEZ NIVEN NIVER NIVERN NIVERNIC NIVERROT NIVIEZ NIVINEN NIVIRIT BRAS NIZERDY NIZON CROISSANT ANDRE NIZON KERAMBEUX NIZON KERANGALL NIZON KERANGOFF NIZON KERCAUDAN KERGORNET NIZON KERFUR NIZON KERGANET NIZON KERGAZUEL NIZON KERGOADIC NIZON KERGOADIC VRAS NIZON KERHUIL NIZON KERLARET NIZON KERLARRET NIZON KERMANN NIZON KERMOAL NIZON KERNONEN NIZON LANDEDEO NIZON LUZUEN NIZON ANDRE NIZON TREMALO NIZON VILLENEUVE NODET NODEVEN NOGUELLOU NOHENNEC NOISERAIE NONAOU NONEN NONNAT HUELLA NONNAT IZELLA NONNOT NORMANDIE NORMANDY NORVENNIC NORVEZ NOTERIC NOTERIOU NOUARNEC NOUSTE POULFANC NOUVELLE COTE NOYELLOU NUOC ODE AR GLOUET ODEBLEIS ODE BLEIZ ODEGALET ODEGAR ODE LOUC H ODEVEN ODEVEUR ODE VEUR ODE VRAZ ORATOIRE OREE OROBANCHES OUENNEC PAGOTAIE PALAE PALAIS PALUD PALUD AN HIR PALUD BIAN PALUD BRAS PALUD CREMUNY PALUD CRUGOU PALUD CRUMUNI PALUD DOURDUFF PALUDEN PALUDEN BEL ABRI PALUD GORREBEUZEC PALUD GOURINET PALUD GRONDVAL PALUD KEREMMA PALUD KERGUELLEC PALUD KERGURUN PALUD KERISTENVET PALUD KERISTEVET PALUD PELLAN PALUD PENLAN PALUD TREBANEC PALUD TREBANNEC PALUE PALUE KERGUELLEC PALUE KERSUGAL PANT PAOTR TREOURE PAOU PAPEREZ PAPETERIE MAUDIT KERISOLE PAPIOLET PARADEN PARADEN TY POAS PARADIS PARALUCHEN PAR AR VEROURI PARC AN AGNIEL PARC AN ALE PARC AN AOD BRAZ KERLIOU PARC AN AOT PARC AN AUTROU PARC AN AVEL QUELVEZ PARC AN BROC PARC AN DOSSEN PARC AN HIVER PARC AN ILLIS PARC AN IVERN PARC AN NOAN PARC AN OACH PARC AN OGUEN PARC AN PICARD PARC AN TRAON PARC AN ZANT PARC AR BARENNOU PARC AR BARNOU PARC AR BARRES PARC AR BASTARD PARC AR BELLEC PARC AR BEYER MENEZ NOAS PARC AR BOURG PARC AR BREACH PARC AR BREUR PARC AR BROC PARC AR CHAEL PARC AR CHAPEL PARC AR C HOAT PARC AR CHOAT PARC AR FAVEN PARC AR FEUNTEUN PARC AR GLOAN PARC AR GO PARC AR GROAS PARC AR HAD PARC AR HEZEC PARC AR HOAT PARC AR JOL PARC AR LAND PARC AR LANDE PARC AR LANN PARC AR MEIN GUEN PARC AR MENEZ PARC AR MENGUEN PARC AR MENHIR PARC AR MILINER PARC AR PARQUER PARC AR PONTIC PARC AR PORS PARC AR PORS KERNEAVAL PARC AR PRAT PARC AR RHEUN TREGONDERN PARC AR ROUE PARC AR STANG PARC AR TRAON LEZABANNEC PARC AR VALIS PARC AR VALLY PARC AR VEIL PARC AR VEROURI PARC AR VUR PARC AR YAR PARC AU DUC PARC VEROURI PARC BALAN PARC BANALOU PARC BEAUSEJOUR PARC BIHAN PARC BLEUD PARC BRAS PARC COAT PARC COLIN PARC COSQUER PARC COZ PARC CREIS PARC D ARVOR PARC DAVID PARC EXPOSITIONS LANGOLVAS PARC FANQUIC PARC FEUNTEUN PARC FOEN PARC FOENNEC PARC FRIC PARC FROST PARCFUR PARC GALLO PARC GUEOT KERNAVENO PARC GWEN PARC HAUT PARC HIR PARC HOAT BEO PARC HUELLA PARC HUELTREC PARC ILIS PARC IZELLA PARC JAFFRE PARC KERHAPPS PARC KERINGARD PARC KERISTIN PARC KERIZIT PARC KEROHAN PARC KERVEGUEN PARCKIGOU PARC KROAS PARC LAER PARC LAGADEC PARC MAIS BLANCHE M LANINON PARC LAND PARC LANGOLEN PARC LANN PARC LANN ROHOU PARC LEC PARC LEONNEC PARC LEUR PARC LEVENEZ PARC LOCH GUEN PARC MARC PARC MARCHE PARC MARCHE COAT BEUZ PARC MARCHE SEACH PARC MARVAIL PARC MAURICE PARC MEIL TRESSEOL PARC MELINER PARC MENEZ PARC MENEZ KERSABY PARC MESNIGOALEN PARC MINE PARC MINOR KERDAENES PARC MOROS BIHAN PARC MORVAN PARC NEVEZ PARCOAM PARC OLLIVIER PARCOSQUER PARCOU PARCOU BIHAN PARCOU BRAS PARCOU BRAZ PARCOU BRIS PARCOU GORREKEAR PARCOU JACQ PARCOU LAND PARCOU LEUHAN PARCOU MENEZ PARCOU MENGLAZ PARCOU PENHOAT SALAUN PARCOU ROYAL PARCOU SEGAL PARCOUYARD PARC PALUD TY PALUD PARC PELL KEREVEN PARC POST PARC QUESTEN PARC QUISTINIC PARC ROCH PARC ROUTE PENMARCH PARC RULAN PARC SCOUDOU PARC SEVEAN PARC STANG PARC STER PARC STER CAP COZ PARC TAL AN ILIS BOURG PARC THOMAS PARC TOUL AR BLEIS PARC TREHERN KERLEAN PARC TRI HORN PARC TY LAI PARC VEIL PARC VENARCH PARC VEROURY PARC VIAN PARC YANN PARC YDER PARC ZALE PARGAMOU PARIAL PARICHARD PARIOU PARK LEUR PARK AL LEUR PARK AN AVEL PARK AN DOUR PARK AN DREZEG PARK AN HIVER PARK AN TIEZ PARK AR BELEG PARK AR BELEG LANVISCAR PARK AR C HIVIGER PARK AR C HOAT PARK AR GOZHKER PARK AR GROAZ PARK AR GWIADER PARK AR LEUR PARK AR MENHIR PARK AR ROZ KERHARO PARK AR RUN BOURG PARK AR STANKOU PARK AR STREAT BOURG PARK AVALOU PARK BALAN PARK BRAS PARK BRAZ PARK CLOS PARK L ISLE PARK FORN PARK GOUEL Y 28 HENT KERBORC H PARK GUEN KERSTRAT PARK HENT GLAZ PARK HUELLA PARK HUELLA KERMARIA PARKING INTERMARCHE KERCHAPALAIN PARK KERCO PARK KERGREAC H PARK KREIS AR VOURC H PARK KREIZ PARK LAE PARK LANIG PERES PARK LEUR PARK LEURIOU PARK MARGOD PARK MENN MEUR PARK MERVENT PARK MEUR PARK MONTOUARC H PARK N ALLE PARK NEVEZ PARKOU PARK PEN DUICK PARK ROUZIG PARK VRAS PARLUCHEN PARLU HEN PAROU KERMAVIOU PARVIS BLAISE PASCAL PASSAGE PASSAGE AUGUSTE BRIZEUX PASSAGE BOBBY SANDS PASSAGE PASSAGE CORENTIN CELTON PASSAGE CORNOUAILLES PASSAGE KERGUELEN PASSAGE LEVEE PASSAGE POSTE PASSAGE L ARMEN PASSAGE FEUX FOLLETS PASSAGE FRENES PASSAGE LAVANDIERES PASSAGE LUTINS PASSAGE ONDINES PASSAGE SIRENES PASSAGE DOMREMY PASSAGE DOURIC COZ PASSAGE FOUR PASSAGE HENAN PASSAGE BART PASSAGE KERGUELEN PASSAGE LUCIEN PICARD PASSAGE PRAD AR ZANT PASSAGE HERLE PASSAGE SIRENES PASSAGE SURCOUF PASTOURON PAVILLON 2 COSQUER PAVILLON ALBERT LARHER PAVILLON LOCH PAYAOU PAYS FRUIT PEDEL PELEUZ PELHET PELLAE PELLAIN PELLAN PELLAN BIHAN PELLAND PELLAVON PELLAY PELLEM PELLEN PELLEOC PEMBRAT PEMP HENT PEMPHENT PEMPIC PEMPOUL PEMPRAT PEN PENAHARS PEN HOAT PENAHOAT PENALAN PEN ALAN PEN LAN PEN LAND PENALAND MEROS PENALEIN PENALEN PEN LEN PEN AL LAN PENALLAN PEN ALLAN GARO PEN AL LAN KERIZUR PEN AL LANN PEN AL LAN VIAN PENALLEN PEN ALLEN 2 HENT MENEZ BRIS PENALLEURGUER PEN ALLEUR GUER PENALLEURGUEUR PENAMENEZ PENAMPRAT PEN AN PEN AN DAIL PEN AN DALLAR PENANDAOUENT PEN AN DELCH PEN AN DOUR PENANDOUR PENANDREFF PENANDREFF BIAN PENANEAC H PEN ANEACH PENANEACH TREANNA PENANECH PEN AN ERO PENANGUER PEN AN HEDER PEN AN ILIS KERVILLIC PEN AN NATIL PEN AN NEACH PEN AN NEAC H PEN AN NEC H PEN AN PRAT PENANREST PENANROS PENANROS PHILIBERT PENANROUS PENANROZ PEN AN ROZ BODIVIT PEN AN ROZ KERGOFF PENANRUN PEN AN TEVEN PEN AN THEVEN PEN AN TRAON PENANTRAON PENANVERN PEN AN VERN PENANVEUR PENANVOAS PEN AR PEN RAS PEN AR BAREZ PEN AR BARREZ PEN AR BED PEN AR BEZ PEN AR C HA PEN AR C HARS PEN AR C HASTEL PEN AR C HEAR PEN AR CHLEUZ PEN AR C HLOZ PEN AR C HLUN PEN AR C HOAS PEN AR C HOAT PEN AR CHOAT PENARC HOAT PEN AR C HOAT AR GARS PEN AR C HOAT AR GORE PEN AR C HOAT BIAN PEN AR CHRA PEN AR C HRA PEN AR C HRANN PEN AR CHRASSEN PEN AR C HREAC H PEN AR CLHEUS PEN AR COAT PEN AR CRASSEN PEN AR CREACH PEN AR CREAC H PENARCREACH PEN AR CREAC HENT KERGONOEN PEN AR CREAC LAMBER PEN AR CREAC VRAS PENARCREACH COATAUDON PEN AR CREARC H PEN AR CREC H PEN AR CROAS PEN AR CROISSANT KERSCO PEN AREUN PENAREUN PEN REUN PEN AR FAVOT PEN AR FEUNTEUN PEN AR FOREST PEN AR GARRONT PEN AR GOARIMIC PEN AR GORRET PEN AR GOUER PEN AR GROAS PEN AR GROAS FREDE PEN AR GROAS MAILLE PEN AR GUEAR PENARGUEAR PEN AR GUEL PEN AR GUEN PEN AR GUER PENARGUER PEN AR HADEN PEN AR HARDEN PEN AR HARS PEN AR HELF PEN AR HEUN PEN AR HISTILLY PEN AR HLEUZ PEN AR HOAD PEN AR HOAS PEN AR HOAT PEN AR HOAT BIAN PEN AR HOAT SALIOU PEN AR HOAZ PEN AR HOEL PENARLAN PEN AR LAND PEN AR LEN PEN AR LENN PEN AR LEORZOU PENARLIORZOU PEN AR LIORZOU PEN AR MEAS PEN AR MENEZ PENARMENEZ PEN AR MENEZ BESCOND PENAROS PENAROZ PENAROZ BOHAST PENAROZ KERANGUEVEN PENAROZ LEZOUALC H PEN AR PALUD PEN AR PAVE PEN AR PORS PEN AR POUL PEN AR POUL TREMET PEN AR POUNT PEN AR PRAD PEN AR PRAJOU PEN AR PRAT PENARPRAT PEN AR PUNS PEN AR QUINQUIS PEN AR RAZ PEN AR REUN PEN AR RHUN PEN AR ROCH PEN AR ROS PEN AR ROUDOUS PEN AR ROUZ PEN AR ROZ PENARROZ PEN AR ROZ VRAS PEN AR RUGEL PEN AR RUGUEL PEN AR RUN PEN AR STANG PENARSTANG PEN AR STEIR PEN AR STER PEN AR STREAT PEN AR STREJOU PEN AR STREJOU MICHEL PENARSUIL PENARUGUEL PEN AR VALANEC PEN AR VALLY PEN AR VALY PENARVALY PEN AR VERN PENARVERN PEN AR VEUR PEN AR VILARS PEN AR VOAS PEN AR VOAZ PEN AR VOEZ PEN AR VOGAR PEN AR VOUEZ PEN AR VOUILLEN PEN AR VUR PEN AR YEUN PEN AR YUN PEN AVEL PENAVERN PEN VERN PENAVERN TURLUER PEN 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PENORS PEN OUEZ PEN PARCOU PENPRADOU PENPRAT PEN PRAT PEN PRAT VIAN PENQUEAR PENQUELEN PENQUELEN IZELLA PENQUELENNEC PENQUELLENEC PENQUELLEN IZELLA PENQUELLENNEC PENQUER PENQUER AR BLOAS PENQUER AR BLOAS VIHAN PENQUER BERGOT PENQUER BIAN PENQUER BIHAN PENQUER BRAS PENQUER COAT GALLOU PENQUER COAT SQUIRIOU PENQUER DIFFON PENQUER HUELLA PENQUER IZELLA PENQUER KERDAENES PENQUER KERDUDOU PENQUER KEREVEN PENQUER LESCLOEDEN PENQUER LOPREDEN PENQUER PENFOUL PENQUER DIDY PENQUER TRAON ELORN PENRUGARO PENSCOURIC PENTEVEN PENTRAON PEN TRAON GUER PENTREFF PENTREFF BIHAN PENTREZ KORRIGANS PENTY 9 RUMORVAN PENTY HOR PENTY KERGALL PENTY KERNEING PENTYS POULDOHAN PENVELED UHELA PENVELEGUEN PEN VENEZ PENVENEZ PENVERN PENVERN BIHAN PENVERNIC PENVINY PENYEUN PEN YUN PENZE PENZE DOSSEN PENZE GREYEN PENZE GUERZIC PENZE KERMOAL PENZE KERREUNAN PENZE MESCURUNEC PENZER PENZES PENZORN PEPINIERE PEPINIERE COAT MEUR PERAN PEREN PERENES PERENNES PERENNOU PERENOU PERENTE PERHARIDY PERHEREL PERINKIN PERN PERN COZ PERN NEVEZ PEROHEN PEROS LILIA PEROUNEL PERRIEN BONIS PERROHAN PERROS PERROS LILIA PERROS POULLOUGUEN PERROS TREBERON PERROS VIAN PERROZ PERROZ LILIA ANTOINE PERSISQUIN PERSIVIEN PERSUEL PERTY PERUNIOU PERZEL PESMARCH BOURG BRIGNEAU CLOITRE CYPRES BOISSIERE COUDRAIE FORET GARENNE GRENOUILLERE ILE LANDE METAIRIE MOTTE PETITE ROCHE PETITES SALLES VALLEE GARLOUET KERMEC LAUNAY LETY LICHERN LIVIDIC MANOIR MANOIR KERLEVEN ST LAURENT MINOU PETIT PARIOU PARIS PENHARS PERRIN QUENNEC LUCAS VERGER PEULLIOU PEULVEN PEUPLERAIE PEURIT PHARE PICARD PICARDIE PICART PIC BRAN PICHERI KOZ PICHERI NEVEZ PIFIDIC PIGEON BLANC PILLAC PINCHON PINET PINETTE PISCAVALOC PISCAVELOC PITIVIN PLACAMEN 14 JUILLET 18 JUIN 19 MARS 1962 19 MARS 3 AOUT 44 3 SEPTEMBRE 1944 3 SEPTEMBRE 44 ABBAYE ABBE BROCH ABBE FEUTREN ABER AJONCS D OR ALAIN FOURNIER ALAIN POHER ALBERT 1 ER ALBERT SCHWEITZER ALEXANDRE MASSE ALLENDE ALPHONSE DAUDET ALSACE AMEDEE FREZIER AMIRAL RONARC H AMPERE ANATOLE BRAZ ANCIEN LYCEE ANCIENNE FONDERIE 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RESISTANCE SALLE OMNISPORTS SOLIDARITE TOURBIE TOUR D AUVERGNE VICTOIRE L L HOUARDON L ENFER L EUROPE L EVECHE L HOTEL L ILE AUX MOUTONS L ILE PENFRET LOCRONAN LOURDES METZ MEZDOUN PENHELEN PENVILLERS QUILBIGNON ROSCANVEL AGAPANTHES ALIZES AMANDIERS AMERICAINS ANCIENS HARAS BOUVREUILS BRUYERES CARMES CHENES COLOMBES CYPRES DEPORTES DROITS L HOMME ECOLES ERABLES FFI FRENES FRERES CAVAREC FTPF FUSILLES HETRES JACOBINS MIMOSAS NOISETIERS ORMES OTAGES PECHEURS PHARES ROSIERS ROUGETS SALAGANES SANGLIERS TONNELIERS STRASBOURG TREBEHORET VERDUN VIARMES D IROISE DIXMUDE CAMUS CARAES LOUDOUX ODEYE 118 EME REGIMENT INFANTERIE 11 NOVEMBRE 18 JUIN 1940 19 EME REGIMENT INFANTERIE 19 MARS 1962 8 MAI 1945 BEL AIR BOT BOULOUARD BOURG CALVAIRE CAPITAINE GUYNEMER CHAMP BATAILLE CHAMP FOIRE PLACE DUCHESSE ANNE COMBAT TRENTE COMMERCE COSMOS DOSSEN DOURIC COZ FROMVEUR GENERAL GENERAL GUERDY DUGUESCLIN LYCEE BRIS MARCHALLAC H MARCHE PARVIS PERE MAUNOIR PLACE GUERN POULIET DUQUESNE ROI ARTHUR ROUZ STIVEL VEROURY VILLERS D YS PLACE EDOUARD VAILLANT PLACE NIZON PEN VERED EMILE GUERN EMILE SOUVESTRE EMILE ZOLA ERIC TABARLY ESTIENNE D ORVES EUGENE FOREST EUROPE FAUVETTES FLEMING FLEMMING F SUD FOIRES FORGE FOURNIL FRANCE LIBRE SUD MORVAN SCORNET VILLON FRATERNITE FRERES BOULCH FRERES POUPON FUSILLES RESISTANTS GALLIENI GAMBETTA GARO GAUGUIN GENERAL GENERAL BOLLARDIERE GENERAL GENERAL FLO BRASSENS GLENMOR GLYCINES GOELANDS GOLVES GONIDEC GOUEROU GRANDS CHENES GUERIN GUTHIERN GUY ROPARTZ HALLES HENRI GESTIN HERVO HETRES HOLBETON HOTEL ILES ISIDORE ROUDAUT ISOLE JACK LONDON JACQUES BREL JACQUES CARTIER BART BERRI BRIAND BRIANT COSQUER FOURNIER GONIDEC GOUILL JAURES KERAVEC MAILLOUX LOUIS BERTHELEME PLACE JEANNE JUGAN SCOUR PRIGENT SIMON JIM SEVELLEC JONQUILLES JOSEPH GOEZ JO TANGUY JULES FERRY JULES VERNE KAEREL KARIT KEN BOND KERANDERO KERASTIVEL KERDREL KERGOFF KERGROAS KERGUELEN KERLUDU KERUSCUN KREISKER KREIZ KER LACAZE DUTHIERS LAENNEC FONTAINE LAVOIR GALL L LEON BLUM LIBERATION LIBERTE LLANDEILO LOCHRIST LOUIS BUREL LOUIS SEVERE LOVIGNON LYAUTEY MARCEL STEPHANT MARCHE MARE MARECHAL D LATTRE TASSIGNY MARECHAL FOCH MARECHAL CURIE MARRONNIERS MARTRAY MASSENA MAURICE GILLET MEDARD MELLAC MENDES FRANCE MENEYER MESGLOAGUEN MEUR MICHEL COLOMB MICHEL LAREUR MIMOSAS MONSEIGNEUR FREPPEL MONSEIGNEUR TESTAR COSQUER MONTS D ARREES MOREAU 4 HENT AR VEIL MOUETTES PLACENAN PLACENANT NAPOLEON III APPERT NORMANDIE OCEAN ODET ONTARIO OTAGES PAIX PARC AN TRAON PARK PASCAL GAUGUIN FLEM SERUSIER STEPHAN PENTHIEVRE PENVERN PER JAKEZ HELIAS PERROS AUGUSTE RENOIR BARRE RONSARD LOTI PLASENN AR BRUN PORS CLOS PORT PORZ GUEREC POSTE POUL AR STANG PRAT AR C HREN PUITS QUARTIER MAITRE LANNUZEL QUATRE POMPES RACINE RENARD RENE CASSIN REPUBLIQUE RESISTANCE RESISTANTS RIMBAUD ROBERT GOFF ROCHEBRUNE ROHAN ROSIERS ROSVILY ALOUARN ANTOINE CORENTIN DIDIER SAINTE BARBE SAINTE HAUDE EUTROPE FIACRE GUENOLE HERBOT HUBERT PLACE JOSEPH LAURENT LOUIS MARTIN MATHIEU MELAR MICHEL MONNA PLACE REMY TANGUY THOMAS 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ROSJOURDEN ROSJOURDREN ROSLAN ROSLAND ROSLUDU ROSMADEC ROSMAGUER ROSMELLEC ROSMELLIC ROSMENGLAS ROSMEUR ROSMORDUC ROSNEVINEN ROSNIVINEN ROSNOEN ROSNONEN ROSONOUAL ROSPEILLET ROSPER ROSPERNES ROSPICO ROSPICO KERANGLAS ROSPICO KERASCOET ROSPICO KERDAVID ROSPIRIOU ROSPLEN ROSQUELEN ROSQUERNO ROSQUEROU ROSQUERVEL ROSQUIJEAN ROSQUIJEAU ROSQUILIOROC ROSQUILL ROSQUILLEC ROSQUILLIOROC ROSSERMEUR ROSSOLIC ROSTAND ROSTEGOFF ROSTELLEC ROSTERNIC ROSTEVEN ROSTIEGUEN ROSTIVIEC ROSTOMIC VRAS ROSTRENNEC ROSTUDEL ROSTURIEN ROSVALLEN ROSVEGUEN ROSVEIGN ROSVERN ROSVEUR ROS VIHAN ROSVOAL ROSVOALIC ROTUDAL ROUAL ROUAL BIAN ROUAL HUELLA ROUALLOU ROUALOU ROUANTELEZ ROUAOU ROUAS ROUDAOU ROUDERRIEN ROUD GUEN ROUDOU ROUDOUAN ROUDOUARN ROUDOUDERC H ROUDOUDERC CADOU ROUDOUFILY ROUDOU FLOCH ROUDOUFRANC ROUDOUGOALEN ROUDOUGUENVEZ ROUDOUHIR ROUDOUIC ROUDOUIC VIHAN ROUDOUIG ROUDOUIG VIHAN ROUDOULEVRY ROUDOUMEUR ROUDOUR ROUDOURIC ROUDOUROU ROUDOUROUS ROUDOUROUX ROUDOUS ROUDOUS AN AOUR ROUDOUS HIR ROUDOUSHIR ROUDOUS MEAN ROUDOUZIC ROUE ROUENNOU ROUGE ROUISTIN ROUMOULOUARN ROUSENCOAT ROUSPOULOUDIC ROUSSEAU ROUSSICA ROUTE ABER ROUTE ABER ILDUT ROUTE ABER KERBRIANT ROUTE ANCIENNE ROUTE ANGES ROUTE ANSE BOURG ROUTE ANSE BOURG ROUTE ANSE ROZ ROUTE AN TRI DI ROUTE ARBRE CHAPON ROUTE ARGENTON ROUTE ARGOAT ROUTE ARGOAT PRAT AN DOUR ROUTE ARMOR ROUTE ARMORIQUE ROUTE AUDIERNE ROUTE AUDIERNE MENEZ KERVIGNAC ROUTE BAIE ROUTE BALAENNOU ROUTE BANINE ROUTE BANNALEC ROUTE BARNAO ROUTE BEDIEZ ROUTE BEG AN TI GARD ROUTE BEG AR GROAS ROUTE BEL AIR ROUTE BELLEVUE ROUTE BENDY ROUTE BENODET ROUTE BERRIEN ROUTE BERTHEAUME ROUTE BEUZIT ROUTE BINDY ROUTE BIZERNIC ROUTE BOCABEUR ROUTE BODENAN ROUTE NEVET ROUTE NEVET ROUTE BOISSIERE ROUTE BONNE NOUVELLE ROUTE BOTAVAL ROUTE BOTCABEUR ROUTE BOTLAN ROUTE BOTSORHEL ROUTE BOUDOUREC ROUTE BOUIS ROUTE BOURG ROUTE BRASPART BOURG ROUTE BRASPARTS ROUTE BRELES ROUTE BRENDAOUEZ ROUTE BRENDUFF ROUTE BRENILOUR ROUTE BRENNANVEC ROUTE BRENNILIS ROUTE BREST ROUTE BREST ARGENTON ROUTE BRIEC ROUTE CADOL ROUTE CALE ROUTE CALE ARGENTON ROUTE CALLAC ROUTE CALVAIRE ROUTE CAMASQUEL 37 CROEZOU ROUTE CAMPING ROUTE CANFROUT ROUTE CANVEZ ROUTE CAOUET ROUTE CAOUT ROUTE CAP CHEVRE ROUTE CARHAIX ROUTE CARN ROUTE CARRIERES ROUTE CARRIERES GUERNILIS ROUTE CAST ROUTE CASTELLIC ROUTE CENTRALE ROUTE NAUTIQUE ROUTE CHANOINES ROUTE CHAPELLE ROUTE CHAPELLE PENHORS ROUTE CHAPELLE GRACES ROUTE ROUTE D EAU ROUTE CHATEAULIN ROUTE CHATEAUNEUF ROUTE CHATEAUNEUF FAOU ROUTE VERT ROUTE CHP ST TREMEUR KERLEGUER ROUTE ROUTE CIRCUIT TREGOR ROUTE CLARTE ROUTE CLEDER ROUTE CLEGUER ROUTE CLEGUERIOU ROUTE CLEUSOER ROUTE CLOAREC ROUTE CLOHARS ROUTE CLOHARS FOUESNANT ROUTE CLOITRE ROUTE CLOITRE PLEYBEN ROUTE COAT ROUTE COATELLEN ROUTE COATEVEZ ROUTE COATILIOU ROUTE COATIVY ROUTE COAT LIVROACH ROUTE COLLOREC ROUTE COLLOREC KERSEACH ROUTE COLONEL FONFERRIER ROUTE COLONIE ROUTE CONCARNEAU ROUTE CORAY ROUTE CORNICHE ROUTE CORNICHE TREZ HIR ROUTE CORN LAN AR BLEIS ROUTE CORPS GARDES ROUTE COSQUER ROUTE COSQUER 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GUISSENY ROUTE KERABIVIN ROUTE KERALLIOU ROUTE KERAMEIL ROUTE KERANDRAON ROUTE KERANNA YVES ROUTE KERASCOET 3 ROUTE KERAUDY ROUTE KERBALANNEC ROUTE KERBENEAT ROUTE KERBULIC ROUTE KERDALEC ROUTE KERDAOULAS ROUTE KERDREN ROUTE KERERAULT ROUTE KEREZOUN ROUTE KERFANY ROUTE KERGADONNA ROUTE KERGANTEN ROUTE KERGLIB ROUTE KERGOFF ROUTE KERGREACH ROUTE KERGREVEN ROUTE KERGUELEN ROUTE KERGUESTEC ROUTE KERGUINOS ROUTE KERGULAN ROUTE KERHALLON ROUTE KERHOULO ROUTE KERHUON ROUTE KERIARS ROUTE KERIEL ROUTE KERIGOU ROUTE KERINEC ROUTE KERIORET ROUTE KERJEAN ROUTE KERLARAN ROUTE KERLAVIC ROUTE KERLEDAN ROUTE KERMARIA ROUTE KERMEN PARCOU BRAS ROUTE KERMOEC ROUTE KERMOOR ROUTE KERNEVENT ROUTE KERNEVEZ HUELLA ROUTE KERNIGUEZ ROUTE KEROGAN ROUTE KEROUEZEC ROUTE KEROUGAR ROUTE KEROUMEN ROUTE KEROURVOIS ROUTE KERSAINT ROUTE KERSCAO ROUTE KERSCAVEN ROUTE KERSOLF ROUTE KERSTRAT ROUTE KERSULEC ROUTE KERVALLAN ROUTE KERVEDAL ROUTE KERVEOGAN ROUTE KERVERN ROUTE KERVEZINGAR HUELLA ROUTE KERVEZINGAR IZELLA ROUTE KERVICHEN ROUTE KERVILLERM ROUTE KERVOULIDIC ROUTE KERVRAHU ROUTE KILIOU ROUTE KROAS NEVEZ ROUTE L ABER ROUTE CALE ROUTE CLARTE ROUTE CORNICHE ROUTE CORNICHE BRULY ROUTE CROIX ROUTE CROIX ROSIER ROUTE CROIX ROUGE ROUTE FONTAINE ROUTE FORET ROUTE ROUTE BODIZEL ROUTE GREVE ROUTE GREVE BLANCHE ROUTE HAUTE CORNICHE ROUTE LAITERIE ROUTE DELAMARE DEBOUTTEVILLE ROUTE MARTYRE ROUTE LAMBELL ROUTE MER ROUTE LANDERNEAU ROUTE LANGOUGOU ROUTE LANNUGAT ROUTE LANORVEN ROUTE LANVEN ROUTE LANVEN BEUZEC CAP CAVAL ROUTE LANVENEEN TY GROEZ ROUTE LANVEOC ROUTE PLAGE ROUTE POINTE MOUSTERLIN ROUTE POINTE TREVIGNON ROUTE L ARBRE CHAPON ROUTE LARC HANTEL ROUTE L ARMORIQUE ROUTE SALLE VERTE ROUTE TORCHE CROAS AN GLOANEC ROUTE TORCHE POULCOQ ROUTE LAVALLOT ROUTE VILLENEUVE ROUTE L ECHANGEUR ROUTE L ELORN ROUTE LESIVY ROUTE LESQUIVIT ROUTE LEUR AR MARC VIHAN ROUTE LEUR AR MARCHE ROUTE LEZEBEL ROUTE L ILE CHEVALIER ROUTE L ILE TUDY ROUTE DELLEDAN ROUTE D ELLIANT ROUTE LOCAL ROZ ROUTE LOCH WENN ROUTE LOCRONAN ROUTE LOCTUDY ROUTE LOPERHET ROUTE MAEL CARHAIX ROUTE MEIL DREAU ROUTE MELGVEN ROUTE MENEZ BRAS ROUTE MENEZ PEULVEN ROUTE MENEZ ROHOU ROUTE MENEZ ROUZ ROUTE MERRIEN ROUTE MESCLEUZIOU ROUTE MESLEIOU ROUTE MILIZAC ROUTE MOELAN KEREOL ROUTE MORLAIX ROUTE MOUSTERLIN KERGUIVER ROUTE D ENCREMER ROUTE NEVEZ ROUTE DENEZ ROUTE DEOLEN ROUTE PARIS ROUTE PEN AN TOUL ROUTE PEN AN TRAON ROUTE PENCRAN ROUTE PENDRUC ROUTE PENFELD ROUTE PENFRAT ROUTE PENGUELEN ROUTE PENHARS ROUTE PENHEP ROUTE PENHUEL ROUTE PENMARCH ROUTE PENVERN ROUTE PLASCAER ROUTE PLEVIN ROUTE PLEYBEN ROUTE PLOBANNALEC ROUTE PLOGONNEC ROUTE PLOMELIN ROUTE PLOMEUR ROUTE PLONEOUR ROUTE PLONEVEZ ROUTE PLONIVEL ROUTE PLOUARZEL ROUTE PLOUMOGUER ROUTE PLOUNEVEZEL ROUTE PLOUVORN ROUTE PLOUZANE ROUTE PLUGUFFAN ROUTE AVEN ROUTE CABIOCH ROUTE L ABBE ROUTE PONTOULLEC ROUTE POL ROUTE QUEAU ROUTE PORAON ROUTE PORS CARN ROUTE PORS CARO ROUTE PORSCREGUEN ROUTE PORZ AR ROUTE PORZ AR VIR KERURUS ROUTE PORZAVIR KERURUS ROUTE PORZ KERALLIOU ROUTE PORZ NEVEZ ROUTE POSTOLONNEC ROUTE POULDOUR ROUTE POULGUEN ROUTE PRAD STER ROUTE PRAT BIHAN ROUTE QUEFFEN ROUTE QUELEN ROUTE QUELOURN ROUTE QUILIMAR ROUTE QUILINEN ROUTE QUILLIMERIEN ROUTE QUIMPER ROUTE QUIMPERLE NIVINEN ROUTE RIEC ROUTE ROCH KILIOU ROUTE ROCH NIVELEN ROUTE ROSPORDEN ROUTE ROSTRENEN ROUTE ROUDOUALEC KERVOAZEC ROUTE ALBIN ROUTE ANTOINE ROUTE CAVA ROUTE SAINTE ANNE ROUTE SAINTE ANNE PORTZIC ROUTE SAINTE CHRISTINE ROUTE SAINTE SEVE ROUTE GERMAIN ROUTE JACQUES ROUTE TROLIMON ROUTE PHILIBERT HUELLOU ROUTE POL ROUTE RENAN ROUTE SEBASTIEN ROUTE THONAN ROUTE URBAIN ROUTE ANGES ROUTE SAVARDIRY ROUTE SCAER ROUTE CHATEAUX ROUTE DUNES ROUTE MANOIRS ROUTE STREAT ZOUM ROUTE TARAIGNON ROUTE TOULBROCH ROUTE TRAON BIAN ROUTE TRAOUIDAN ROUTE TREBRIVAN ROUTE TREFFIAGAT ROUTE TREGUNC ROUTE TREHUBERT KERSIDAN ROUTE TREMAIDIC ROUTE TREMEOC ROUTE TRENEN ROUTE TREVIGNON TREGONAL ROUTE TREVISQUIN ROUTE TREYERES ROUTE TREZULIEN ROUTE TROLIGUER ROUTE TRONOEN ROUTE TY FORN ROUTE TY NAY ROUTE TY RHU ROUTE VALY LEDAN ROUTE VOURCH VIAN ROUTE D HANVEC ROUTE DIBENNOU ROUTE DIEVET ROUTE DIRAK AN INIZI ROUTE DIRINON ROUTE D IRVILLAC ROUTE DOELAN ROUTE DOELAN TY FORN ROUTE DOENNA ROUTE DORLAN ROUTE DOSSEN ROUTE DOUARNENEZ ROUTE DOUR BIHAN ROUTE DOURDUFF ROUTE DOURMEUR ROUTE BAS RIVIERE ROUTE BELON ROUTE PINS ROUTE BOLHOAT ROUTE BOUGUEN ROUTE BRIEUX ROUTE CAMPING ROUTE CAP CHEVRE ROUTE CAP HUELLA ROUTE CAP IZELLA ROUTE CLEUYOU ROUTE COADIC ROUTE CONQUET ROUTE CORNIOU ROUTE DELLEC ROUTE DOURDUFF ROUTE FAOU ROUTE FAOUET ROUTE FRET ROUTE GONIO ROUTE GUELEN ROUTE LARGE ROUTE GUILLY ROUTE GUILVINEC ROUTE HUELGOAT BOURG ROUTE JUCH ROUTE LAEZRON ROUTE LENDU ROUTE LOCH ROUTE MENEZ ROUTE MINOU ROUTE CHAT ROUTE LECK ROUTE ROUTE DUNES ROUTE NIVERN ROUTE PASSAGE ROUTE PENQUER ROUTE PERGUET ROUTE GUELEN ROUTE MINOU ROUTE ROUTE BODIVY ROUTE PORT ROUTE POULDU ROUTE QUEFFLEUTH ROUTE QUINQUIS LECK ROUTE RAKER ROUTE RIS ROUTE RIS D YS ROUTE ROCHER CROIX ROUTE ROCHER L IMPERATRICE ROUTE STANGALA ROUTE STER 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KERBIGOT ROUTE KERBOMNES ROUTE KERBOULIC ROUTE KERBRANN ROUTE KERBRAT ROUTE KERBRIANT ROUTE KERBRIS ROUTE KERBULIC ROUTE KERBUNCOU ROUTE KERCHAPALAIN ROUTE KERC HEN ROUTE KERDANIEL ROUTE KERDANIOU ROUTE KERDENIEL ROUTE KERDEO ROUTE KERDIGAZUL ROUTE KERDOSTIN ROUTE KERDRES ROUTE KERDRUC ROUTE KERENOT ROUTE KEREUNEUT UHELLA ROUTE KEREVER ROUTE KERFEUNTEUN ROUTE KERFEURGUEL ROUTE KERFILY ROUTE KERFLOUZ ROUTE KERGABET ROUTE KERGADEL ROUTE KERGADIOU ROUTE KERGANAVAL ROUTE KERGARADEC ROUTE KERGARANTEZ ROUTE KERGARIOU ROUTE KERGOADIC ROUTE KERGOAT ROUTE KERGONDA ROUTE KERGONIOU ROUTE KERGONNIOU KERIZEL ROUTE KERGORS LESTREVET ROUTE KERGOUTOIS GOARIVA ROUTE KER GOZ ROUTE KERGUELL ROUTE KERGUERAL ROUTE KERGUIDAL ROUTE KERHALL ROUTE KERHAMON ROUTE KERHENOR ROUTE KERHORNOU ROUTE KERHUEL ROUTE KERIAZO ROUTE KERIDER ROUTE KERIFORN ROUTE KERINCUFF ROUTE KERINEC ROUTE KERINER ROUTE KERIVES ROUTE KERIVOAL ROUTE KERIZELLA ROUTE KERJEAN ROUTE KERJEGU ROUTE KERJEZEQUEL ROUTE KERLANN ROUTE KERLANOU ROUTE KERLAUDY ROUTE KERLEAO STANG VENELLE ROUTE KERLECUN ROUTE KERLEGUER ROUTE KERLEO VIHAN ROUTE KERLEVEN ROUTE KERLIVER ROUTE KERLOC H ROUTE KERLOM ROUTE KERLOSQUET ROUTE KERLOU ROUTE KERLOUAN ROUTE KERMADORET ROUTE KERMARIA ROUTE KERMARZIN ROUTE KERMAZEGAN ROUTE KERMEBEL ROUTE KERMENGUY ROUTE KERMENHIR ROUTE KERMERRIEN ROUTE KERMEUR ROUTE KERMEUR BRAS KERVIGODES ROUTE KERMINE ROUTE KERMORVAN ROUTE KERMORVAN BLANCS SABLONS ROUTE KERNAOUR ROUTE KERNEGUES ROUTE KERNEOST ROUTE KERNEVEZ ROUTE KERNEYET ROUTE KERNILIS ROUTE KERNIOC ROUTE KERNIOUAL ROUTE KERNIVINEN ROUTE KERNIZAN ROUTE KERNIZAN KERBERNES ROUTE KERNON ROUTE KERODORET ROUTE KEROHANT ROUTE KEROLLAND ROUTE KEROUANNEN TREZ HIR ROUTE KEROUGUEN ROUTE KEROUMAN ROUTE KEROUTER ROUTE KERPHILLIPOT ROUTE KERRETEUR ROUTE KERRIEN ROUTE KERRIOU ROUTE KERROM ROUTE KERSCOFF ROUTE KERSCOULET ROUTE KERSEHEN ROUTE KERSENVAL ROUTE KERSERVEN ROUTE KERSIVIANT ROUTE KERSONIS ROUTE KERSPRIGENT ROUTE KERSTEPHAN ROUTE KERSTRAD ROUTE KERTANGUY TALAOURON ROUTE KERTHOMAS ROUTE KERUGUEN ROUTE KERURUS ROUTE KERUZOU ROUTE KERVAEZ ROUTE KERVANQUEN ROUTE KERVAZAEN ROUTE KERVEAN ROUTE KERVEDAL ROUTE KERVEGUEN ROUTE KERVELLIC ROUTE KERVENNEC ROUTE KERVENNI ROUTE KERVENNOU ROUTE KERVENNY ROUTE KERVEOC H ROUTE KERVEOCH ROUTE KERVERRY ROUTE KERVESCONTOU ROUTE KERVIAN ROUTE KERVIC ROUTE KERVIGEN ROUTE KERVIGORN ROUTE KERVILIGEN ROUTE KERVILLO ROUTE KERVILLOU ROUTE KERVILON ROUTE KERVINIC ROUTE KERVIZIEN ROUTE KERVIZIN ROUTE KERVOAL ROUTE KERVOAZEC ROUTE KERVOERET ROUTE KERVOEZEC ROUTE KERYANN ROUTE KERZEHEN ROUTE KERZERVEN ROUTE KERZEST ROUTE LABABAN ROUTE LABER ROUTE LACROIX ROUTE FEUILLEE ROUTE FEUILLEE BOURG ROUTE FEUILLEE KERHORNOU ROUTE HAUT ROUTE MARTYRE ROUTE LAMBERVEZ ROUTE LAMPAUL ROUTE LAMPAUL GUIMILIAU ROUTE LAMPAUL GUIMILIAU KERMAT ROUTE LAMZOZ ROUTE LAN ROUTE LANARVILY ROUTE LANDERNEAU ROUTE LANDEVENNEC ROUTE LANDIVISIAU ROUTE LAND KERGUIPP ROUTE LANDOUZEN ROUTE LANDUDAL ROUTE LANGOLEN ROUTE LANGONERY ROUTE LANGURU ROUTE LANHOULOU ROUTE LANHURON ROUTE LANLEYA ROUTE LANMEUR ROUTE LANNEDERN ROUTE LANNEGUIC ROUTE LANNIEN ROUTE LANNILIS ROUTE LANNILIS KERRENVEL ROUTE LANNOUENNEC ROUTE LANNOURZEL IZELLA ROUTE LANNOURZEL UHELLA ROUTE LANN ROSTED ROUTE LANRIVOARE ROUTE LANVENAEL BEUZEC CAP CAVAL ROUTE LANVOY ROUTE LAOUAL ROUTE LARAON ROUTE ROCHE MAURICE ROUTE LARRET ROUTE LARVOR ROUTE LAUBERLAC H ROUTE LAUNAY ROUTE LAUNAY VERN ROUTE LAVOIR ROUTE FOGOT ROUTE LEGUERRIOU ROUTE LEINEURET IZELLA ROUTE LEINGUERN ROUTE LEON ROUTE LESCOAT ROUTE LESCOGAN ROUTE LESCONIL ROUTE LESCUZ ROUTE LESMENGUY ROUTE LESMINILY ROUTE LESNEVEN ROUTE LESQUIDIC ROUTE LESTREVET ROUTE LESTREVET MENEZ NOREN ROUTE LESVEZ ROUTE LETTY ROUTE LEUR AR HARDIS ROUTE LEUR GOZ ROUTE LEURVEAN ROUTE LEZ ROUTE LEZAOUVREGUEN ROUTE LIGEN ROUTE LINIOU ROUTE LIVIDIC ROUTE LOCAL AMAND ROUTE LOCH AR BIG ROUTE LOCH AR VATEN ROUTE LOCHRIST ROUTE LOCQUILLEC ROUTE LOCQUIREC ROUTE LOCRONAN ROUTE LOCRONAN TY PLANCHE ROUTE LOCRONAN VORCH HUELLA ROUTE LOCTUDY ROUTE LODONNEC ROUTE LODONNEC EZER ROUTE LOGONNA ROUTE LOGONNA DAOULAS ROUTE LOGONNA KERSANTON ROUTE LOPEREC ROUTE LOPERHET ROUTE LOQUEFFRET ROUTE LORIENT ROUTE LOSMARC H ROUTE MADELEINE ROUTE MAEL CARHAIX ROUTE MALANDIN ROUTE MANOIR BEULIEC ROUTE MANOIR GOANDOUR ROUTE MARCHE ROUTE MARTYRE ROUTE MEAN ROUTE MEAN KADOR ROUTE MECHOU ROUTE MECHOU BALAENNOU ROUTE MEIL DREAU ROUTE MELON ROUTE MENDY ROUTE MENEZ ROUTE MENEZ AR VEL CERTAU ROUTE MENEZ HAIE ROUTE MENEZ MEUR ROUTE MENEZ ROHOU ROUTE MENGLEUS ROUTE MENHIR ROUTE MER ROUTE MESCADOROC ROUTE MESCREN ROUTE MESLAN ROUTE MESLAND ROUTE MESPAUL ROUTE MESSELLOU ROUTE MESTREZEC ROUTE MEZANCOU ROUTE MEZANGOU ROUTE MEZ LAN ROUTE MEZ PRIZ ROUTE MILIEU ROUTE MILINOU ROUTE MILIZAC ROUTE MOELAN ROUTE MONTAGNE ROUTE MONT MICHEL ROUTE MONTS D ARREE ROUTE MORLAIX ROUTE ROUTE KERNABAT ROUTE CHANTRE ROUTE ROUTE KERAVEN ROUTE MER ROUTE MEUR ROUTE NEUF ROUTE PROVOST ROUTE MOULINS KERMARIA ROUTE TREVIEN ROUTE VERT ROUTE MOUSTER ROUTE MOUSTERCOAT ROUTE MOUSTERLIN ROUTE 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PLAGE KERIZELLA GUERN ROUTE PLAGE CAON ROUTE PLATEAU ROUTE PLEGAVERN ROUTE PLESTIN ROUTE PLEYBEN ROUTE PLOENEZ ROUTE PLOEVEN ROUTE PLOGASTEL ROUTE PLOMEUR ROUTE PLONEIS ROUTE PLONEVEZ ROUTE PLONEVEZ FAOU ROUTE PLONEVEZ PORZAY ROUTE PLONIVEL ROUTE PLOUAY ROUTE PLOUDALMEZEAU ROUTE PLOUDINER ROUTE PLOUDIRY ROUTE PLOUESCAT ROUTE PLOUEZOCH ROUTE PLOUGASNOU LAND KERVERN ROUTE PLOUGRAS ROUTE PLOUGUERNEAU ROUTE PLOUIDER ROUTE PLOUMOGUER ROUTE PLOUNEOUR MENEZ ROUTE PLOURIN ROUTE PLOUVIEN ROUTE PLOUYE ROUTE PLOUZANE ROUTE POINTE ROUTE POINTE RAZ ROUTE POINTE JUMENT ROUTE POINTE LAURENT ROUTE AR BIS ROUTE AR FLOC H ROUTE AR GUIP ROUTE AR STANG ROUTE AVEN ROUTE AVEN PENQUER ROUTE BODIVY ROUTE CROIX ROUTE BUIS ROUTE EON ROUTE PONTEREC ROUTE GONAN ROUTE PONTHOU ROUTE PONTIC ROUTE L ABBE ROUTE L ABBE GUIB ROUTE L ABBE KERANDRO ROUTE LORBIC ROUTE MENOU LOVROU ROUTE MEUR ROUTE PONTOULLEC ROUTE SCORFF ROUTE TALEC ROUTE TOULLEC ROUTE PORSACH ROUTE PORSAC KERVELAN ROUTE PORSACH CROAS AR GALL ROUTE PORSACH 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PILLAT ROZARPOTEN ROZ AR POULLOU ROZ AR SCOUR ROZ AR VEIL ROZ AR YAR ROZ AVEL ROZAVEL ROZ AVEL BRIGNOLEC ROZ AVEL KERVEN ROZ AVEL LESCORRE ROZ BALAN HENT KEROUZIEN ROZ BRAS ROZ BRASS RUAT ROZ BREIZELLOU ROZBRULLU ROZ COAT ROZ COAT DELLEC ROZ COAT KERJOA ROZ ROZELLEC ROZELLIG ROZEMBELLEC ROZENN AN AVEL ROZERNIC ROZERVILLE ROZ FEUNTEUN ROZ GARNOEN ROZ GLAS ROZ GLAZ ROZ GODEN ROZ GUIN ROZ GWEN ROZ HIR ROZHUEL ROZ HUELLA ROZIC ROZIC AN ARCH ROZIC AR PINS ROZIG ROZIGOU ROZIOU ROZIVIN ROZKANNOU ROZ KERBERON ROZ KEREON ROZ KERGREACH ROZ KERGUIAN ROZ KERNARET ROZ KERVINIOU ROZ KERVOUYEN ROZ LANGUIDOU ROZ LANVAIDIC ROZ LEURIOU ROZ LEZONGARD ROZLINOU ROZ LODOU ROZ MEIN GLAS POULDOHAN ROZMEL ROZMEUR ROZ MOGUEROU ROZ NAOUR ROZ NEVEZ ROZON ROZONNEC ROZPLA ROZ QUELEN ROZ QUIMERCH ROZ RAGN GUILLY VIAN ROZ RHUN ROZ SIMON ROZ SPERNEZ ROZUC ROZUEL ROZ VALAN ROZVEGUEN ROZ VEIN ROZVENNI ROZ VEUR ROZ VIAN ROZ VIHAN ROZ WENN RTE BENODET RTE GUILVINEC 29 ROBINER RTE PLOUIDER RTE FOUESNANT RTE HUELGOAT 5 RTE NEVEZ RTE PORT FORET RTE QUIMPER RTE ROSCOFF RUAT RUBANAL RUBARDIRI RUBEO RUBETEREL RUBIAN RUBINOU RUBIOU RUBRAN RUBRAT RUBUEN RUBUSCUFF RUBUSKU RUBUZAOUEN RUCARADEC RUCAT KERAVEL RUDENEZ RUDEVAL RUDEVEL RUDOLOC RUDONGE RUDULLACH 11 NOVEMBRE 1918 11 NOVEMBRE 12 KERMABON 12 SILLONS 14 JUILLET 16 AOUT 1944 17 KERMABON 18 JUIN 1940 18 JUIN 19 MARS 1962 19 MARS 62 1 ERE DFL 2 DIVISION BLINDEE 3 3 EME RAP 3 FRERES DARIDON 3 FRERES LEJEUNE 3 FRERES LEROY 3 FRERES VIENNE 4 SAISONS 4 VENTS 5 AOUT 1944 7 AOUT 1944 8 MAI 1945 8 MAI 45 9 AOUT ABALOR ABBADIE ABBAYE ABBE CADIOU ABBE CONAN ABBE L EPEE ABBE JOSEPH TANGUY ABBE BORGNE ABBE GALL ABBE GUEN ABBE LETTY ABBE LUGUERN ABBE QUEAU ABBE RICOU ABBE ROUDOT ABBES PETTON HABASQUE ABBESSE ABBES TANGUY ABBE TANGUY ABEILLES ABER ABER BENOIT ABER BENOUIC ABER ILDUT ABERS ABER WRACH ABRI MARIN ACACIAS ADIGARD ADOLPHE ALAVOINE ADOLPHE ASSI ADOLPHE GOAZIOU ADOLPHE PORQUIER PLOUJEAN AIGRETTES AIME COTTON AIMEE TALLEC AIME TALLEC AIRE L EVEQUE AIRELLES AJONCS AJONCS D OR ALAIN ALAIN BARBE TORTE ALAIN BERTHOU ALAIN BOUCHARD ALAIN CANIART ALAIN CAP ALAIN COLAS ALAIN DANIEL ALAIN BERGEVIN ALAIN KERGRIST ALAIN FERGENT ALAIN FOURNIER ALAIN GERBAULT ALAIN ALAIN LAY ALAIN MARTRET PREVILLE ALAIN QUINIOU ALAIN SIGNOR LAURENT JUSSIEU ALAVOINE ALBATROS ALBENIZ ALBERT ANDRIEUX ALBERT CAILLAREC ALBERT CAMUS ALBERT MUN ALBERT EINSTEIN ALBERT BRUN ALBERT ALBERT LOUPPE ALBERT POCHAT ALBERT ROLLAND ALBERT ROUSSEL ALBERT SAMAIN ALBERT SCHWEITZER ALBERT THIBAUDET ALBERT THOMAS ALBERT YVINEC ALBINONI ALDERIC LECOMTE GUEN ALENO ALOUARN ALEXANDRE BALEY ALEXANDRE BRETHEL ALEXANDRE DUMAS ALEXANDRE FLEMING ALEXANDRE LEMONNIER ALEXANDRE MASSERON ALEXANDRE MILLERAND ALEXANDRE PRADERE NIQUET ALEXANDRE RIBOT ALEXANDRE TRAUNER ALEX INIZAN ALEXIS CARREL ALEXIS CLAIRAUT ALEXIS MOIGNE ALEXIS OLIVIER VERCHIN BINET COURCY MUSSET VIGNY DODDS JARRY RAY MUSSET NOBEL ALGESIRAS ALGUES ALICE COUDOL ALIZES ALLAE ALLAIN AL LANN VERTE ALLENDE ALLIES ALOUETTES ALPHONSE DAUDET ALPHONSE LAMARTINE ALPHONSE RAZER ALPHONSE SALAUN ALSACE ALSACE LORRAINE AMBROISE CROIZAT AMBROISE PARE AMBROISE THOMAS AMEDEE BELHOMMET AMEDEE MENARD AMEDEE ROLLAND AMERIGO VESPUCCI AMERS AMIRAL BAUDIN AMIRAL BAUGUEN AMIRAL CHARNER AMIRAL COUEDIC AMIRAL COURBET AMIRAL CRAS AMIRAL COATLOGON AMIRAL COUEDIC AMIRAL GUEYDON AMIRAL KERSAINT AMIRAL GRANDIERE AMIRAL D ESTAING AMIRAL D ESTREES AMIRAL D ORVILLIERS AMIRAL COUEDIC AMIRAL EMERIAU AMIRAL GRIVEL AMIRAL GUEPRATTE AMIRAL JAUREGUIBERY AMIRAL CRAS AMIRAL JULIEN COSMAO AMIRAL JURIEN GRAVIERE AMIRAL LABROUSSE AMIRAL LACAZE AMIRAL QUERRE AMIRAL LINOIS AMIRAL MORARD GALLE AMIRAL MUSELIER AMIRAL NICOL AMIRAL NIELLY AMIRAL REVEILLERE AMIRAL ROMAIN DESFOSSES AMIRAL RONARC H AMIRAL RONARCH AMIRAL SALAUN AMIRAL THIERRY D ARGENLIEU AMIRAL TROUDE AMIRAL VALLON AMIRAL VRIGNAUD AMIRAL ZEDE AMIRAUX AMPERE AMUNDSEN AN AOD AN AOD VENELLE AN AOD WENN ANATOLE FRANCE ANATOLE BRAS ANATOLE BRAZ ANCIENNE VOIE FERREE ANCIENS COMBATTANTS ANCIEN VOIE FERREE 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ARGENTRE BERTRAND FRECHE BERTRAND RUSSEL BERVEN BESQUIEN BETHANIE BETHEREL BEUZIT HUELLA BEUZIT IZELLA BIDERGUEN BIENVENUE BIGOT MOROGUES BIHAN BILOU BIR HAKEIM BISSON BLAISE CENDRARS BLAISE PASCAL BLAVEAU BLERIOT BLES D OR BLEUETS BLEUN BRUG BOBBY SANDS BOCAGE BOD ILIO BOD ILLIO BODINIO BODONOU BOENNEC BOHARS AR COAT BOIELDIEU BOILEAU BOIS AMOUR D AMOUR KERESTAT LAMBEZELLEC PIN KERESTAT NEUF BOISSIERE BON COIN BONNE NOUVELLE BONNE RENCONTRE BONNETS ROUGES BORDA BORIS VIAN BOSCO BOSQUET BOSSUET BOT ON BOT QUELEN BOTREL PORSGUEN BOTSCO BOUCHERIES BOUDOULEC BOUESTARD TOUCHE BOUET BOUGAINVILLE BOUILLON BOUILLOUR BOULOUARN BOURDELLE BOURDONNAIS BOURG BOURG BOURGS BOURGNEUF BOURHIOL BOURHIOL VRAS BOURLA BOURNAZOU BOURRELIER BOURVIL BOUS BOUSSINGAULT BOUT BOUTINOU BOUVET BRAHMS BRAILLE BRANDA BRANLY BRASPARTS BREBIS BREHAT BREIZ IZEL BREMODER BREMOND D ARS BRENILOUR BREST BREST BEURRERIE BRETAGNE BREZEHAN BRIEC BRISCOUL BRIZEUX BROCELIANDE BRODEUSES BROSSOLETTE BROUAN BROUSSAIS BROUZIC BRUAT 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C HENRY SIMON CHERCHEUR CALMETTE GUERIN CHERE CHERTEMPS SEUIL CHEVALIER D ASSAS CHEVALIER PERCEVAL CHEVREUILS CHEVREUL CHOISEUL CHOPIN CHOQUET LINDU CHOULDRY CHRISTOPHE COLOMB CHRISTOPHE MORVAN CIEUX CINQ CROIX CITE KERMARIA CITERNE CLARTE CLAUDE BERNARD CLAUDE BOURGELAT CLAUDE DEBUSSY CLAUDE FARRERE CLAUDE FORBIN CLAUDE GOASDOUE CLAUDE LANCELOT CLAUDE LAE CLAUDE PRAT CLAUDE MONET CLAUDE PERRAULT CLAUDE POUILLET CLEDER CLEGUER CLEGUER BIHAN CLEGUER TREZ HIR CLEMENT ADER CLEMENT MAROT CLEMEUR CLOCHETTES CLOITRE CLOS CLOSMEUR CLOUET CLOZ NEVEZ COAT AN ABAT COAT AN ED COAT AR BOURG COAT AR FAOU COAT AR GUEVEN COAT AR MAILLOUX COAT AR PIQUET COAT BIAN COAT BIHAN COAT BRAS COATDIC COAT FAO COATIGARIOU COATILOUARN COAT IVIN COAT KAER COAT KEAR COAT KERROUS COAT KERVEAN COAT LOSQUET COAT MEZ COAT ONN COAT PIN COATUDAL COATUDAVEL COATUDUAL COETIVY COETLOGON COIN COLIN COLAS COLBERT COLETTE COLLEGE COLLINE COLONEL BERTHAUD COLONEL BERTHAUD BOURRIERES COLONEL CURIE COLONEL SOYER COLONEL DRIANT COLONEL FABIEN COLONEL FAUCHER COLONEL FONTFERRIER COLONEL PICOT COLONEL RAOUL LANCIEN COLONEL SICAUD COLONNES JUSTICE COLS VERT COLS VERTS COMBATTANTS U F COMBOUT COMMANDANT AVRIL COMMANDANT CARFORT COMMANDANT CHARCOT COMMANDANT COUSTEAU COMMANDANT EDMOND CARON COMMANDANT FERNAND COMMANDANT LAMY COMMANDANT JEUNE COMMANDANT THUILLIEZ COMMANDANT L HERMINIER COMMANDANT LOUIS FRIANT COMMANDANT LUCAS COMMANDANT MAILLOUX COMMANDANT MALBERT COMMANDANT MAURICE CHALLE COMMANDANT MOGUEROU COMMANDANT NATALINI COMMANDANT SOMME PY COMMANDANT TISSOT COMMANDANT TOUL COMMANDANT VIOLANT COMORANS COMTESSE BLANCHE COMTESSE CARBONNIERES COMTESSE NOAILLES COMTESSE SEGUR COMTESSE RODELLEC PORTZIC CONCARNEAU CONDE CONDILLAC CONDORCET CONEN LUC CONNETABLE RICHEMONT CONSEIL CONSEILLEROU CONSTANT LANCIEN CONSTANT ROUDAUT COQ HARDI COQUELICOTS CORENTIN BARON CORENTIN BOZEC CORENTIN CARIOU CORENTIN CARRE CORENTIN CELTON CORENTIN GUILLOU CORENTIN BERRE CORENTIN FLOCH CORENTIN PERENNES CORMORANS CORN AR GAZEL 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KERJEGU KERJOAIC KERJOUANNEAU KERLAN VIAN KERLECU KERLEGUER KERLEREC KERLIDEC KERLOBRET KERLOUARNEC KERLOUIS KERMABON KERMADIOU KERMARIA KERMAT KERMENGUY KERMERRIEN KERMEUR KERMOALEC KERMOGUER KERMONFORT KERMOOR KERNABAT KERNALEGUEN KERNEACH KERNEGANT KERNEVELECK KERNEVENAS KERNIGUEZ KERNOURS KEROC HIOU KEROMNES KERONTEC KERORIOU KEROSE KEROTER KEROUDOT KEROURGUE KEROURIEN KERPRIGENT KERRAROS KERRARUN KERREZ KERRICHARD KERSAINT KERSAINT GILLY KERSALE KERSAOS KERSCOFF KERSEBET KERSEDRENN KERSENE KERSILES KERSILEZ KERSODY KERSQUINE KERSTRADO KERSULUAN KERSUNEZ KERTANGUY KERVANOUS KERVAO KERVAZE KERVEDAL KERVEGUEN KERVELLEC KERVEN KERVEZENNEC KERVIGNAC KERVIGODES KERVILER KERVILIEN KERVILLERM KERVILLOU KERVILON KERVILY KERVINIOU KERVITOUS KERVOAZOU KERYET KER YS KERZINCUFF KERZOUAR KERZUDAL KERZUGUEL KOAD EULOCH KROAZ BOULIG 2 EME DIVISION BLINDEE BAIE BARRIERE L ABATTOIR L ABBE STEPHAN L ABBE TANGUY L ABER L ABER BENOIT L ABER ILDUT L ABER WRACH BEURRERIE BLANCHISSERIE BOISSIERE CALE NEUVE CARRIERE CHALOTAIS CHAPELLE CHAPELLE CUZON CITADELLE LA COLLINE COMMUNAUTE COOPERATIVE CORDELIERE CORNICHE CORVETTE CREOLE CROIX DELACROIX CROIX FLEURS CROIX NEUVE CROIX ROUGE DIGUE DOUFFINE DUCHESSE ANNE L FEE MORGANE FONTAINE FONTAINE AU LAIT FONTAINE AUX LOUPS FONTAINE BLANCHE FONTAINE FONTENELLE FORET FORGE FOSSE AUX LOUPS FOUASSERIE FRANCE COMBATTANTE FRATERNITE GALICE GARENNE GROTTE HALLE HAUTIERE JOIE LANDE LANDELLE LIBERATION LIBERTE L ALMA LOUISIANE MADELEINE MAISON BLANCHE MAISON PAILLE MARINE MARNE LAMARTINE MARTINIQUE LAMBOUR MEDEE MELPOMENE METAIRIE METAIRIE NEUVE MONTAGNE MOTTE LAMPAUL LANADAN HUELLA L ANCIENNE FORGE L ANCIENNE LANDEVENNEC LANDIVISIAU L ANJOU LANNILIS LANNINGUER LANNION NOUVELLE CALEDONIE LANRIEC LANRIVINEC L ANSE SAUPIN LANVEN LANVEOC LANVIAN PAIX PALESTINE PAPETERIE PASSERELLE PECHERIE PENFELD CHESNAIE MONTAGNE PALUD PISCINE PLAGE PLAINE POINTE PORTE PORTE ROUGE POSTE PRAIRIE PROVIDENCE PSALETTE L AQUEDUC ROMAIN RADE REPUBLIQUE RESISTANCE L ARGOAD L ARGOAT L ARGONNE RIVE L ARMORIQUE SABLIERE SALETTE SALLE SOMME SOURCE L ASTROLABE SURVEILLANTE TANNERIE TERRE NOIRE L ATLANTIQUE TOUR TOUR AUVERGNE TOUR D AUVERGNE TOURELLE TRINITE TROMENIE LATTRE TASSIGNY L AUBEPINE LAUBERLACH L AULNE L AUNIS LAUTREAMONT VALLEE VENDEE L VICTOIRE VIERGE NOIRE D YS VILLEMARQUE VILLENEUVE L EAU BLANCHE L L VOILE L ECOSSE LEDENES L EGALITE L L ELLE L ELORN L ENFER LEN GOZ LESCAO LESCORS LESNEVEN L ESPOIR LESQUIFFINEC L EST LESTONAN VIAN L ESTUAIRE L ETAIN L ETANG LEUR AR VEIL LEURGUERIC L EUROPE L EXODE L HARTELOIRE L HIPPODROME L HIRONDELLE L HOPITAL L HOPITAL FREMEUR L HORN L HOSPICE L HYERES LIEGE L ILE AUX MOINES L ILE AUX MOUTONS L ILE BENIGUET L ILE CALLOT L ILE D ARZ L ILE BALANEG L ILE BATZ L ILE BREHAT L ILE GROIX L ILE DESIRADE L ILE SEIN L ILE D HOEDIC L ILE D HOUAT L ILE LOCH L ILE GARO L ILE LONGUE L ILE GALANTE L ILE MAYOTTE L ILE MOLENE L ILE RONDE LILLE L IROISE LISKEARD L ISOLE LISTROUARN LITTIRY D ELLIANT L OBSERVATOIRE L OCEAN LOCRONAN LOCTUDY L ODET LOHIGOU L ORATOIRE L OREE LORIENT LORIOU L ORPHELINAT LORRAINE LOSCOAT LOST LENN LOSTALLEN L OUEST LUDUDU LUDUGRIS L UNIVERSITE LYON L YSER MADAGASCAR MAISSIN MANER AR C HOAT MARREGUES MEERBUSCH MEJOU BRAZ MELGVEN MELLAC MEN CREN MENEZ BIHAN MENEZ GAD MENEZ GROAS MENEZ KERGOFF MENEZ NOAS MENEZ MENEZ TY AR GALL MEN FALL MERVILLE MESCOUEZEL MESCOUEZEL VIAN MESDOUN MESGALON MESLOAGUEN MESMERRIEN MESNIGOALEN MESNOS MESSIDOU MESTRIDEN MESTUAL METZ MEZANTELLOU MILIZAC MINVEN DEMI QUARTIER MISSILIEN MOELAN MOGUERIEC DEMOISELLES MOLENE MORLAIX MOSTAGANEM NANCY NARVIK NEUVILLE NEVARS DENIS PAPIN NORMANDIE DENTELIERES DENTELLES DENTELLIERES DENTELLIERES LECHIAGAT PARC AR GROAS PARC AR HOAT PARC STEIR PARIS PARK AR MANER PARK MENEZ DEPARTOUT PENANGUER PEN AN PEN AN TRAON PEN AR C HOAT PEN AR CREACH PEN AR GARENNE PEN AR LIORZOU PEN AR MENEZ PEN AR PEN AR QUINQUIS PEN AR RUN PEN AR STANG PEN AR STEIR PEN AR STREAT PEN AR VALY PEN CARN PENFRET PENGUEREC PENHARS PENHOAT PENMARCH PENNANROZ PENNARUN PENQUEAR PENTHIEVRE PENZANCE PENZE PERINAGUEN PERSIVIEN PICARDIE PLOBANNALEC PLOEUC PLOUENAN PLOUESCAT PLOUGASNOU PLOUJEAN PONTAMIS PONTANIOU AR LAER AVEN BANAL PONTIVY L ABBE NIGNON ODET PORS AN TREZ PORS AOR PORS LAND PORS LEON PORSMOGUER PORSPODER PORSTREIN PORT BOUC DEPORTES PORTZMOGUER PORZAMBARS PORZ AR GOSKER PORZH AMIRAL BRIZ POUL AR BACHET POUL AR FEUNTEN POUL AR HORET POULBRIANT POUL BRIEL POULBRIQUEN POULCANASTROC POULEDER POULLEAC H POULOUPRY POULPATRE POULPRY POULRANET PRAD GWER PRAT AN PRAT AR RATY PRAT GOUZIEN PRAT GUEN PRAT PODIC PROVENCE QUELENNEC QUELERN QUELIVERZAN QUILLIMADEC QUIMPER QUIMPERLE QUIZAC RAOZOC REDON REICHSTETT REIMS RENNES RIBEUZE ROC GLAS ROCHEFORT ROHAN RONANGUIP ROSCANVEL ROSCOFF ROSCONGAR ROSMADEC ROSTOMIC ROUBAIX ROUERGUE ROYAN ROZ AR CHASS ROZOLEN ROZ VILY RULIANEC RUSALIOU RUVEI 2 FRERES COLCANAP 2 FRERES GOUERE 2 FRERES GUEZENNEC 3 FRERES BLAISE 3 FRERES COZIAN 3 FRERES JEUNE 3 FRERES ROY 4 VENTS ABEILLES ABERS ABERS CROAS QUENAN ACACIAS AIGUILLETTES ALPHONSE BRIEUC ELOI EXUPERY GERMANS GUENOLE JALMES NAZAIRE SAINTONGE PHILIBERT POL LEON RIOU SAUVEUR THEGONNEC SAISY AJONCS AJONCS D OR ALBATROS ALGUES ALLIES SALONIQUE ALOUETTES SALTASH ANCIENS ABATTOIRS ANEMONES SANTEC ANTILLES ARCHIVES ARTIFICIERS AUBEPINES AUGUSTINS AULNES SAVOIE AZALEES BAIGNEURS BECASSINES BENEDICTINES BERGERONNETTES BLES D OR BLEUETS BOLINCHEURS BONNETS ROUGES BOUCHERIES BOUCHERS BOULEAUX BOULISTES BOUTONS D OR BOUVREUILS BREBIS BRODEUSES BRUYERES BUTINEUSES CABLES SOUS MARINS CAMELIAS CAPUCINES CAPUCINS CARMES DESCARTES CASTORS CEDRES CELTES CERISIERS CHAMPS CHARDONNERETS CHARDONS BLEUS CHARMES CHATAIGNIERS CHENES CHEVALIERS CLOCHETTES DESCLOUZEAUX COLIBRIS COLOMBES COLS VERTS COLVERTS COOPERATIVES COQUELICOTS COQUILLAGES CORDIERS CORMORANS CORSAIRES COURBES COURLIS CYGNES CYPRES DAHLIAS DARDANELLES DEMINEURS DENTELLIERES DEPORTES DOUVES DUNES ECOLES ECOSSAIS ECUREUILS EGLANTIERS EGLANTINES ELFES EMBRUNS ERABLES FAUVETTES 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KERVEGUEN KERVELEGAN KERVELEGEN KERVELEUGANT KERVEN KERVENALEN KERVENOC KERVENOUEL KERVERGER KERVERN KERVEROT KERVEZENNEC KERVIAN KERVIDRE KERVIGORN KERVIHAN KERVILLER KERVILON KERVILZIC KERVINIC KERVINOUEL KERVIZIGOU KERVOAC KERVOASTELLOU KERVOAZEC KERVOAZOU KERVOELEN KERVRIOU KERYEQUEL KER YS KERZAVID KERZAVID TREZ HIR KERZELLEC KESTEL KESTELL KLEBER KLEGUER KOATDIC KOAT KRENN KONRAD ADENAUER KORRIGANS KOSTED AN AOD KREIS AR PIN KREISKER KREIS KER KREIZ AN AVEL KREIZKER KROAS AN AOTER KROAS AR POULLOU KROAS AR SKRILL KROAS PRENN KROAS SALOU KROAZ AR BLEON KROAZ HIR KROAZ PRENN KUZ HEOL RUELA L ABBE BALOUIN L ABBE QUEAU LABBES L ABER LABICHE BRUYERE BUTTE LAC CHALOTAIS CHAPELLE CHAUSSEE LACORDAIRE LACOSTE CROIX DONNART LAENNEC LAENNEC GENDARMERIE LAEZ AR STER LAEZ AR VOURCH FAYETTE LAFAYETTE FONTAINE LAGAD YAR GALISSONNIERE LAGATJAR GRANDIERE LAGRANGE L AHTABASKAN LAITERIE LAKANAL LANDE LIBERATION MADELEINE LAMARRE LAMARTINE LAMBABU LAMBADER LAMENNAIS LAMMENAIS MONTAGNE LAMORICIERE MOTTE FEODALE MOTTE PICQUET LAMOTTE PICQUET LANCELOT LANCELOT LAC LANDE LANDEGAROU LANDEJULIEN LANDELEAU LANDERNEAU LANDES LANDIERS LANDIVISIAU LANELVOEZ LAN FRANOU LANGEVIN LANGOLEN LANGOR LANGOUSTIERS LANGOUSTIERS KERHERNEAU LANGOUSTIERS LESCOFF LANGOZ LANGROAS LANGUIDEN LANGUIDOU LAN ILIS LAN INIZAN LANN LANNEANOU LANNEC LANNELVOEZ LANNEVAIN LANNIC LANNIEN LANNOU LANNOUENNEC LANNOURON LANN RUZ LANORGARD LANOU LANREDEC LANRIAL LANRIJEN LANRIVOARE LANTRENNOU LANVAR LANVENEC LANVENEGEN LANVEUR LANVOUEZ LAP PAIX PALUD PALUE COSQUER LAPEROUSE ROCHE ROCHE CINTREE ROCHE SUR FORON LARS L ATHABASCAN LATOUCHE TREVILLE TOUR D AUVERGNE LATTRE TASSIGNY LAURENT CHABRY LAURENT GONIDEC LAURENT GENDRE LAURENT ROUX LAURENT PENNEC LAURIERS VALBELLE LAVANDIERES LAVILLO LAVOIR LAVOIR LOCHRIST LAVOIR LECHIAGAT LAVOIRS LAVOISIER LAZARE CARNOT BAIL MAIGNANT BAIL MEIGNANT BAS BERRE BIHAN BRETON BRIX LECONTE LISLE CORBUSIER DEAN LEDENEZ LEDRU ROLLIN DUC FLEMM LEGGE GONIDEC LEGOUT GERARD GRECO LEGRIS DUVAL GUEN KERANGALL GUENNEC LEHAN LEHOU LEIGN AR MENEZ LAE MEJOU BIHAN MINIHY LEMORDANT MOYNE BORDERIE LENN LENN AR JOA LENNON LEOCADIC PENQUER LEOCADIE PENGUER LEOCADIE PENQUER LEO DELIBES LEO LAGRANGE LEON LEONARD VINCI LEON AUGUSTIN HERMITTE LEON AUGUSTIN L HERMITTE LEON BLOY LEON BLUM LEON CHANDORA LEON FRAPIE LEON GAMBETTA LEON HARMEL LEON JOUHAUX LEON L HERMITE LEON NARDON LEONTINE DRAPIER CADEC REUN RONS LERVILY LESAGE LESCONIL LESCREAC H LESCUZ LESMINILY LESNE LESNEVEN LESNOA PRES VERTS LESQUIFFINEC SULER L ETANG TITIEN LEUHAN LEUKER KERAMEIL LEUQUER GUEOR LEURDANET LEURE LEUR GOZ LEURIOU LEURRE LEURVEN VERRIER LEVOT LEZ LEZAROUAN LEZ HUEL LEZINGARD LEZIREUR LEZOUZARD L HOSTIS LIA LIAIC LIAUTEY LIBENTER LIBERATION LIBERTE LICHEN LICHENS LIEGE LIEUTENANT LIEUTENANT COLONEL FLATTERS LIEUTENANT JOURDEN LIEUTENANT VAISSEAU PARIS LILAS LIN LINGUEZ LINIOUS LINO VENTURA LIORS MEUR LISZT LITTRE LIVIDIC LLANDOVERY RUELLE D ARMOR RUELLE GORRETS RUELLE DOUR DOUN RUELLE PARC LEUR LOCH LOCH 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MARCEL MARC MARCEL MASSE MARCEL MILIN MARCEL PAGNOL MARCEL MARCEL PELLAY MARCEL PIROU MARCEL POTIN MARCEL PREVOST MARCEL PROUST MARCEL SALEUN MARCEL SCUILLER MARCEL SEMBAT MARCHANDS MARC HARID FULUP MARCHE MARCHE DELATTRE MARC SANGNIER MARC SCOUARNEC MARC SEGUIN MARECHAL CLAUZEL MARECHAL LATTRE MARECHAL LATTRE TASSIGNY MARECHAL FOCH MARECHAL FRANCHET D ESPEREY MARECHAL JOFFRE MARECHAL JUIN MARECHAL KOENIG MARECHAL MARECHAL PELISSIER MARENGO MARGUERITE DURAS MARGUERITES MARGUERITE YOURCENAR MARIA CHAPDELAINE MARIANO ANNE STEPHAN CHAPALAIN CURIE KERSTRAT CORDELIERE JACQUELINE DESOUCHES LENERU LITTRE LOUIS CLOAREC LOUISE BERNARD MILIN MARIGNAN MARINE MARINS FL MARINS PECHEURS MARION FAOUET MARIOTTE MARK TWAIN MARNE MARQUISE KERGARIOU MARTHE MAISONFORT MARTIN LUTHER KING MARTYRS MARX DORMOY MARYAN DESCHART MARYSE BASTIE MASSENA MASSENET MASSILLON MATELOT POCHIC MATHIEU DONNART MATHILDE DELAPORTE MATHILIN AN DALL MATHURIN MEHEUT MATHURIN THOMAS MAUDUIT DUPLESSIS MAUPERTUIS MAURICE BARRES MAURICE BLONDEL MAURICE BON MAURICE BROGLIE MAURICE DENIS MAURICE TRESIGUIDY MAURICE DONNAY MAURICE DUHAMEL MAURICE GENEVOIX MAURICE GUERIN MAURICE FLEM MAURICE LUC MAURICE SCOUARZEC MAURICE SCOUEZEC MAURICE MATHEY MAURICE NOGUES MAURICE MAURICE PIQUEMAL MAURICE RAVEL MAURICE STERVINOU MAX FAUCHON MAXIME CAMP MAXIME MAUFRA MAX JACOB MAX POL FOUCHET MAX RADIGUET MEANTOUL MECHOU MECHOUROUX MECHOUROUX PELLA MEDECIN GENERAL KERAUDREN MEDECIN GENERAL BERRE MEGALITHES MEIN TORET MEIN TORRET MEJOU MEJOU AN ILIS MEJOU BIHAN MEJOU GUEN MEJOU KER MEJOU VEIL MELANIE ROUAT MELLAC MENAIS MEN AN HID MEN CREN MEN CRENN MENDELSSOHN MENDES FRANCE MEN DON MEN MENE BARRIERE MENEC MENEYER MENEZ MENEZ AR C HLAN MENEZ AR PIQUET MENEZ AR VOURCH MENEZ BERROU MENEZ BIAN MENEZ BIHAN MENEZ BONIDON MENEZ BONIDOU MENEZ BOSS MENEZ BRAS MENEZ DREGAN MENEZ FROST MENEZ GORED MENEZ GOUYEN MENEZ HOM MENEZ HUELLA MENEZIC MENEZ IZELLA MENEZ KERGREACH MENEZ KERMADEC MENEZ KERSUGARD MENEZ KERVEADY MENEZ LUZ MENEZ MOOR MENEZ PLAINE MENEZ PLEN MENEZ ROUZ MENEZ ROZ MENEZ MENEZ TY AR GALL MENEZ TY LOR MENEZ VEIL MENFIG MENGLENOT MEN GUEN MENGUEN MENHIR MENHIR COUCHE MENHIRS MEN HOLLO MEN HOLO MENIC MEN MEUR MEN PLAD MER MERENCE LISRIM MERENCE LISRIN MERENCE LISRINS MERLIN MERMOZ MERMOZ CADOL MESANGES MESARNOU MESCOTY MESCURUNEC MESCURUNEC PENZE MESDOUN MESLE MESMENIOU MESMEROT MESNY MESPANT MESPERLEUC MESRENCE LISRIN MESSIBIOC MESSILIO MESSINOU MESSIRE KEREBEL MESTIOUAL METAIRIE METAIRIE KERNEVEL MEUNIERES MEZARNOU MEZ CREIS MEZEOZEN MEZMORVAN MEZOU AR MEIN MEZOU BRAZ MICHEL MICHEL ANGE MICHEL BAKOUNINE MICHEL BALTAS MICHEL BEHIC MICHEL CALVEZ MICHEL COLOMBE MICHEL CREACH MICHEL CORNOUAILLE MICHELET MICHEL FLOCH MICHEL HENRY MICHEL HERVE MICHEL KERBIRIOU MICHEL BARS MICHEL GARS MICHEL NOBLETZ MICHEL SAOUT MICHEL MARION MICHEL MORAND MICHEL OGEE MICHEL OLIVIER MICHEL YVONNOU MIDI MILIN MILIN AVEL MILLET MIMOSAS MIMOSAS TREZ HIR MINES MINEZ BERROU MINOTERIE MIORCEC KERDANET MIQUEL MIRABEAU MOABREN MOAL MOELAN 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BRELIVET BRETONNEAU BRIAND BROSSOLETTE BROSSOLETTE BOURG BROSSOLETTE YVES CAPITAINE CARDUNER CAUSSY CORLE CORNEILLE CORRE CROC CURIE DANTEC BEAUMARCHAIS BELAY COUBERTIN DREUX FERMAT RONSARD DORNIC DOUILLARD DUPOUY CURIE GESTIN GOURLAOUEN GUEGUEN JAFFRET JEFFROY JESTIN JULIEN GILBERT KERNEIS LAROUSSE DAUX GUEN LEOSTIC LEROUX L HOSTIS LOHEAC LOSTIS LOTI MAC ORLAN TOUBOULIC OZANNE PATEROUR PELISSIER PENDELIO PERNES PERON PICARD POSTOLLEC PUGET QUILLEC RENE FELIX DAGORNE RICHARD RIQUET SANQUER SEMARD SEVRE TALCOAT TANNEAU TERMIER TREPOS VELLY VOLANT ZACCONE PILLOT PILOTE TREMINTIN PINS PINSONS PIRANDELLO PISCINE PITI GUEGUEN PITRE CHEVALIER PLACITRE PLAGE PLAGES PLAGES KERLURON PLAINE PLAN D EAU PLATEAU PLESTIN PLOMEUR PLONEVEZ FAOU PLOUEGAT PLOUESCAT PLOUEZOCH PLOUIGNEAU PLOURIN PLOZEVET PLUVIERS PLYMOUTH POINTE POINTE LECHIAGAT POINTE JULIEN POITOU POL AURELIEN POL COURCY POLITZER POL PASQUET POMMIERS PONANT PONCELET PONCELIN PONCHOU PONTADIG LOUET AR AR GROAS AR GUIP AR RESTAURANT AVEN BELLEC CALLEC COAGUEL COZ D ARM BUIS DINOU DU CHATEL DUET PONTIGOU ILIS JEGU L ABBE LEN LENN MENOU NELIS NELLIS NEUF NEVEZ PER PERONIC PONT POL ROHEL PONTRUCHE SAINTE MARTHE PONTSCORFF POPULAIRE PORIGUENOR PORLAZOU PORS AN TOUCH PORS AR BAYEC PORS AR MARC H PORS AR STREAT PORS AR VILLIEC PORS BOURDEL PORS BRAS PORSCAVE PORSCAVE LAMPAUL PORSCAVEN PORS CLOS PORSCUIDIC PORS EGRAS PORS GROUEN PORSGUEN PORSGUERN PORS HAOR PORS KERVEN PORS KORENTIN PORS LAOUEN PORS LEON PORS LOUARN PORS MANOIR PORSMILIN PORS MISCLIC PORSMOGUER PORSPAUL PORSPODERIOU PORSPODIROU PORSPOL PORS POULHAN PORS RIAGAT PORSTREIN LAPIERRE PORT PORT LARVOR PORTE PORTE VEZINS PORTEZ PORT FORET PORT LARVOR PORT POULDU PORT MANECH PORT PORTSALL PORTSALL GOZ PORTZMOGUER PORZ AR VAG PORZAY PORZ CASTEL POSTE POTERIE POUL AN OD POUL AN ODD POUL AN VERN POUL AR POUL AR BLOUCH POUL AR GANOL POUL AR GOASY POUL AR GOAZY POUL AR GRAZ POUL AR MANCHEC POUL AR PALUD POUL AR RANET POUL AR SAF POUL AR STANG POUL BLEIZ POUL BRIANT POULBROHOU POUL CAILLO POULDREUZIC POULDU POULFANC POUL FOEN POULFOS POULGUEGUEN POULHERVE POULHI POULIE POULIZAN POUL LAND POULLAOUEC POULLEACH POULLEN POULLIC AL LOR POULLOU POULLOU AOT POULLOUROU POULLUEN POULMALOTTEN POULMARC H POULMIC POULMOAN POULOU POULOUPRY POULPEYE POULPRY POULQUER POUL RHU POUL ROARCH POULY POULZERBE POULZERBE TREZ HIR POUNT AR C HANTEL PRAD PRAD KELENN PRADOC PRAIRIE PRAIRIES PRAJOU GWEN PRAT PRAT AN AOD PRAT AN ASQUEL PRAT AN EOL PRATANIROU PRAT AR CHARMILLE PRAT AR C HREN PRATAREUN PRAT AR FEUNTEUN PRAT COULM PRAT CUIC PRAT LAOUALC H PRAT LOUALC H PRAT MENOC PRAT MENOCH PRAT MEUR PRAT PANN PRAT PER PRE PRE BAS PREFET ALDERIC LECOMTE PREFET COLLIGNON PREFONTAINE PRES PRESBYTERE PRESIDENT ALLENDE PRESIDENT KENNEDY PRESIDENT SADATE PRES J F KENNEDY PRESQU ILE PRES TREVIDY MAISON PRIEURE PRIMAUGUET PRIMEL PRIMEVERES PRINCE JOINVILLE PRINCESSE DAHUT PRINCIPALE PRISON PROFESSEUR ALAIN PROFESSEUR CHRETIEN PROFESSEUR LANGEVIN PROFESSEUR LERICHE PROFESSEURS CURIE PROGRES PROGRES POULGOAZEC PROMENADE PROSPER MERIMEE PROSPER PROUX PROSPER SALAUN PROUST PROVENCE PROVIDENCE PUEBLA PUIG RITALONGI PUIT PUITS PUVIS CHAVANNES PYRAMIDE PYRAMIDES QIMPER QUAI QUARTIER GOLVES QUARTIER MAITRE BONDON QUARTIER NEUF QUATRE BRAS QUATRE VENTS QUEBEC QUEGUINER QUELARGUY QUELERN QUEMENES QUEMEURLANEC QUERE QUERRIEN QUESTEL QUILINEN QUILLIEN QUILLIMERIEN QUILLIVIC QUIMPER QUIMPER KERVERN QUIMPERLE QUINQUIS RABELAIS RACINE RADENOC RAGUENES RAIL RAMEAU RAMPE RAMZ KOZ RANDON RANTERBOUL RAOUL ANTHONY RAOUL DAUTRY RAOUL DUFY RAPHAEL RAPHAEL XAVIER QUIDEAU RAVAGNON RAYMOND ARON RAYMOND COUILLANDRE RAYMOND GUENET RAYMOND JEZEQUEL RAYMOND CORRE RAYMOND PALU RAYMOND POINCARE RAYMOND STEPHAN REAUMUR RECTEUR MOAL REFUGE REGIMENT NORMANDIE NIEMEN REIMS REMBRANDT REMOULOUARN REMPARTS REMPARTS HAUTS QUELERN REMY BELLEAU REMY COPPIN REMY LAY RENANGUIP RENE AUTRET RENE BARJAVEL RENE BAZIN RENE BERTHELOT RENE BIHANNIC RENE CAILLE RENE CASSIN RENE CLAIR RENE CLEMENT RENE COADOU RENE COGUEN RENE DANIEL RENE CHATEAUBRIANT RENE KERALLAIN RENE DESCARTES RENE GAUBIN RENE GOUBIN RENE GUILLAMOT RENE GUY CADOU RENE GUY KADOU RENE HUIBAN RENE LAENNEC RENE BERRE RENE BOMIN RENE HAMP RENE MAO RENE MADEC RENE QUILLIVIC RENE QUINTRIC RENE TURQUET RENE YVES CRESTON RENE ZAZZO RENOIR REPUBLIQUE RESISTANCE RESKEN RESTAURANT RESTIGOU RETALAIRE REUN AR CHOAT REUNION REVELOUR REVEREND PERE MASSE RHEUN RHODODENDRONS RHU BIAN RHUN AR VUGALE RHUN PREDOU RICHELIEU RICHER RICOU RIDINY RIDOU RIVE RIVERIEUX RIVOLI RUERNO ROBERT KERGROADES ROBERT DOISNEAU ROBERT FROGER ROBERT JOUBIN ROBERT JOURDREN ROBERT GUINER ROBERT MAO ROBERT PLANQUETTE ROBERT RICCO ROBERT SCHUMAN ROBERT SCHUMANN ROBERVAL ROBESPIERRE ROBINSON ROCHAMBEAU ROC AVEL ROC EOL ROC MELEN ROC VENN ROCH AR RUGUEL ROCH AVEL ROCH ROCHE ROCHE CINTREE ROC HELLOU ROCH EOL ROCHERS ROCHES ROCHES BLANCHES ROCHES DIABLE ROCHE TREMBLANTE ROCH GLAZ ROCH KROUM ROCHON ROCH VENELLE RODIN RODOLPHE KOECHLIN ROGER DIQUELOU ROGER MARTIN GARD ROGER PENEAU ROGER QUINIOU ROGER SALENGRO ROGER SIGNOR ROGER THOMAS ROGER VERCEL ROHENNIC ROI GRADLON ROI MARC H ROITELETS ROLAND DORE ROLAND GARROS ROLZAC H ROMAIN DERRIEN ROMAIN DESFOSSES ROMAIN GARY ROMAIN MALASSIS ROMAIN ROLLAND ROMAIN ROLLAND LESCONIL ROMARIN ROMY SCHNEIDER RONSARD ROPARTZ MORVAN ROQUEFEUIL ROQUEFEUILLE ROSAMPONT ROSAMPOUL ROSBIGOT ROSCAROC ROSCOGOZ ROSEAUX ROSE VENTS ROSE VENTS LANMEUR ROSEMONDE GERARD ROSENBERG ROSERAIE ROSES ROSES VENTS ROSIERE ROSIERS ROSIERS BOURG ROSILY MESROS ROSMADEC ROSNOEN ROSPORDEN ROSSINI ROSTIVIEC ROSVILY ROUDOUALLEC ROUDOUR ROUDOUS ROUEDOU ROUERGUE ROUGET L ISLE ROUGET LISLE ROUZ ROZ ROZAMBIDOU ROZ AN DOUR ROZ AON ROZAON ROZ AR GALL ROZ AR GOFF ROZ ARM ROZ AR ROZ AR VERN ROZAVEL ROZ AVEL ROZAVOT ROZENDERO ROZEVEN ROZIC ROZIERE ROZILIO RUBALAN RUBEREL RUBIAN RUDYARD KIPLING RUELLOU RUERNO RUFA RUFOLIGOU RUGOLVA RUGUNAY RUJEAN RUKERDREIN RULAZ RULIANEC RUMEUR RUN AR GROAS RUNAVALEN RUNESCOP RUSTEPHAN RUVEIL SABATIER SABLE BLANC SABLES BLANCS SABLES BLANCS LARVOR SABOTIER SABOTIERS SACRE COEUR SADI CARNOT SAINT ALOR ALPHONSE ANNE ANTOINE AUGUSTIN BREVALAIRE BRIEUC BUDOC CHRISTOPHE CONOGAN CORENTIN DEMET DOMINIQUE DREYER SAINTE ANNE SAINTE ANNE EUTROPE SAINTE AZENOR SAINTE BARBE SAINTE BERNADETTE SAINTE BEUVE SAINTE BRIGIDE SAINTE CATHERINE SAINTE EVETTE EGAREC SAINTE GENEVIEVE SAINTE GWEN SAINTE MARGUERITE SAINTE MARINE SAINTE MARTHE SAINTE MERE ERGAT ESPRIT SAINTE THERESE EVETTE EXUPERY FIACRE GERMAIN GILDAS GILLES GUENAL GUENOLE GUETHENOC GURLOES HERVE IDUNET JACOB JACQUES SAINT BAPTISTE SALLE JEROME JOSEPH JULIEN JUST KE LAURENT LEGER LEONARD LO LUC LUCAS LUC QUIMERCH MALO MARC MARTIN MATHIEU MAUDET MAURICE MELAINE MICHEL PATERNE ROUX PHILIBERT SAINT PIRIC POL POL LEON POL ROUX QUENTIN QUIJEAU RATIAN RENAN RIOC ROCH SACREMENT SAENS SALOMON SANE SAUVEUR SEBASTIEN SESNY SEVERIN SEVERIN TY NEVEZ SIMON TELO THEGONNEC THELAU THELO THIVISIAU THOMAS THUAL THUDEG THUDON THURIEN TREVEL TUGDUAL VALENTIN VINCENT VINOC YVES YVES TREZ HIR SALETTE SALOMON ROUX SALOU SALUT AR VERHEZ SALVADOR ALLENDE SANE SANT DIVY SANTEC SANTIK SANT KIREG SANTOS DUMONT SAO HEOL SAPINIERE SAPINS SARAGOZ SARAH BERNHARDT SARDINIERS SAULES SAVARY SAVOIE SAVORGNAN BRAZZA SCAER SCARS SCHILLER SCHUBERT SCIERIE SCOEN SCRAFIC SEBASTIEN AR BALP SEBASTIEN BALP SEBASTIEN DURAND SEBASTIEN GUIZIOU SEBASTIEN BALP SEBASTIEN VELLY SEBASTOPOL SEDILLOT SEIGNEUR GALON SEIZ AVEL SELL WAR KER SEMAPHORE SEPT ILES SERGENT BAILLANCE SEVERN SEVIGNE SHAKESPEARE SHERIDAN SIBIRIL SIECK SILINOU SIMON SIMON NANTUA SIMONE SIGNORET SIMONE WEIL SIMON GARENGEAU SIRENE SIROCCO SISLEY SOEUR SOEUR PAULINE SOLDAT DUVAL SOLDAT DUVAL LESCOFF SOLFERINO SOMME SORBIERS SOURCE SOURCES SOUS MARIN EURYDICE SOUTH BRENT SPERNOC SPHINX SPORTS SQUIRIOU SQUIRIOU HUELLA SQUIRIOU HUELLA KERHEOL SQUIVIT STADE STADE LECHIAGAT STADES STALINGRAD STANCOU STANCOULINE STANG STANG ALLESTREC STANGALON STANG AN DOUR STANG AR DOUR STANG AR GUER STANG AR LIN BEUZEC CONQ STANG AR ROZEN STANG REO STANG TREMEUR STANG VENELLE STANG YEN STAOL STEIR STELLAC H STENDHAL STEPHANE MALLARME STER STER AN ALLE STER GOR STERNES STERNES TREZ HIR STEROUDOU STERVINOU STIFF STIVEL STIVELL STRASBOURG STRAUSS STREAT AL LANN STREAT AN AOD STREAT AN DREZ STREAT AR ARVORIZ STREAT AR BOULAC H STREAT AR GOZ STREAT AR RELEGOU STREAT BALAN STREAT DOUN STREAT GLAZ STREAT HIR STREAT JOLY STREAT KAN AN AVEL STREAT NEVEZ STREJOU STYVEL SUFFREN SULER SULLY SULLY PRUDHOMME SURCOUF SURCOUFF SUZANNE PARCEVAUX SUZANNE PARCEVEAUX SUZANNE GUIGANTON SUZANNE POUMEROL SUZANNE VALADON SYNDICAT AGRICOLE TACHEN AR CHOAT TACHEN FOIRE TACHEN GUEN TACHEN MORCH TACHEN PONTIC TAILLEURS TAINE TALAMON TAL AR C HEF TAL AR GROAS TALARMAIN TAL AR VEIL TAL AR VILIEN TALBOTEC TAL LENN TAMARIS TANGUY CHATEL TANGUY CHATEL KERSAINT TANGUY JACOB TANGUY MALMANCHE TANGUY PHILIP TANGUY PRIGENT TANNEUR TANTE YVONNE TARIEC TARPORLEY TEILHARD CHARDIN TELEGRAPHE TELEPHERIQUE TEMPLE TEMPLIERS TERRAIN SPORTS TERRAIN SPORTS TERRAINS SPORTS TERRE VANNES TERRE NEUVAS TERRES NEUVAS TEVEN AR REUT TEVENNEC THEBAULT THENENAN MONOT THEODORE BOTREL THEODORE BANVILLE THEODORE HARS THEODORE MONOD THEOPHILE BLIN THEOPHILE GAUTIER THEOPHILE LOUARN THEVEN THEVEN BIHAN THEVEN BRAS THEVEN KERBRAT THEZAC THIERS THOMPSON THONIERS TI KOZ TI LANN TILLEULS TILLIENNOU TINO ROSSI TORPILLEUR BOURRASQUE TORPILLEUR MAILLE BREZE TORPILLEUR ORAGE TOSSENOU TOUCHE TREVILLE TOUL TOUL AL LAER TOUL AN AVEL TOUL AN DOUR TOUL AN HERRY TOUL AR HOAT TOUL AR PARC TOUL AR RANIGUET TOUL AR ZONER TOUL BATTEL TOUL FOEN TOULIBIL TOULINGUET TOULL AR RANIG TOULL BATEL TOULLOU PRI TOULOUPRY TOUR TOUR D AUVERGNE TOUR TOURNEPIERRES TOUROT TOURTERELLES TOURVILLE TOUSSAINT GARREC TRAIN BIRINIK TRANSVAAL TRANSVAL TRAON TRAON AR LIN TRAON AR MANER TRAON AR PIN TRAON AR POUL TRAONBEZEDEN TRAON L ELORN TRAON MANER TRAONOUEZ TRAPPIC COAT TRAVERSE TRAVERSE TRAVERSE RIVE TRAVERSE L TRAVERSE BOUCHERIES TRAVERSIERE TREBEUZEC TREDERN LEZEREC TREFLEZ TREGLONOU TREGOR TREGUIDO TREGUNC TREHOT CLERMONT TREIZ AN DOURIC TREMENACH TREMENARCH TREMEUR TREMINTIN TREMOR TRENINEZ TREOMPAN TREUSCOAT TREVAREZ TREVOEDAL TREVOEDAL LOCQUERAN TREVORC H TREVOUX TREZ TREZIEN TRIELEN TRINITE TRISTAN TRISTAN CORBIERE TRISTAN CORBIERES TRISTAN ISEULT TROADEC TROALEN TROENES TROERIN TROEUJEUR TROFEUNTEUN TROHOAT TROIS FERMES TROIS FORGET TROIS FRERES CHARETTEUR TROIS FRERES DARIDON TROIS FRERES LEJEUNE TROIS FRERES LEROY TROIS FRERES TANGUY TROIS GUENNEC TROIS ROIS TROMENIE TROMEUR TROUGOUREZOU TROUIDY TROUZ AN DOUR TULIPES TULIPES LODONNEC TUMULUS TUMULUS POULGUEN TURANCHOU TURENNE TURGOT TY ARVOR TY BAUL TY BEUZ TY BRAS TY COAT TY COZ TY DEVET TY DEVET TY DON TY DOUAR TY DOUR TY FORN TY FORUM TY GARDE TY GLAZ TY GUEN TY JOPIC TY LAE TY LANN TY LAPIN TY LOSQUET TY MELAR TY MEUR TY NEVEZ TY NEZ TY PELLA TY PIC TY PISTEL TY PRY TYRIEN GLAS TY RUS TY VARLAES URBAIN QUELEN USINE UTRILLO VAD VALANEC VALENTIN VALENTIN HAUY VAL FLEURI VALLEE VALLON VALMY VALORY VALY VALY COZ VAN GOGH VANNEAUX VANNIERS VAN PARYS VARCQ VARQUEZ VASCO GAMA VAUBAN VAUCOULEURS VAUVENARGUES VEDRINES VEILLENEC VEIN VELASQUEZ VELNEVEZ VENELLE VENELLE CARELLOU VENELLE DERO VENGLEUX ROUX VENTRON VENTS VER VERDELET VERDEREL VERDI VERDUN VERGER VERGERS VERGNIAUD VERLAINE VERN CREIS VERN MESPAUL VERONESE VEROUDY VERSANT SUD VEULEURY VEZEN DAN VIADUC VIALA VIANS VICE AMIRAL EXCELMANS VICTOIRE VICTOR AUBERT VICTOR COUSIN VICTOR EUSEN VICTOR HUGO VICTORIEN SARDOU VICTOR GUERN VICTOR MARTIN VICTOR MASSE VICTOR PENGAM VICTOR ROSSEL VICTOR SCHOELCHER VICTOR SEGALEN VICTOR SUREL VIEIL AMER VIEILLE COTE VIEILLE FORGE VIEILLE RETRAITE VIEILLES MURAILLES VIEILLE USINE VIERGE VIEUX BOURG VIEUX CHENES VIEUX FOUR VIEUX KERBIGUET VIEUX MANOIR VIEUX VIEUX PORT VIEUX PUITS VILAR VILLAGE KERAMPROVOST VILLAREN VILLARET JOYEUSE VILLEBOIS MAREUIL D YS VILLEMARQUE VILLER VILLIERS L ISLE ADAM VILLON VILLOURY VINCENT AURIOL VINCENT ROSANCOAT VINCENT D INDY VINCENT JEZEQUEL VINCENT LABBE VINCENT SCOTTO VIOLETTES VIRGINIE HERIOT VIS VIVIER VIVIER LECHIAGAT VIZAC VOAS CAM VOELAZ VOIE FERREE VOILIERS VOLNEY VOLTAIRE VORC LAE VORCH LAE VORLEN VOURCH VUILLEMIN VULCAIN WALDECK ROUSSEAU WALLIS FUTUNA WATTEAU WINSTON CHURCHILL WOAS KAM XAVIER BICHAT XAVIER LANGLAIS XAVIER GRALL YAN D ARGENT YAN NAOD YANN D ARGENT YANN LUIS DOORNIK YED YEUN YOUEN DREZEN YOUENN DREZ