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Model-based clustering for aCGH data using variational EM Alain, Guillaume 2009

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Model-based clustering for aCGH data using variational EM by Guillaume Alain  B.Sc., The University of Ottawa, 2004 M.Sc., The University of Ottawa, 2006  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate Studies (Computer Science)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2009 c Guillaume Alain 2009  Abstract DNA copy number alterations (CNAs) are genetic changes that can produce adverse effects in numerous human diseases, including cancer. Copy number variations (of which CNAs are a subset) are a common phenomenon and not much is known about the nature of many of the mutations. By clustering patients according to CNA patterns, we can identify recurrent CNAs and understand molecular heterogeneity. This differs from normal distance-based clustering that doesn’t exploit the sequential structure of the data. Our approach is based on the hmmmix model introduced by [Sha08]. We show how it can be trained with variational methods to achieve better results and make it more flexible. We show how this allows for soft patient clusterings and how it partly addresses the difficult issue of determining the number of clusters to use. We compare the performance of our method with that of [Sha08] using their original benchmark test as well as with synthetic data generated from the hmmmix model itself. We show how our method can be parallelized and adapted to huge datasets.  ii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  List of Tables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  Acknowledgments  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii  Introduction . . . . . . . . . . . . . . 1.1 Context and motivation . . . . 1.1.1 Copy number variations 1.1.2 Synthetic example . . . 1.1.3 The hmmmix generative 1.1.4 Biclustering . . . . . . 1.2 Our contributions . . . . . . . 1.3 Outline of the thesis . . . . . .  . . . . . . . . . . . . . . . . model . . . . . . . . . . . .  . . . .  . . . . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  2 Description of the current model . . . . . . . . . . . . . . . . . . 2.1 Priors for the parameters . . . . . . . . . . . . . . . . . . . . . . 2.2 Inference with hmmmix-hard . . . . . . . . . . . . . . . . . . . . 2.3 Objective functions and model selection . . . . . . . . . . . . . . 2.4 Determining the number of clusters with the silhouette coefficient 2.5 Bayesian information criterion and minimum description length  . . . .  3 Generalization of hmmmix . . . . . . . . . . . . . . 3.1 Note on forwards-backwards . . . . . . . . . . . . 3.2 Notation and assumptions for variational methods 3.3 Training . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Updates and initialization . . . . . . . . . 3.3.2 Hidden chains . . . . . . . . . . . . . . . . 3.3.3 Partial patients assignments . . . . . . . . 3.3.4 Observation parameters . . . . . . . . . . . 3.4 Projecting to hard assignments . . . . . . . . . . . 3.5 Flow of approximations . . . . . . . . . . . . . . .  . . . . . . . . . .  . . . . . . . .  . . . . . . . . . .  . . . . . . . .  . . . . . . . . . .  . . . . . . . .  . . . . . . . . . .  . . . . . . . .  . . . . . . . . . .  . . . . . . . .  . . . . . . . . . .  . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  1 1 1 2 5 6 7 7  . . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  9 9 11 11 14 16  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  18 19 21 22 22 23 24 26 27 27  . . . . . . . . . .  iii  4 Results and implementation . . . . . . . . . . . . . 4.1 A minimum description length experiment . . . . 4.2 Gap statistic . . . . . . . . . . . . . . . . . . . . . 4.3 Benchmark with synthetic data not from hmmmix 4.4 Comparison with hmmmix-hard on FL data . . . 4.5 Alternative distance-based clustering . . . . . . . 4.6 Multiple chromosomes . . . . . . . . . . . . . . . . 4.7 Complexity analysis . . . . . . . . . . . . . . . . . 4.8 Scalability . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Hardware issues . . . . . . . . . . . . . . . 4.8.2 Parallelization . . . . . . . . . . . . . . . . 4.9 Free energy variant . . . . . . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  29 29 30 32 33 36 37 37 38 39 41 41  5 Conclusion and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46  Appendices A Derivation of update formulas for parameters . . . . . . . . . . . . . . . . . . 49 B Measures of performance for clusterings . . B.1 Silhouette coefficient . . . . . . . . . . . . . . B.2 Jaccard coefficient . . . . . . . . . . . . . . . B.3 Gap statistic . . . . . . . . . . . . . . . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  52 52 52 53  iv  List of Tables 4.1  Comparing hmmmix-soft to hmmmix-hard using Jaccard coefficient. The values are averaged over 50 generated datasets and the standard deviations are included to give an idea of how spread results are. . . . . . . . . . . . . . . . . 33  v  List of Figures 1.1  1.2  1.3 2.1  These two figures show an example for 10 patients of the kind of data that we are interested in modeling. This data has been sampled from the hmmmix model itself so it may not be representative of real aCGH data, but it serves to illustrate the problem that we want to solve. Along the horizontal axis we p have the measurements Y1:T that normally span across multiple chromosomes. For the sake of simplicity, we used sequences of length T = 100 with just 1:P as usually measured one chromosome. We have on top (a) the raw data Y1:T experimentally, while at the bottom (b) we color-coded the sequences based 1:P , the CNVs identified in patients. Red segments on the true values of Z1:T corresponds to deletions and green segments to higher copy numbers. We are interested in learning this information. These hidden values are known to us here because we generated the data. The data has been generated by the hmmmix model using G = 3 clusters. The true cluster assignments place patients {1, 2, 3, 5}, {4, 6, 7} and {8, 9, 10} together, counting from the bottom up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1:G On top we show the recurrent CNAs that characterize the disease profiles M1:T found. Red corresponds to losses and green to gains. On the bottom we show 1:P that are inferred by the hmmmix-hard algorithm from [Sha08] the states Z1:T as well as the final clustering for the patients. The cluster indices are shown in the right margin (based on the profiles indices above) and the patients have been reordered so that those in the same cluster are together. . . . . . . . . . . 1:P The hmmmix model. When generating data, the parameters π 1:G , A1:G , µ1:P 1:K , λ1:K are fixed. The parameters αL , αN , αG are not shown here. . . . . . . . . . . . .  3  4 5  The silhouette coefficient (vertical axis) depending on the number of clusters used (horizontal axis) for different artificial datasets. The number G is the true number of clusters in the datasets and L is the length of the alterations inserted. The solid green curve is the average of the results for 50 datasets and the dotted blue curves show the standard deviation. The silhouette coefficient is a good way to detect the real number of clusters for those artificial datasets when the values of (G,L) are small, but it’s not reliable in cases (f), (h), (i) as the curves don’t peak at the true values of G. . . . . . . . . . . . . . . . . . . 15  vi  2.2  We show here the log-likelihood for data generated by the hmmmix model using G = 10 clusters, P = 100 patients and sequences of length T = 10 000. The dotted curve shows the log-likelihood alone while the solid curve shows the loglikelihood with the BIC penalty term. The model was trained by hmmmix-hard with initial clustering by k-medoids (using G = 6, . . . , 14 clusters). To get a smooth curve, we averaged the results over 30 trials. If BIC was a good way to recover the true number of groups, we would expect the solid curve to peak at G = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17  3.1  If we take a slice at time step t of the graphical model found in figure 1.3, we can see how the hidden chains Mt1 , Mtg , MtG are dependent given the observed value Ytp . We omitted parameters here to simplify the graph. This coupling of the hidden chains is what makes exact inference intractable. . . . . . . . . . . . 19 Our algorithm updates variational approximations and parameters iteratively. This picture shows an overview of the order in which the updates are performed. Within rectangular boxes, we have cycles of updates that should be performed until the quantities stabilize within a certain tolerance. . . . . . . . . . . . . . . 22 The relations between the different parts of the algorithm in terms of approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  3.2  3.3 4.1  4.2  4.3  4.4  We show here the results for the log-likelihood of the data penalized by the MDL term (see equation 4.1). We use G = 6, 10 hidden chains respectively to generate the data and distributed P = 100 patients among those G clusters. The length of the sequences is T = 10 000. In green we have the results from hmmmix-soft and in black we have those from hmmmix-hard. The solid lines represent the values obtained with a random initialization while the dotted curves represent those obtained by k-medoids. These results come from averaging 30 trials. Ideally, we would want the curves to grow monotonously with the number of groups used (clusters) up until the true value of G. At that point, we would like the curves should be flat, indicating that adding further clusters will not help us to model the data. . . . . . . . . . . . . . . . . . . . . The original results from [Sha08] for hmmmix-hard on the follicular lymphoma dataset. The patients are reordered for easier visualization. The green segments correspond to a higher copy number while red segments correspond to deletions. Final results for the follicular lymphoma dataset after running hmmmix-soft 1:G , G1:P . The profiles identified are displayed and deriving hard values for M1:T 1:P are shown below for every patient. The on top and the best values of Z1:T patients have been reordered so that patients from the same cluster are listed consecutively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The execution time for the basic implementation of hmmmix-soft (dotted blue curve) compared to a low-memory implementation (solid black curve). . . . . .  31  34  35 40  vii  Acknowledgments This work is based on Sohrab Shah’s work with my M.Sc. supervisor Kevin Murphy. I thank them both for giving me the chance to work on this project. I am indebted to Kevin Murphy for taking me as student and to Sohrab Shah for providing me with an opportunity to work on a relevant biological problem without having background knowledge in the field.  viii  Introduction The purpose of this thesis is to expand on Sohrab Shah’s work on detecting shared chromosomal aberrations in cohorts of patients from aCGH data. His approach, presented in [Sha08] and [SJJ+ 09], which he names the hmmmix model, treats the gene-probes sequences as hidden Markov chains and clusters the patients to discover shared patterns of observations. In this thesis, we investigate the idea that Shah’s model would perform better and would be more flexible if it were trained using variational methods (see chapter 10 of [Bis06]). Exact inference with the model is intractable and it is currently trained using iterative conditional modes (ICM). We also discuss techniques to estimate the best number of clusters to be used to model the data. We start in this chapter by explaining the context of the problem and giving a synthetic example based on the hmmmix model. The model itself is described briefly in section 1.1.3. Our particular contributions and the plan for the thesis follow right after.  1.1 1.1.1  Context and motivation Copy number variations  DNA copy number variations (CNVs) are a natural phenomenon in which certain gene segments are replicated a higher or lower number of times compared to a reference genome. These changes can be responsible for various diseases such as cancer, CHARGE syndrome [JAVDD+ 06], Parkinson [SFJ+ 03] and Alzeihmer [RLHR+ 06], but not all CNVs are associated with disease. Those that are dangerous are referred to as copy number alterations (CNAs). The introduction of array-based comparative genomic hybridization (array CGH) is responsible for the recent explosion in data, but one of the problems is that not much information is available about the nature of the identified CNVs. They are so prevalent that the proportion of any given chromosome susceptible to CNVs varies from 6% to 19% [RIF+ 06].  1  Different features of a population can be studied by looking at the CNV patterns of the individuals. An interesting experiment is presented in [RIF+ 06] where they cluster populations based on the CNV genotypes. They show how their results match those of similar experiments that put African, European and Asian populations in different clusters. The same thing that be done for CNAs that result from human disease. Patients sharing a disease may exhibit the same characteristic CNAs and that knowledge may be used to identify people in the early stages of that disease. In that case, the pathogenic CNVs need to be sorted out from the abundant healthy CNVs that constitute background noise. By analyzing aCGH data for multiple patients simultaneously, we can have a more robust method to identify the recurrent CNA patterns. We can then cluster all patients based on these patterns. This can be used to shed some light into the mechanisms of certain types of cancers and it can be used to make better informed medical decisions.  1.1.2  Synthetic example  To illustrate the problem better before discussing possible solutions, we give here an example with synthetic data generated by the hmmmix model from [Sha08]. We have P = 10 1:P corresponding to gene expression patients for which we have sequences of observations Y1:T measurements, where T = 100 is the length of the sequences. We show in figure 1.1(a) the raw data sequences. When dealing with more than one chromosome, we can concatenate them in sequence as long as we’re careful to differentiate between them during learning (see section 4.6). For all values Ytp , we postulate that there is a corresponding hidden state Ztp ∈ {loss, neutral, gain} whose discrete value determines the emission model from which Ytp is drawn. These three states correspond respectively to CNA deletions, no alterations, and CNAs with higher copy numbers. We show in figure 1.1(b) the hidden states Ztp by color-coding the observations 1:P are never observed directly and the reason that we from figure 1.1(a). The values of Z1:T have access to them is because we are dealing with artificial data for which we generated the 1:P . hidden values before generating the observed data Y1:T 1:P in real experimental data, but we also want to We want to infer those hidden values Z1:T discover shared patterns among patients. The idea is that there are certain underlying CNAs common to the members of certain cohorts of patients and that these CNAs are identifiable 1:P given the right model. In this particular case, the patients have from the observations Y1:T been generated according to G = 3 disease profiles (also referred to as cohorts, groups or g clusters). We define one hidden Markov chain M1:T per disease profile g = 1, . . . , G. The CNAs that characterize the different profiles are modeled by the discrete hidden Markov  2  450 400 350  patients  300 250 200 150 100 50 0  0  10  20  30  40  50  60  70  80  90  100  80  90  100  chromosomes (a) experimental measurements 450 400 350  patients  300 250 200 150 100 50 0  0  10  20  30  40  50  60  70  chromosomes (b) experimental measurements tagged by color using ground truth  Figure 1.1: These two figures show an example for 10 patients of the kind of data that we are interested in modeling. This data has been sampled from the hmmmix model itself so it may not be representative of real aCGH data, but it serves to illustrate the problem that p we want to solve. Along the horizontal axis we have the measurements Y1:T that normally span across multiple chromosomes. For the sake of simplicity, we used sequences of length 1:P as usually T = 100 with just one chromosome. We have on top (a) the raw data Y1:T measured experimentally, while at the bottom (b) we color-coded the sequences based on the 1:P , the CNVs identified in patients. Red segments corresponds to deletions true values of Z1:T and green segments to higher copy numbers. We are interested in learning this information. These hidden values are known to us here because we generated the data. The data has been generated by the hmmmix model using G = 3 clusters. The true cluster assignments place patients {1, 2, 3, 5}, {4, 6, 7} and {8, 9, 10} together, counting from the bottom up.  3  chains taking one of three values {loss, background, gain}. p g We want to cluster the patients based on the values inferred for Z1:T and M1:T , while refining our estimates of those values based on the clustering in progress. This way, we hope to recover the recurrent CNAs responsible for the diseases, while at the same time determining which patients are afflicted by which type of disease. We show at the top of figure 1.2 the CNA 1:P for profile information learned by using the method of [Sha08]. The particular values of Z1:T the patients are shown at the bottom of figure 1.2. The patients have been reordered after being clustered into the 3 profiles identified. The cluster numbers are in the right margin.  1:G Figure 1.2: On top we show the recurrent CNAs that characterize the disease profiles M1:T 1:P found. Red corresponds to losses and green to gains. On the bottom we show the states Z1:T that are inferred by the hmmmix-hard algorithm from [Sha08] as well as the final clustering for the patients. The cluster indices are shown in the right margin (based on the profiles indices above) and the patients have been reordered so that those in the same cluster are together.  By clustering the patients using a model that takes into consideration the sequential structure of the data, we are able to exploit the fact that CNAs come in segments. This provides a good advantage over distance-based methods that can only take advantage of this fact indirectly by smoothing the imputed states after inference. Bear in mind that in this section we presented data sampled from the very model that [Sha08] introduced to solve the problem. This circular demonstration is intended here as a means to illustrate the problem tackled by [Sha08] and us, and not as a justification for the hmmmix model itself. The hmmmix model has been validated by experiments with real data presented 4  in [Sha08] and [SJJ+ 09]. In this thesis we are mostly interested in finding ways to use the model in a better way.  1.1.3  The hmmmix generative model  The hmmmix model is a generative model that has G hidden Markov chains and P patients, with every patient assigned to one of the chains (we assume that P G). Figure 1.3 shows the graphical representation of the model.  1:P Figure 1.3: The hmmmix model. When generating data, the parameters π 1:G , A1:G , µ1:P 1:K , λ1:K are fixed. The parameters αL , αN , αG are not shown here.  We now describe how the model (conceptually) generates data. For every group g = 1, . . . , G, we have a corresponding hidden Markov chain of length T whose states are represented by g M1:T . These chains have K discrete states that, for the purposes of this work, can take one of K = 3 discrete values corresponding to different copy number alterations : {1=loss, g 2=background, 3=gain}. For every g, we sample one full sequence M1:T from a hidden Markov g g chain with corresponding prior π and transition matrix A . These parameters should be chosen to encourage the normal (k = 2) state and discourage state transitions.  5  For every patient p = 1, . . . , P , we assign that patient to one of the groups {1, . . . , G} with equal probability. We let Gp be the index identifying to which hidden Markov chain p belongs. g We then take the associated chain M1:T (for Gp = g) and we sample a sequence of hidden p p states Z1:T associated to that patient. Note that the Z1:T sequence is not a Markov chain. p The discrete value of Zt is sampled from a multinomial whose parameters are given by one of K predefined vectors αL , αN , αG depending on the state {1=loss, 2=background, 3=gain} of the source Mtg . p(Ztp = k|Mtg = j, Gp = g) =  αj (k) αj (1) + αj (2) + αj (3)  (1.1)  We usually define the parameter vectors as αL = (aL , 1, 1), αN = (1, aN , 1), αG = (1, 1, aG ) for an appropriate choice of aL , aG ≥ aN ≥ 1. These parameters determine how likely the patients are to have the same hidden states as the ones found in the group to which they belong, that is, the “purity” of a given column. We referred to the copy number state Mtg = 2 as background but to the state Ztp = 2 as neutral. By using a small value of αN ≥ 1, the hidden states Mtg = background become agnostic about the corresponding states Ztp of patients p belonging to profile g. This is meant to encourage a certain sparsity for the CNAs identified. The states Ztp = neutral, however, simply represent the fact that the corresponding observations Ytp will be most likely found to higher than those from Ztp = loss and lower than those from Ztp = gain. 1:P given Z 1:P according to We generate the observations Y1:T 1:T  p(Ytp |Ztp = k) = Student Ytp |mean=µpk , precision=λpk , degrees of freedom=νkp where  Γ ν+1 2 Student (y|µ, λ, ν) = Γ ν2  λ λ 1 + (y − µ)2 νπ ν  − ν+1 2  (1.2)  (1.3)  In [Sha08] the constant αN is referred to as α0 . The model handles data from probes expected to be in the background state a bit differently, but we don’t do this distinction here during training. The hmmmix model is also defined in [Sha08] using a “compound DirichletMultinomial” where the essential parameters are marginalized. We chose to omit this step as the resulting distribution simplifies into a basic multinomial.  1.1.4  Biclustering  As mentioned in the previous sections, we are be using a model-based approach instead of a distance-based approach in order to exploit the structure of the data. The method of “biclustering” introduced by [CC00] in this context is another popular tool for expression 6  data analysis. It tries to find relations between genes and conditions, represented by the rows and columns of a matrix. In [MTSc+ 07] is a fast algorithm to perform biclustering based on the sequential structure of the data. It operates under the assumption that the biclusters found will be contained in contiguous columns. This is similar to the construction of hmmmix that encourages the hidden chains representing CNAs to have segments where the state is constant. However, it assumes that the data is discrete and noise-free. We will not be discussing biclustering in this thesis, though.  1.2  Our contributions  Our contributions to the hmmmix model are three-fold. First, we use variational methods to train the model. The most interesting consequence of this is the possibility to partially assign patients to multiple clusters. We expect this to improve on hard clustering, just as EM for Gaussian mixtures improves on K-means. We derive the new update formulas to train the model and we analyze its runtime performance. Secondly, we show how variational methods open the way to other initialization methods than the k-medoids method used in [Sha08]. This was a serious limitation in their model as it required the number of clusters to be selected a priori. We show how the Bayesian information criterion (BIC) fails in this particular setting and how the minimum description length (MDL), the silhouette coefficient and the gap statistic can be used instead. We also discuss the addition of scaling constants to the variational entropy terms in order to smooth out the objective function. Finally, we give a low-memory implementation made to handle large datasets. We compare the performance of both on small and large datasets. We have also rewritten the forwardsbackwards algorithm in C to speed up by a factor between 10 and 100 the original code from [Sha08], which was written in Matlab.  1.3  Outline of the thesis  In chapter 2, we describe how the hmmmix model is trained in [Sha08]. We present an experiment with the silhouette coefficient and one with BIC to determine the best number of clusters to use.  7  In chapter 3 we show how it is possible with variational methods to extend the model to remove some of the constraints that [Sha08] has. We derive the formulas relevant to our method. In chapter 4 we compare our method with that of [Sha08] using his benchmarks on artificial data and by comparing visually the results on real data. We present an experiment with MDL and the gap statistic to determine if the true number of clusters can be recovered practically with either [Sha08]’s method or our own. We also analyze the computational/memory complexity of our algorithm and discuss how it scales to accommodate the very large datasets that are common in biology. Finally, in chapter 5, we conclude with a general discussion and point to potential improvements.  8  Chapter 2  Description of the current model In this section we explain how the hmmmix model is used in [Sha08] and [SJJ+ 09]. We follow certain naming conventions in order to tell apart the model in [Sha08] from ours, and to tell the difference between the statistical models themselves and the various methods to train them. What we refer to as hmmmix is a generative model with various parameters. Exact inference in that model is very hard and various approximations can be done to simplify this task. In [Sha08] a certain technique is used for that, and we will refer to it as hmmmixhard as it performs hard cluster assignments to patients and uses the Viterbi algorithm to impute values to the hidden Markov chains. In this thesis, we start again from the hmmmix model but we solve the learning problem by using variational inference. We call our method hmmmix-soft in reference to the partial cluster assignments for patients. The underlying model is still the hmmmix model but we approach it differently. The core of the hmmmix model was presented in section 1.1.3. We introduce priors in section 2.1 for some of the parameters. An important component of the model comes in choice of the emission distributions for the observations and the method we use learn the parameters. For this, [Sha08] borrows from [Arc05] but the actual formulas from the latter have to be adapted to our particular case. We discuss in section 2.3 the choice of objective function in the perspective of model selection to identify the best number of clusters to use. We present in section 2.4 an experiment to determine the number of clusters using the silhouette coefficient. In section 2.5, we show why BIC cannot be used for this task.  2.1  Priors for the parameters  There are two places in this model where priors are placed on parameters. The first place is when estimating the transition matrices involved in the hidden chains. We put a Dirichlet prior on every row of Ag with parameters picked to encourage same-state transitions. This 9  essentially corresponds to putting pseudocounts on the transition matrices. More details on this are found in section 3.1 when we put the forwards-backwards algorithm in context. 1:P The second place where priors are used when we are estimating the parameters θ = {µ1:P 1:K , λ1:K } p p 1:P |Z 1:P , θ) = that take part in p(Y1:T t p p(Yt |Zt , θ). There are slight divergences in nota1:T tion between [Sha08] and [Arc05] as well as some imprecision concerning the roles that the hyperparameters play. We decided to follow [Arc05] whenever possible and to be explicit about the density functions whenever confusion could arise. This is important as we will be using these definitions in section 3.3.4 when addressing how to learn the parameters from hmmmix-soft. It should be understood that these priors are there to give the model some 1:P flexibility and there might be better ways to train the values of µ1:P 1:K , λ1:K when we have domain-specific knowledge.  We factor the prior as p(µ, λ) = p(µ|λ)p(λ). On the precisions λpk , we put a Gamma prior with hyperparameters γkp , Skp also indexed by k, p : p λpk |γkp , Skp = Gamma λpk |γkp , Skp where Gamma (x|a, b) =  1 a a−1 b x exp (−bx) . Γ(a)  (2.1)  (2.2)  On the means µpk , we put a normal prior with hyperparameters mpk , ηkp : p µpk |mpk , ηkp , λpk = N µpk |mean=mpk , precision=ηkp λpk  (2.3)  where N (x|mean=µ, precision=λ) =  λ λ exp − (x − µ)2 . 2π 2  (2.4)  It should be noted that, in [Sha08], the hyperparameters mpk , νkp are only indexed as mk , νk . From personal correspondence with the author, we learned that hmmmix-hard actually used patient-specific parameters so we thought it would be better to present the model in this fashion. As those values are not learned during inference, we can always assign the same values across patients and it will be equivalent to using the mk , νk indexing. Finally, there is a certain amount of preprocessing done on the experimental data before it is used to learn the hmmmix model. This preprocessing step has more to do with instrument calibration than with statistics. The effects of that are reflected in the values of mpk that we use and they should be consistent with our choice of labels for the hidden states. With a good prior respecting at least the equality mp1 ≤ mp2 ≤ mp3 , it should come naturally that µp1 ≤ µp2 ≤ µp3 without having to enforce it explicitly.  10  2.2  Inference with hmmmix-hard  The patients are initially clustered within a specified number of groups with a method de1:P and then uses the scribed in [Sha08]. This method imputes values to the variables Z1:T k-medoids algorithm to get the initial assignments. From then on, the hmmmix-hard algorithm proceeds by the iterative conditional modes (ICM) algorithm, updating the hidden 1:G , the patients assignments G1:P and the parameters µ1:P , λ1:P until convergence. chains M1:T 1:K 1:K These updates are done by assigning patients to the single group whose hidden chain “explains” best the observations. The updates to the hidden chains are done by imputing values 1:G using the Viterbi algorithm to get the most likely sequence of states explaining the to M1:T observed values for the patients assigned to that group. We refer the reader to [Sha08] for the actual formulas. To estimate the parameters, the priors from [Arc05] are used. We are using the same priors in section 3.3.4, although we use a slightly different approach. The reader can again refer to [Sha08] and [SJJ+ 09] for the essential formulas, to [Arc05] for the original derivations or simply to section 3.3.4 of this thesis. There are two reasons why the k-medoids algorithm is used instead of the classic k-means. The first is that k-means is not defined on discrete data, and the second is that k-medoids does the clustering with a distance matrix that computes distances between points only P (P2−1) times. The cost of each evaluation is prohibitive when we are dealing with long sequences of genes. The cost of running k-means (or EM) to initialize the algorithm can become as expensive as running the algorithm itself, so multiple restarts of k-means are not an option. With k-medoids, the same distance matrix can be used for multiple restarts of the algorithm.  2.3  Objective functions and model selection  For the rest of this section, we will denote by C the number of clusters used (currently denoted by G) to avoid potential confusion with the hidden variables G1:P . The meaning of G is unambiguous in a context where it is always accompanied by Gp or G1:P when referring to clusters to which patients are assigned, but here we want to be able to drop some indices from the current discussion to avoid clutter. We are interested in determining what is the optimal number of clusters C to use. This becomes a model selection problem. Conceptually, when learning the hmmmix model we are learning the parameters A1:C , π 1:C , θ  11  that maximize the likelihood max  A,π,θ  p(Y, M, G|θ, A, π). M  (2.5)  G  By using priors on the parameters, we are actually maximizing max  A,π,θ  p(Y, M, G, θ, A, π), M  (2.6)  G  1:C and patient but the important feature is that we are integrating over the unknown chains M1:T assignments G1:P . 1:C and G1:P while maximizing By proceeding by ICM, hmmmix-hard imputes values for M1:T the likelihood with respect to the parameters A, π, θ. We are essentially treating the hidden 1:C , G1:P as though they were parameters like the rest, and maximizing variables M1:T  max max max p(Y, M, G, θ, A, π),  A,π,θ M  G  (2.7)  instead of (2.6). There are two issues with this. First, we have to ask ourselves if we are interested in the quantity (2.7) itself instead of the one at line (2.6). Secondly, ICM might be a terrible way to maximize (2.7) in the context of the hmmmix model. Without the possibility of maximizing (2.7) exactly, it’s hard to determine how efficient hmmmix-hard is (we return to that topic in 4.1). We think that hmmmix-soft can perform better in situations where hmmmix-hard gets stuck quickly in bad local maxima. Coming back to the first issue, the original problem is to identify CNAs and cluster patients to discover shared patterns. The objects that we study are marginalized in (2.6) and we are left with the parameters A, π, θ that are not of much use. We are very much interested in finding out where the CNA segments are. That information is present in the imputed chains 1:C that we get from hmmmix-hard, but not in the MAP estimates of A, π. M1:T Instead of taking the maximum over the parameters in (2.6), we could integrate them out as p(Y, M, G, θ, A, π). A,π,θ M  (2.8)  G  However, we have two objections against integrating out the parameters, not mentioning the objection about using G M instead of maxG maxM expressed earlier. First, the forwards-backwards algorithm works with fixed constants A, π and not with distributions. If we didn’t have a good prior for the transition matrices A and we tried to evaluate (2.8) by sampling values of A, most of the transition matrices obtained would be completely incompatible with our data that we expect to contain only rare transitions between states. If 12  we had a prior that accounted for this, then the posterior distribution would be peaked and a pointwise estimate of A would be fine. This is exactly what we get by using maxA,π instead of A,π . The other objection about marginalizing over the parameters θ is that we are using specific hyperpriors tailored to our problem. For example, we set the hyperpriors m1:P 1:K to the values 1:P that we hope the parameters µ1:K will take in the absence of overwhelming evidence. This defeats the philosophical ideal of integrating over the parameters to keep the model as agnostic as possible. Those objections notwithstanding, with equation (2.8) we can compare the likelihood of the 1:C . The likelihood is given data under the models with different numbers of hidden chains M1:T by 1:C p(Y, A, π, θ, M1:T , G).  p(C|Y ) ∝ p(Y |C) = A,π,θ  1:C M1:T  (2.9)  G  As argued earlier, another equally valid but more convenient alternative is 1:C p(Y, A, π, θ, M1:T , G).  p(C|Y ) ∝ p(Y |C) = max A,π,θ  1:C M1:T  (2.10)  G  With the hmmmix-soft algorithm introduced in chapter 3, we use variational approximations in order to maximize a lower bound approximation on (2.10). We get distributions 1:C ) and h(G1:P ) that behave somewhat like the quantities M 1:C , G1:P imputed by ICM h(M1:T 1:T by hmmmix-hard to maximize (2.7). We update them in the same fashion as hmmmix-hard and when they are peaked they are similar to the values from hmmmix-hard. We explain in section 3.4 one method to convert the variational approximations from hmmmix1:C , G1:P . This allows us to assess hmmmix-soft’s performance in soft to imputed values M1:T terms of maximizing (2.7) as hmmmix-hard does, and it also illustrates why the latter can be improved. If we could work with (2.9) instead of (2.10), we could use the distribution p(C|Y ) ∝ p(Y |C) to determine the true number of clusters with the Bayesian equivalent of Occam’s razor. Assuming that this were also true with (2.10), nothing guarantees that our variational lower bound on (2.10) obtained by hmmmix-soft is tight (it should not be since we made assumptions about the factoring). It would be hard to argue that, upon getting that p(C = 10|Y ) = 0.3 and p(C = 11|Y ) = 0.28 we should select C = 10 and risk being one cluster short when the penalty for being one short is potentially worse than for being one over. The Bayesian approach to p(C|Y ) ∝ p(Y |C) doesn’t work with equation (2.7) because more clusters just lead to overfitting. In the worst case, we could wind up with one patient per 13  cluster. In section 2.5 we discuss the use of BIC to solve this problem, but before that we show in section 2.4 how [Sha08] use the silhouette coefficient to determine the best number of clusters.  2.4  Determining the number of clusters with the silhouette coefficient  One of the important limitations of the hmmmix-hard approach is that we have to specify in advance the number of clusters to be used. This is essential for the k-medoids algorithm and the resulting clustering contains only hard assignments to clusters. To get around this issue of determining the number of clusters, [Sha08] use the silhouette coefficient as a measure of quality (see appendix B for definition). They impute the values of 1:P and use them as input for k-medoids. The quality of the initial clustering the hidden Z1:T 1:P . naturally depends on the quality of the imputed values of Z1:T We ran our own simulation to see if we could rely on the silhouette coefficient to estimate the number of clusters in the case of the synthetic data generated by [Sha08]. We used their code to generate data with different numbers of clusters G = 3, 5, 10 and different lengths L = 25, 50, 75 for the copy number alterations. We looked at the silhouette coefficients obtained when selecting the best clustering from k-medoids with G = 2, . . . , 25 clusters. We used the same parameters as [Sha08] to explore the nine combinations and averaged over 50 datasets each time. The results are in Figure 2.1. From this we see that in many of the simpler cases, the silhouette coefficient with k-medoids identifies correctly the true number of clusters. It peaks at the true values and the variance in the results is relatively small. However, in 2.1(f) the results are unreliable due to high variance, in (h) the peak is hard to identify even when averaging multiple runs and in (i) the silhouette coefficient is off track, probably because the k-medoids algorithm fails to produce any good clustering. This is somewhat counter-intuitive as we would have expected better results with larger values of L. It might be related with the fact that the k-medoids clustering 1:P instead of the Y 1:P and the quality of those values Z 1:P is done with imputed values of Z1:T 1:T 1:T might decrease when the length of the inserted CNAs increases. These graphs were produced from artificial datasets obtained by taking real data and inserting mutations in them of appropriate lengths. Experimental data does not have a true number of clusters and the value found by the silhouette coefficient does not guarantee that we will do any good for any other kind of measure of performance (such as imputing the Z, the M or predicting Y ). 14  0.7  0.5  0.6  0.45  0.35  0.4 0.3 0.2  0.4  Silhouette coefficient  Silhouette coefficient  Silhouette coefficient  0.3 0.5  0.35 0.3 0.25  0.25  0.2  0.2 0.15  0.1 0 0  0.15 5  10 15 nbr of clusters used  20  0.1 0  25  10 15 nbr of clusters used  20  0.1 0  25  (b) G=3,L=50 0.24  0.4  0.22  0.6  0.35  0.4 0.3 0.2 0.1 0  Silhouette coefficient  0.45  0.7  0.5  0.3 0.25 0.2  10 15 nbr of clusters used  20  0.1 0  25  (d) G=5,L=25  20  25  20  25  20  25  0.2 0.18 0.16  5  10 15 nbr of clusters used  20  0.12 0  25  (e) G=5,L=50  0.35  10 15 nbr of clusters used  0.14  0.15  5  5  (c) G=3,L=75  0.8  Silhouette coefficient  Silhouette coefficient  (a) G=3,L=25  5  5  10 15 nbr of clusters used  (f) G=5,L=75  0.18  0.16  0.17 0.14  0.25  0.2  Silhouette coefficient  0.16 Silhouette coefficient  Silhouette coefficient  0.3  0.15 0.14 0.13 0.12 0.11  0.15  0.12  0.1  0.08  0.1 0.1 0  5  10 15 nbr of clusters used  (g) G=10,L=25  20  25  0.09 0  5  10 15 nbr of clusters used  (h) G=10,L=50  20  25  0.06 0  5  10 15 nbr of clusters used  (i) G=10,L=75  Figure 2.1: The silhouette coefficient (vertical axis) depending on the number of clusters used (horizontal axis) for different artificial datasets. The number G is the true number of clusters in the datasets and L is the length of the alterations inserted. The solid green curve is the average of the results for 50 datasets and the dotted blue curves show the standard deviation. The silhouette coefficient is a good way to detect the real number of clusters for those artificial datasets when the values of (G,L) are small, but it’s not reliable in cases (f), (h), (i) as the curves don’t peak at the true values of G.  15  2.5  Bayesian information criterion and minimum description length  The Bayesian information criterion (BIC) is introduced in [Sch78] and defined as 1 MLE log pH (X1 , . . . , Xn |θˆH ) − k log(n) 2  (2.11)  MLE ) is the maximum value for where {X1 , . . . , Xn } is the data observed, log pH (X1 , . . . , Xn |θˆH the log-likelihood under model H and k is the number of parameters that model H has. To select the best model H1 , H2 , . . ., we pick the one that maximizes BIC. The size penalty for the model acts like Occam’s razor that selects the simplest model that explains the data.  There is something unsatisfying about the silhouette coefficient in section 2.4 because it does not fit into the statistical framework of the model. With BIC we add a penalty term to the log-likelihood of the data in order to do model selection, but we can’t do the equivalent for the silhouette coefficient. In fact, when [Sha08] compare hmmmix-hard to existing methods with their benchmark test that uses synthetic data, they omit this step and assume that the true number of clusters G is known by their k-medoids clustering algorithm. A first objection against BIC is that it requires us to train models of various sizes to determine the best number of clusters to use ultimately. This is prohibitive in our case because we are dealing with large datasets. One approach could be to reuse partial results from models of different sizes to train others, but this first issue isn’t the real problem with BIC and the hmmmix model. The fundamental problem is the fact that the parameters of hmmmix are redundant. As 1:G and explained in section 2.3, the hmmmix-hard algorithm treats the hidden chains M1:T patient assignments G1:P as parameters. The BIC penalty term is then 1 1 (GT + GK 2 + P + KP ) log(P T ) ≈ GT log(P T ). 2 2  (2.12)  Considering that T dominates all other terms (in our particular application), the penalty term is essentially 12 GT log(P T ). g However, this is misleading because the chains M1:T can be encoded with much less than T parameters due to the fact that state transitions are uncommon. Their rare occurrence is part of the assumptions that we make based on the fact that we are modeling CNAs. Individual patients have all kinds of CNVs in their chromosomes, but the CNAs that we want to identify in patient cohorts are supposed to be sparse.  Hypothetically, we could encode the chains imputed by hmmmix-hard with one parameter for the initial state and two parameters for each transition, indicating when the transition 16  occurs and to what state we are switching. This would translate into representations using a varying number of parameters, with “most” sequences (in the probabilistic sense) being represented by very few parameters. With this in mind, we should leave BIC behind and use the minimum description length (MDL) instead. We discuss MDL in section 4.1 after hmmmix-soft is introduced. We conclude here with a numerical example to support our case against BIC. We train hmmmix-hard using data sampled from the hmmmix model itself. The values of the hyperparameters used to sample the data are also given to hmmmix-hard. Using G = 10 groups as ground truth, we compare in figure 2.2 the log-likelihood contributions for model sizes G = 6, . . . , 16 to the same log-likelihood that includes the BIC penalty term. We used kmedoids to initialize the clusters. The plot shows that the BIC penalty term is of the wrong 1:P | · · · ) as it makes the order of magnitude. It overshadows the log-likelihood term log p(Y1:T solid curve in figure 2.2 decrease monotonically without anything special happening around the true value of G = 10. 6  −2  x 10  loglikelihood of data  −2.2 −2.4  loglikelihood loglikelihood with BIC  −2.6 −2.8 −3 −3.2 6  8 10 12 number of groups used for clustering  14  Figure 2.2: We show here the log-likelihood for data generated by the hmmmix model using G = 10 clusters, P = 100 patients and sequences of length T = 10 000. The dotted curve shows the log-likelihood alone while the solid curve shows the log-likelihood with the BIC penalty term. The model was trained by hmmmix-hard with initial clustering by k-medoids (using G = 6, . . . , 14 clusters). To get a smooth curve, we averaged the results over 30 trials. If BIC was a good way to recover the true number of groups, we would expect the solid curve to peak at G = 10.  17  Chapter 3  Generalization of hmmmix Inference is difficult in hmmmix model mostly due to the fact that the hidden chains are coupled because of explaining away (see figure 3.1). This makes learning intractable unless we are willing to make concessions. [Sha08] imputes values to the hidden G1:P to break the ties between the hidden chains. With the chains uncoupled, the Viterbi algorithm is used to select the most likely sequence of states for hidden chains separately. The hmmmix-hard model gets good results when used with a decent initialization of patients assignments, but it tends to get stuck in very bad local maxima when not initialized properly. In this chapter, we will show how we can use variational methods to make it more flexible and better in some cases (but never worse). The actual benchmarks are found in the next chapter. An interesting consequence of our approach is that we will be dealing with soft patient assignments. The difference between hard patients assignments and soft patients assignments is very similar to the case of k-means versus Gaussian mixture models learned by EM. A very insightful paper on variational methods used for a related model called factorial hidden Markov chains is [GJ97]. We are dealing with a model a bit more involved than in [GJ97] so we have to re-derive the formulas for our particular case. At the core of the hmmmix-soft algorithm is a variational lower bound for the log-likelihood of the data that we want to maximize. This is a technique that maximizes the log-likelihood itself, but we are more interested in the variational approximations to the hidden chains because they are supposed to represent the underlying biological phenomenon that we are studying. The final log-likelihood quantity whose value we optimize marginalizes over these hidden chains. Having optimal values for the transition matrices A1:G is not half as interesting as knowing where the state transitions are located. Again, we can draw a comparison with the case of Gaussian mixture models trained with EM where we are very interested in seeing which data points lie in which cluster, but we might not even care about the final log-likelihood. The relationships between the different quantities optimized are discussed in section 2.3. A crucial optimization step in section 3.3.2 requires a good eye to identify certain quantities 18  Figure 3.1: If we take a slice at time step t of the graphical model found in figure 1.3, we can see how the hidden chains Mt1 , Mtg , MtG are dependent given the observed value Ytp . We omitted parameters here to simplify the graph. This coupling of the hidden chains is what makes exact inference intractable. as being the log-likelihood of a hidden Markov chain with given observations. To simplify the exposition of our model, we felt that it would be appropriate to devote section 3.1 to derive certain formulas in a context where we have only one chain. That way, it will be easier to untangle expressions involving G chains. Section 3.1 should simply be viewed a collection of useful lemmas. Section 3.2 sets up the context for our model. In section 3.3 we derive the formulas for updating the parameters. We show in section 3.4 one possible way to get good imputed values from the variational approximations. In section 3.5, we sketch a map of the approximations that were used to make the original model tractable and discuss in more depths those pertaining 1:P to the parameters µ1:P 1:K , λ1:K .  3.1  Note on forwards-backwards  Let M1:T be a Markov chain with K states, with transition matrix A and initial state priors given by a multinomial of parameter π. Moreover, for all hidden states Mt at time t ∈ {1, . . . , T }, let Rt [k] be the log-likelihood of the observed data for state k ∈ {1, . . . , K}. This encapsulation of the log-likelihood values enables us to focus on the chains without thinking about the rest of the model at this point. The complete data log-likelihood for M1:T can be written as T  K  K  I(Mt = k)Rt (k) + t=1 k=1  T  K  K  I(M1 = k) log πk + k=1  I(Mt−1 = j, Mt = k) log A(j, k). t=1 j=1 k=1  Later on we will encounter this expression integrated over the whole state space M1:T with a probability distribution h(M1:T ). It is important to be able to see that, if we want to 19  maximize with respect to h(M1:T ) the following quantity : T  K  h(M1:T )  I(Mt = k)Rt (k) + const1 t=1 k=1  +    K  h(M1:T )   T  K    K  I(Mt−1 = j, Mt = k) log A(j, k)  I(M1 = k) log πk + t=1 j=1 k=1  k=1  −  h(M1:T ) log h(M1:T ) + const2 , (3.1)  that maximal value is achieved when h(M1:T ) follows the same distribution as the one induced on a first-order Markov chain by observations with log-likelihood values Rt [k]. This is true when h(M1:T ) is free to be any joint distribution on (M1 , . . . , MT ) ∈ {1, . . . , K}T . This follows from the general principle that, for any given distribution P , the unique distribution Q that minimizes KL(Q P ) is Q = P if we don’t impose any constraints on Q. The optimal h(M1:T ) can be obtained by running the forwards-backwards algorithm (see [Bis06] chap 13.2). It can be expressed analytically with the help of a vector of marginals h(M1 ) and a set of T − 1 square matrices called the two-slice marginals that are denoted by ξt−1,t (j, k) = h(Mt−1 = j, Mt = k). We can sample from h(M1:T ) easily with those quantities and we get the smoothed marginals h(Mt ) as a bonus from the forwards-backwards algorithm. The transition matrix A can have a conjugate prior that takes the form of pseudocounts APC (j, k) ≥ 0 with a pdf proportional to K  K  A(j, k)APC (j,k) .  p(A|APC (j, k)) ∝ j=1 k=1  This can be achieved more formally by taking the rows of A to be distributed independently according to Dirichlet(APC (j, 1), . . . , APC (j, K)) for rows j = 1, . . . , K. The MLE with respect to A of equation (3.1) is given by the forwards-backwards algorithm as an average of the twoslice marginals ξt−1,t . We can find the MAP estimate of A with pseudocounts simply by adding those to the two-slice marginals before renormalizing the rows (to get a proper transition matrix). AˆMAP (j, k) = k  T t=2 ξt−1,t (j, k) + APC (j, k) T t=2 ξt−1,t (j, k ) + APC (j, k  )  (3.2)  These are the quantities that maximize (3.1) with respect to A when all other variables are fixed. A similar reasoning shows that the MLE for the initial state priors π1:K is given by the normalized smoothed marginals M1 (1:K). 20  3.2  Notation and assumptions for variational methods  The most important hidden quantities in our model are the patients assignments G1:P , the 1:G and the hidden states Z 1:P of the patients. We will use f to refer to the original chains M1:T 1:T pdf from the hmmmix model and h to refer to the pdf of the variational approximation to the hidden variables. Thus, we will be interested in maximizing a quantity of the form L(h) = h(W ) log f (Y, W |θ, A, π)dW − h(W ) log h(W ) where W represent here the hidden variables, Y are the observed variables and θ, A, π are the parameters of the function f . The tractability of our model comes from restricting the possibilities of h to a subfamily 1:G , G1:P ) = h(M 1:G )h(G1:P ) holds. This allows us to maximize where the factorization h(M1:T 1:T 1:G ) and h(G1:P ). Moreover, we will assume that we can further factorize the iteratively h(M1:T g 1:G ) = 1:P ) = p factors as h(M1:T g h(M1:T ) and h(G p h(G ), both of those assumptions being quite reasonable since patients are not supposed to be related and neither should the CNA g profiles be. We will not assume that h(M1:T ) = t h(Mtg ) but, for purposes other than for g estimating the transition matrices Ag and initial state priors π g , the distributions h(M1:T ) will g propagate through the model in the same way as if we assumed that h(M1:T ) = t h(Mtg ). 1:P from the model. This We decided to marginalize out analytically the hidden variables Z1:T 1:P ), but Z 1:P still play a part in the formulas. is why we don’t have a variational term h(Z1:T 1:T The expected complete data log-likelihood to be maximized is L(h) =  1:G 1:P 1:G 1:G h(M1:T )h(G1:P ) log f (Y1:T , M1:T , G1:P , A1:G , π 1:G , θ)dG1:P dM1:T −  h log h. (3.3)  where we use θ as shorthand for  1:P (µ1:P 1:K , λ1:K )  and where f decomposes as  1:P 1:G 1:P 1:G 1:G 1:G 1:G f (Y1:T , M1:T , G1:P , A1:G , π 1:G , θ) =f (Y1:T |M1:T , G1:P , θ)f (M1:T |A , π )f (G1:P )  (3.4)  1:P 1:P 1:P 1:P 1:P f (A1:G |APC )f (π 1:G )f (µ1:P 1:K , λ1:K |m1:K , η1:K , γ1:K , S1:K ).  It will be convenient also to let Cgp be the probability under h(G1:P ) that patient p is assigned to cluster g. This is well-defined because we assume that the assignments to clusters between p patients are independent. It also means that G g=1 Cg = 1 for all patients p. 1:G ) log h(M 1:G ) that we will denote by H[h(M 1:G )] Some formulas contain entropy terms − h(M1:T 1:T 1:T to lighten the notation. Due to assumptions in our variational approximation, we have that G 1:G H[h(M1:T )]  = g=1  g H[h(M1:T )]  P  and  1:P  H[h(G  H[h(Gp )]  )] =  (3.5)  p=1  21  3.3 3.3.1  Training Updates and initialization  1:G ), h(G1:P ) We train our model by updating iteratively the variational approximations h(M1:T 1:P and the model parameters A1:G , π 1:G , µ1:P 1:K , λ1:K . Due to the particular structure of the hmmmix model (as represented in figure 1.3), the parameters A1:G , π 1:G and the parameters 1:P 1:G 1:P ) are fixed. Traditional variational EM µ1:P 1:K , λ1:K are independent when h(M1:T ) and h(G 1:P would have us update all the parameters A1:G , π 1:G , µ1:P 1:K , λ1:K simultaneously, but for our par1:G ) and A1:G , π 1:G together. ticular problem it is more natural to perform the updates to h(M1:T We illustrate in figure 3.2 in what order the updates are performed.  Figure 3.2: Our algorithm updates variational approximations and parameters iteratively. This picture shows an overview of the order in which the updates are performed. Within rectangular boxes, we have cycles of updates that should be performed until the quantities stabilize within a certain tolerance. To initialize the algorithm, we start by assigning patients to clusters using k-medoids or by setting h(G1:P ) to uniform random values. Throughout the experiments that we did, results were always better with k-medoids than with a random initialization. The extent to which the k-medoids initializations are better depends on the particular case, though. We set the 1:P 1:P means µ1:P 1:K = m1:K and let the precisions λ1:K be relatively small to give the model some initial slack. We then start the updating scheme with the hidden chains and we iterate as shown in figure 3.2 until convergence. If we start with good initialization values for the patients clustering (e.g. from k-medoids), it is important that we do not lose these values by updating them prematurely while we still 1:G ) and θ. In that particular case, we recommend skipping a few have crude values of h(M1:T updates to h(G1:P ) while the rest of the model adjusts itself to the initial patient assignments. The arrows from figure 3.2 suggest one possible order in which we can do the updates. We 22  have tried a few variants but they are roughly equivalent.  3.3.2  Hidden chains  1:G ), we can omit the terms not involving If we focus on maximizing (3.3) with respect to h(M1:T 1:G to get M1:T 1:G 1:P 1:G 1:G h(M1:T )h(G1:P ) log f (Y1:T |M1:T , G1:P , θ)dG1:P dM1:T  +  1:G 1:G 1:G 1:G 1:G 1:G h(M1:T )h(G1:P ) log f (M1:T |A , π )dG1:P dM1:T + H[h(M1:T )].  (3.6)  We start by reformulating the first term in the previous expression (3.6) to separate as much as possible the contributions of the G groups. We have that 1:G 1:P 1:G 1:G h(M1:T )h(G1:P ) log f (Y1:T |M1:T , G1:P , θ)dG1:P dM1:T  =  1:G h(M1:T )  1:G log f (Yt1:P |Mt1:G , G1:P , θ)dG1:P dM1:T  h(G1:P )  (3.7) (3.8)  t  =  1:G h(M1:T )  1:G I(Gp = g) log f (Ytp |Mtg , Gp = g, θ)dG1:P dM1:T  h(G1:P ) p  t  1:G Cgp log f (Ytp |Mtg , Gp = g, θ)dG1:P dM1:T  1:G h(M1:T )  =  t  =  g  (3.9)  g  (3.10)  p 1:G Rt,g [Mtg ]dM1:T  1:G h(M1:T ) t  (3.11)  g  where, in that last line, we defined Rt,g [Mtg ] =  Cgp log f (Ytp |Mtg , Gp = g, θ).  (3.12)  p  Continuing on (3.11), we get 1:G Rt,g [Mtg ]dM1:T =  1:G h(M1:T ) t  g  g h(M1:T ) g  g Rt,g [Mtg ]dM1:T .  (3.13)  t  We now come back to the original expression (3.6) using (3.13) and the entropy term decomposition (3.5) to rewrite our formula of interest (3.6) as g h(M1:T ) g  g g Rt,g [Mtg ] + log f (M1:T |Ag , π g ) dM1:T −  g g g h(M1:T ) log h(M1:T )dM1:T .  t  (3.14) 23  g Because the distributions f M1:T |Ag , π g from the hmmmix model are Markov chains, we can see that (3.14) is essentially a sum of G independent hidden Markov chains as found in section (3.1). It follows that, to maximize our variational lower bound (3.3) with respect to 1:G ), we should run forwards-backwards on the chains independently for g = 1, . . . , G with h(M1:T Rt,g [Mtg = 1, . . . , K] as observation log-likelihoods. The difference from “regular” forwardsbackwards is that we use these quantities Rt,g [k] derived from the current variational approximations h(G1:P ). g When implementing the algorithm, we represent internally h(M1:T ) as a collection of two-slice marginals ξt−1,t (1:K, 1:K) along with a vector for the smoothed distribution of the first state h(M1 ).  It can easily be shown that, to maximize the original variational lower bound (3.3) with respect to (A1:G , π 1:G ), we can use the two-slice marginals and h(M1 ) to find the new MLE values. We can add pseudocounts to A1:G by proceeding as explained in section 3.1. 1:G ) and (A1:G , π 1:G ) because they deWe alternate between optimization with respect to h(M1:T pend on each other. We pick a tolerance threshold to determine when the values have stabilized to avoid excessive computations in the first steps in the algorithm. Once we are satisfied with 1:G ) the values MLE values of (A1:G , π 1:G ), we no longer need the two-slice marginals of h(M1:T and it is sufficient to keep only the smoothed marginals {h(Mtg )|t = 1 . . . T, g = 1 . . . G} as a K-by-T-by-G matrix.  3.3.3  Partial patients assignments  We now turn to optimizing our lower bound L(h) with respect to the distributions h(G1:P ) = p p h(G ). As the variational parameters h(Gp ) are essentially parameters for multinomial distributions (of one draw), we can represent them by values Cgp corresponding to the weight of the partial assignment of patient p in group g. Dropping all unrelated terms from (3.3), we get L(h) = =  1:G 1:P 1:G 1:G h(M1:T )h(G1:P ) log f (Y1:T |M1:T , G1:P , θ)f (G1:P )dM1:T dG1:P + H G1:P  h(G1:P )  (3.15)  1:G 1:P 1:G 1:G h(M1:T ) log f Y1:T |M1:T , G1:P , θ dM1:T + log f (G1:P ) dG1:P + H G1:P  (3.16) =  h(G1:P )  p g I(Gp = g) log f Y1:T |M1:T , Gp = g, θ  1:G h(M1:T ) p  +  g  h(G1:P ) log f (G1:P ) + H G1:P  (3.17) 24  =  g p 1:G h(M1:T )I(Gp = g) log f Y1:T |M1:T , Gp = g, θ  h(G1:P ) p  g  +  (3.18)  g p g h(M1:T ) log f Y1:T |M1:T , Gp = g, θ  h(Gp )  =  h(G1:P ) log f (G1:P ) + H G1:P  p  g  h(Gp ) log f (Gp ) +  + p  H (Gp )  (3.19)  p  There are different ways to handle the h(Gp ) log f (Gp ) terms. The simplest of those is to use a uniform prior that allows us to ignore them. Another possibility is to take a Dirichlet(τ, . . . , τ ) prior on f (Gp ). This prior is such that we have, for every p, that H (Gp ) +  h(Gp ) log f (Gp ) = −  Cgp log Cgp − g  Cgp (τ − 1) log Cgp + cst(τ ) g  p  = τ H (G ) + cst(τ )  (3.20)  With this prior, we develop (3.19) further. It can be written as a sum of P independent expressions, so the maximization problem is solved separately for the many h(Gp ). For every patient p, we get expressions of the form h(Gp ) log(σgp ) −  h(Gp ) log h(Gp )dGp  (3.21)  where g p g g h(M1:T ) log f Y1:T |M1:T , Gp = g, θ dM1:T  σgp = exp τ −1  .  (3.22)  The τ −1 in 3.22 comes from the τ coefficient in (3.20) by which we can divide the whole of expression (3.19). It follows that the desired distributions of h(Gp ) are softmaxima scaled by a factor of τ −1 > 0. A stronger prior belief that cluster assignments should be uniform (corresponding to a larger τ ) translates into more uniform softmaxima terms due to lighter evidence (being divided by τ ). The updated quantities for h(G1:P ) are given by Cgp =  g  =  exp τ −1  exp τ −1 g  g p g g h(M1:T ) log f Y1:T |M1:T , Gp = g, θ dM1:T  exp τ −1  exp τ −1  T t=1 T t=1  g p g g h(M1:T ) log f Y1:T |M1:T , Gp = g , θ dM1:T g K k=1 h(Mt  p g = k) log f Y1:T |M1:T , Gp = g, θ  g K k=1 h(Mt  p = k) log f Y1:T |Mtg , Gp = g , θ  (3.23)  (3.24)  25  The uniform prior on f (Gp ) corresponds to picking τ = 1. By adding a prior we are changing the hmmmix model slightly. We discuss this further in section 4.9. This τ parameter should be seen as a tool to add flexibility to the model. It can be used to tune the convergence speed of the whole algorithm. Note also that we only need the smoothed distributions h(Mtg ) to update the Cgp and we don’t use the two-slice marginals. This has to do with the fact that the 1:P are not connected as Markov chains. hidden states Z1:T  3.3.4  Observation parameters  1:P When comes to the time to maximize L(h) with respect to θ = {µ1:P 1:K , λ1:K }, we use the priors 1:P in a slightly from section 3.3.1 of [Arc05]. We decided to handle the hidden variables Z1:T p different way than [Arc05], though, by integrating over Zt when evaluating the quantity Rtp [k]. As [Arc05] covers a variety of models with different flavors, we define everything explicitly in here. The technique used to get the parameter update formulas is not part of our contributions, but it is still necessary for us to give out the details of the work for our particular case.  We want to maximize the component of the lower bound of L(h) that depends on θ. This quantity is L(θ) = = ≥  1:G 1:P 1:G h(M1:T )h(G1:P ) log f (Y1:T |M1:T , G1:P , θ)f (θ)  1:G h(M1:T )h(G1:P ) log  1:P 1:P 1:P 1:G 1:P f (Y1:T |Z1:T , θ)f (Z1:T |M1:T , G1:P )dZ1:T + log f (θ)  1:G 1:P 1:G 1:G h(M1:T )h(G1:P )f (Z1:T |M1:T , G1:P )dM1:T dG1:P  (3.25)  1:P 1:P 1:P log f (Y1:T |Z1:T , θ)f (θ) dZ1:T  (3.26) Now we will maximize a lower bound on L(θ) instead of maximizing that quantity itself. A 1:P ) to legitimate alternative would have been to start out with a variational parameter h(Z1:T 1:G )h(G1:P ), but we opted against this approach. In any case, the effects of the go along h(M1:T relaxation from (3.25) to (3.26) are limited to the optimization of the parameter θ. This is the main reason why our final formulas are slightly different from those of [Arc05]. 1:P The derivations for the update formulas of the parameters µ1:P 1:K , λ1:K are not too insightful. We provide them in appendix A and give the essential formulas here. These formulas involve 1:P temporary variables u ¯pt (k) that can be discarded once we settle on a choice of µ1:P 1:K , λ1:K . We select a tolerance threshold and we iteratively update the quantities according to the following formulas until they stabilize. From our experience, this generally took at most a dozen iterations. νkp − 1 u ¯pt (k) = p (3.27) 2 λk Ytp − µpk + νkp  26  p ¯pt (k)Ytp + ηkp mpk t ρt (k) u p ¯pt (k) + ηkp t ρt (k) u 2 p ¯pt (k) Ytp − µpk + ηkp µpk − t ρt (k) u p p t ρt (k) + 2γk − 1  µpk =  (λpk )−1  3.4  =  (3.28) mpk  2  + 2Skp  (3.29)  Projecting to hard assignments  With hmmmix-soft we are maximizing the variational lower bound L(h) (equation (3.3)) with 1:G ), h(G1:P ) respect to the parameters A1:G , π 1:G , θ. We get variational approximations h(M1:T as well as MAP estimates for A1:G , π 1:G , θ. With hmmmix-hard we get imputed values for the 1:G and patient assignments G1:P . There are a few simple methods that can be used chains M1:T to get hard values from the variational approximations, but not all of them are equally good. We don’t necessarily want to get those values, so the method explained here is not really the “final step” of hmmmix-soft, but rather just something that could be done if desired. The natural way to get hard values instead of a posterior over chains is to use the Viterbi algorithm like in hmmmix-hard instead of the forwards-backwards used by hmmmix-soft. The quantities Rt,g [Mtg = k] (equation 3.12)) can be used by the Viterbi algorithm in the same way that they are used by forwards-backwards as log-likelihoods of observations. With the 1:G obtained, we then use equation (3.24) to compute the quantities C p . We assign values of M1:T g p each patient p to the cluster g = argmaxg Cg . The code from the regular hmmmix-soft updates 1:G values if we encode them as 1-of-K binary vectors that can be reused with the hard M1:T 1:G ) term. essentially represent a point-mass distribution taking the role of the usual h(M1:T 1:G ,G1:P obtained in this fashion are not guaranteed to be good maximizers of The values M1:T  max p(Y, M, G, θ, A, π),  M  G  A,π,θ  (3.30)  and that quantity could probably be improved by a few rounds of ICM. If we launched hmmmix-hard again at this point, the whole hmmmix-soft procedure would serve as some kind of initialization method for hmmmix-hard. We compare in section 4.1 the values of (3.30) achieved by hmmmix-hard to those obtained by this method after running hmmmix-soft. Experimentally, those obtained with hmmmix-hard are higher than those from hmmmix-soft, as one might expect.  3.5  Flow of approximations  On top of the original assumptions about the factorization of our variational distribution g 1:G , G1:P ) = h(M 1:G )h(G1:P ) = p h(M1:T g p h(M1:T )h(G ), we used a few heuristics to learn 1:T 27  1:P the parameters θ = µ1:P 1:K , λ1:K . An overview of all the approximations used is shown in figure 3.3.  Figure 3.3: The relations between the different parts of the algorithm in terms of approximations. We never showed that the values of u ¯pt (k), µpk , λpk that we get by iterative substitutions in section 3.3.4 were a maximum and not an minimum. However, we can always compute the value of either (3.25) or (3.26) to determine if we want to accept a proposed update or to reject it. In our implementation, we compute the values of f (Ytp |Ztp = k, θ) and store them in a matrix of size (K,T,P). By storing ρpt (k) also in a matrix of size (K,T,P), it relatively straightforward to evaluated the two quantities used as safeguards against bad parameter updates. The original log-likelihood lowerbound (3.25) is given by sum(sum(log(sum(rho KTP .* YgivenZ KTP,1))))  (3.31)  and the lower bound (3.26) on (3.25) is given by sum(sum(sum(rho KTP .* log(YgivenZ KTP)))).  (3.32)  If we trust the parameter updates, we can speed up the algorithm greatly by not computing the log-likelihood. As explained in section 4.7 and confirmed by our runtime profiling, the performance bottleneck comes from the evaluations of p(Ytp |Ztp = k, θ).  28  Chapter 4  Results and implementation In this chapter we present three experiments done to evaluate the performance of hmmmixsoft. We then analyze the complexity of the algorithm and discuss the potential hardware issues that arise when the size of datasets exceeds the amount of available memory. We point out how the algorithm could be parallelized and we come back to the topic of the free energy scaling constant first introduced in section 3.3.3.  4.1  A minimum description length experiment  We argued in section 2.5 that BIC was not a good way to find the number of clusters as it fails to take consideration the fact that the chains that model CNAs have rare state transitions. The minimum description length (MDL) criterion was suggested as alternative. We present here an experiment done with synthetic data sampled from the hmmmix model itself. Like in section 2.5, we are using using G = 10 chains, P = 100 patients and sequences of length T = 10 000. We supply hmmmix-hard and hmmmix-soft with the true hyperparameters with which the data was generated. By the MDL principle, we should seek to minimize the quantity 1:P 1:G 1:P 1:G 1:G 1:G 1:P − log2 f (Y1:T |M1:T , G1:P , µ1:P , π , G1:P , µ1:P 1:K , λ1:K ) + L(M1:T , A 1:K , λ1:K )  where L(M, A, π, G, µ, λ) is the code length necessary to encode the parameters given the known fixed hyperparameters. As pointed out by [HY01], we don’t need to describe the coding scheme explicitly to use the MDL principle. The cost of sending an event x with probability P (x) is − log2 P (x). This means that we want to minimize 1:P 1:G 1:P 1:G 1:G 1:G 1:P − log f (Y1:T |M1:T , G1:P , µ1:P , π , G1:P , µ1:P 1:K , λ1:K ) − log f (M1:T , A 1:K , λ1:K | hyperparams).  No prior is defined in the original hmmmix model for the patient assignments G1:P so we exclude log f (G1:P ) from the code length. We also exclude the term log f (A, π|pseudocounts) for practical reasons (after verifying that its contribution was negligible). Thus, we compare hmmmix-hard and hmmmix-soft on their ability to maximize the quantity 1:P 1:G 1:G 1:G 1:G log f (Y1:T |M1:T , G1:P , θ) + log f (M1:T |A , π )  29  1:P 1:P 1:P 1:P 1:P + log f (µ1:P 1:K , λ1:K |m1:K , η1:K , γ1:K , S1:K ).  (4.1)  1:G and patients We use the method described in section 3.4 to get hard values for the chains M1:T 1:G ) and h(G1:P ). assignments G1:P from hmmmix-soft instead of variational distributions h(M1:T We are not only interested in comparing the two methods here, but we are also interested in seeing if the MDL values obtained by each method can be used to find the true number of clusters used to generate the data.  For both hmmmix-hard and hmmmix-soft, we use a random initialization of the patients (and 1:P , in the case of hmmmix-hard ) as well as a k-medoids initialization. The hidden states Z1:T random initializations are shown as solid curves in figure 4.1 while the k-medoids variants are shown as dotted curves. We plot the values of the quantity (4.1) obtained by each method. The values obtained by hmmmix-hard are significantly higher than those obtained by hmmmixsoft. In all cases, we get an improvement by using k-medoids to initialize the patient assignments. In the case of figure 4.1(a), it’s quite clear that we can identify the true number of clusters with the results of hmmmix-hard or hmmmix-soft with k-medoids. The results are more ambiguous in the case of G = 10 and they would very unreliable in all cases if we didn’t average them over 30 trials to get smooth curves.  4.2  Gap statistic  The gap statistic introduced by [TWH01] is another way to estimate the true number of clusters. One of its special features is its ability to detect when there is no need for more than one cluster, but this is not the reason why we are using it. The results that we got with the gap statistic are not convincing enough to justify a detailed exposition of the theory. Instead, we present in appendix B the method used in our particular case and we refer the reader to the original [TWH01] or to [Koe08] for a good summary. At the core of the gap statistic is the notion of within cluster spread spread(E) =  1 2|E|  dist(x, y).  (4.2)  x,y∈E  For a given clustering with C ∈ {1, . . . , CMAX } clusters, we define the spread of the final set of clusters as the sum of the spreads of the individual clusters. We then compare the results for all the values of C studied. We should expect spreads values to decrease fast as we increase C up to the point when C = G, where G is the real number of clusters. After C > G, the decrease should be less significant because there are no more clusters with far away points to break up. 30  6  MDL penalized loglikelihood  −2  x 10  −2.1 hmmmix−soft (random init) hmmmix−soft (k−medoids init) hmmmix−hard (random init) hmmmix−hard (k−medoids init)  −2.2 −2.3 −2.4 −2.5 −2.6 4  6 8 10 12 number of groups used for clustering  14  (a) true number of clusters G = 6 6  MDL penalized loglikelihood  −2.1  x 10  −2.2 −2.3 −2.4  hmmmix−soft (random init) hmmmix−soft (k−medoids init) hmmmix−hard (random init) hmmmix−hard (k−medoids init)  −2.5 −2.6 −2.7 −2.8 4  6 8 10 12 number of groups used for clustering  14  (b) true number of clusters G = 10  Figure 4.1: We show here the results for the log-likelihood of the data penalized by the MDL term (see equation 4.1). We use G = 6, 10 hidden chains respectively to generate the data and distributed P = 100 patients among those G clusters. The length of the sequences is T = 10 000. In green we have the results from hmmmix-soft and in black we have those from hmmmix-hard. The solid lines represent the values obtained with a random initialization while the dotted curves represent those obtained by k-medoids. These results come from averaging 30 trials. Ideally, we would want the curves to grow monotonously with the number of groups used (clusters) up until the true value of G. At that point, we would like the curves should be flat, indicating that adding further clusters will not help us to model the data.  31  Due to the very prohibitive cost of this method, we tested it only on hmmmix-soft with random initializations, using the datasets generated in the same way as in 4.1 but with T = 1000 instead of T = 10 000. We generated datasets with G = 10 clusters each and we explored values of C = 6, . . . , 14. Out of 28 trials, the number of clusters reported by the gap statistic was • 6 : 6 occurrences • 9 : 21 occurrences • 12 : 1 occurrence. Starting hmmmix-soft with C clusters doesn’t guarantee that we end up with C clusters being used in the final patient assignments, though. Any convincing experiment to validate the gap statistic would have to be done on a larger scale and the cost is just too great. The 9 clusters estimate here is close to the target value 10, but if the method was valid we would expect wrong guesses to be more uniform than this. The fact that we obtained 6 occurrences is suspicious because this was the smallest value of C explored.  4.3  Benchmark with synthetic data not from hmmmix  In [Sha08] hmmmix-hard is compared to the current state-of-the-art methods using a benchmark test with artificial data. This data is not generated from the hmmmix model itself (like in sections 4.1 and 4.2), but comes from real aCGH data into which segments of the data has been modified to look like CNAs. We reproduced the experiment from [Sha08] with hmmmix-hard and ran hmmmix-soft on the same data to see how our method would compare. As mentioned in section 2.4, one of the limitations of hmmmix-hard is its reliance on good cluster initialization for the patients. For this reason, we have spent most of our efforts trying to beat hmmmix-hard on the synthetic data benchmark without having to resort to a k-medoids initialization that knows the real number of clusters a priori. This turns out to be really hard with the data used except in cases where k-medoids fails to produce anything useful. The measure of success chosen for this experiment is the Jaccard coefficient (see appendix B). It requires knowledge of the true cluster assignments to be computed and this is why it can only be used on artificial data. The Jaccard coefficient is always in [0, 1], higher being better, and a value of 1 can be achieved if and only if we find cluster assignments equivalent to the  32  true clustering. The order or labels of the clusters does not affect the result, but using the wrong number of clusters gives a smaller Jaccard coefficient. We used the same data as [Sha08] and explored the combinations of values from G = 3, 5, 10 and L = 25, 50, 75 where L corresponds to the length of the mutations introduced. As hmmmix-hard performs quite well in most of the cases with G = 3, 5, we decided to focus on G = 10. Our results can be summarized by table 4.3. Table 4.1: Comparing hmmmix-soft to hmmmix-hard using Jaccard coefficient. The values are averaged over 50 generated datasets and the standard deviations are included to give an idea of how spread results are. L=25 L=50 L=75 k-medoids initialization 0.77 ± 0.11 0.33 ± 0.01 0.15 ± 0.03 hmmmix-hard 0.90 ± 0.13 0.61 ± 0.17 0.17 ± 0.08 hmmmix-soft 0.93 ± 0.11 0.58 ± 0.15 0.35 ± 0.09  We have in the first row the value of the Jaccard coefficient with the initial clustering obtained from k-medoids. We can see that it’s already large in the case of L = 25 and that it’s almost worthless in the case of L = 75. With a completely random initialization of 10 clusters, the Jaccard coefficient is a little bit above 0.10. In the case of L = 75, we can see that hmmmix-soft can improve on the initial clustering while hmmmix-hard cannot. These results are averaged on 50 tries so the difference is statistically significant, but one could argue that 0.35 is still relatively bad. One of the difficulties that we encountered here was that the Jaccard coefficient does not care about the fact that a soft clustering might be desirable or even very informative. Instead of using the method from section 3.4 to get hard patient assignments, we merged duplicate clusters obtained by hmmmix-soft and then assigned patients to their most likely cluster. Some other scoring criterion could be used to deal with partial patient assignments, but the purpose of this experiment was to show that we can perform well in the benchmark test from [Sha08].  4.4  Comparison with hmmmix-hard on FL data  It is difficult to compare quantitatively the performance of hmmmix-soft and hmmmix-hard on real data because often there are no target values available as ground truth. Not only are we missing information, but the parameters from the models don’t necessarily correspond to any real feature. One way to proceed is to predict biological phenomena from the models and 33  verify with experts if your predictions are correct (or useful). The is one of ways in which [Sha08] justified the hmmmix model. With our hmmmix-soft, we would like to be able to achieve results similar to those of hmmmixhard with real data in terms of imputing hidden values. Our measure of success is how close we can get to the results presented in section 5.4.1 of [Sha08] using the same dataset. The dataset used for this experiment is the follicular lymphoma dataset from [CSS+ 09] that has P = 106 patients, with sequences of length T = 24 881 spanning over 22 chromosomes. To 1:P determine the number of clusters, [Sha08] used weighted k-medoids on imputed values of Z1:T to maximize the silhouette coefficient (they got G = 6 clusters). Their original results are included here in figure 4.2.  1:P (a) initial imputed values of Z1:T  1:P (b) final values of Z1:T after hmmmix-hard  Figure 4.2: The original results from [Sha08] for hmmmix-hard on the follicular lymphoma dataset. The patients are reordered for easier visualization. The green segments correspond to a higher copy number while red segments correspond to deletions. The results that we get from the hmmmix-soft algorithm naturally depend on the choices that we make for the free parameters in the model. In particular, choosing αL , αN , αG to be higher forces the hidden chains to “explain” the data better, but it undermines the smoothing effect that the transition matrices are supposed to have in the chains. We experimented with a few different sets of hyperparameters and in figure 4.3 we show one of the closest matches (visually) to the results in [Sha08]. We used a special way to cluster the patients that we describe in section 4.5. We used partial cluster assignments and most of the weight was distributed among 4 main clusters. Although the results shown in figure 4.3 are somewhat similar to those of 4.2(a), we are missing an important CNA segment in chromosome 1.  34  Figure 4.3: Final results for the follicular lymphoma dataset after running hmmmix-soft and 1:G , G1:P . The profiles identified are displayed on top and the best deriving hard values for M1:T 1:P values of Z1:T are shown below for every patient. The patients have been reordered so that patients from the same cluster are listed consecutively.  35  4.5  Alternative distance-based clustering  One of the goals of this thesis is to see what can be done with the hmmmix model once we can use partial patients assignments. For that reason, to get the results from figure 4.3 we didn’t use the k-medoids initialization but we decided instead to see if we could achieve similar results using the classic mixture of Gaussians trained with EM to provide an initial clustering. The 1:P imputed by a method usual k-medoids used in other experiments in done with values of Z1:T 1:P instead to perform the clustering. from [Sha08]. Here we used the observations Y1:T The idea was to get a rough partial clustering that distributed the weights of the patients to multiple clusters in a useful way. We didn’t want patients to be assigned to a single cluster but we didn’t want to have a trivial uniform cluster assignment either. The high dimensionality T of the data tends to make EM assign patients to single clusters. To mitigate the effects of the large T , we introduced a scaling factor to the entropy term similar to the τ from section 3.3.3 (also discussed in section 4.9). To sketch quickly the details of this modification the EM, we refer the reader to the description of the algorithm in chapter 9 of [Bis06]. With p(xn |µk , Σk ) being the likelihood that sample n belongs to Gaussian component k, instead of updating the cluster assignments in the E-step −1 according to γ(znk ) ∝ p(xn |µk , Σk ), we update them with γ(znk ) ∝ p(xn |µk , Σk )τ . Large values of τ spread the cluster responsibilities γ(znk ) more evenly. It’s hard to justify any value for the scaling τ factor a priori, but we investigated values from different orders of magnitude and computed the entropy of the resulting EM clustering distributions (merging duplicate clusters). With the follicular lymphoma experiment of section 4.4 there was a clear and identifiable threshold value at which the patients went from having all their weight assigned to single clusters to being spread around more evenly (but not in a trivial uniform way). That threshold value stayed roughly the same when restarting EM multiple times. The particular value is a property of the data and what worked for the FL dataset might not work elsewhere (the threshold value should change, for example). The cost of running EM on a large dataset is high, but an iteration of EM is cheaper than an iteration of hmmmix-soft itself. If we refrain from doing multiple restarts of EM, it will not be the performance bottleneck. If it’s still somewhat prohibitive, we can perform the initial clustering with EM using only a subsequence of the indices t = 1 : T and we certainly don’t need to iterate until the EM stabilizes within a very small tolerance bound. It would have been possible also to use a mixture of student-t distributions instead of gaussians, but we haven’t explored that alternative. It would be a little more complicated in that case to learn the parameters for the distributions with EM. 36  4.6  Multiple chromosomes  The description of the hmmmix model was simplified a bit here to deal with only one chain for every group/cluster/cohort. When dealing with real data, we have observations associated to multiple chromosomes and we don’t necessarily want to treat them all as one long chain. As mentioned in [Sha08], there are good reasons to believe that the best values for the transition matrices vary from chromosome to chromosome within any given group. Multiple chromosomes can be handled very easily by running the forwards-backwards algorithm separately for each chromosome, keeping track of the parameters MLE (a transition matrix and a vector of beliefs about the initial states) for every chromosome and every group. The complexity analysis from section 4.7 only has terms involving the chain lengths T linearly so we don’t get penalized in terms of computational costs by doing to the forwards-backwards on chromosomes separately. In fact, if we were dealing with an enormous dataset, we could analyze the different chromosomes in parallel when solving for the multi-chromosome equiv1:G ). The update formulas for h(G1:P ) and θ are not even aware of the chain alent of h(M1:T g structure of the hidden Markov chains M1:T . They stay the same when dealing with multiple chromosomes. We can store the data for all the chromosomes one after the other as long as we keep track of the cuts when doing forwards-backwards on the sections.  4.7  Complexity analysis  When analyzing the complexity of each of the components of the algorithm, we have to remember that we are not taking into consideration the number of iterations that they will take to stabilize (within a certain tolerance). Experimentally, we observed that it usually takes a bit less than 10 iterations (update M , update θ, update G, update θ, . . .) to have patients assignments stabilize. By using a large value of τ for the assignment prior (from section 3.3.3), this slowed down the algorithm so that convergence took around 50 such iterations, but we got quantitatively better results (this is just a heuristic rule, though). With this in mind, it is still relevant to analyze the complexity of one such iteration to get a good picture of what is involved. The main loop of the algorithm goes over the following important steps, which correspond to one cycle in the circuit from figure 3.2. • compute the values of Rt,g [k] with equation (3.12) for all chains : O GP T K 2 • do the following until stabilization  37  g – update the chains h(M1:T ) using the values of Rt,g [k], running forwards-backwards 2 : O GT K  – update the parameters Ag , π g for all chains : O GT K 2 g • update the patient assignments h(Gp ) using Rt,g [k] and the current chains h(M1:T ) with equation (3.24): O (GP T K)  • compute ρpt (k) with equation (A.6) : O GP T K 2 • update the parameters θ by updating the following values with formulas (3.27), (3.28), and (3.29) until stabilization : – u ¯pt (k) : O (P T K) – µ1:P 1:K : O (P T K) – λ1:P 1:K : O (P T K) We are usually using K = 3 and G ≈ 10 so the most important terms are P and T . The whole hmmmix-soft algorithm run as O GP T K 2 times the number of iterations that we do externally (usually between 10 and 50). In our experiments, the update steps that involve some indefinite number of iterations internally (to stabilize within a certain tolerance) have never taken more than a dozen iterations. It should also be noted that we need to take logarithms to compute Rt,g [k], but all the other operations involve only basic arithmetic. The runtime profiling confirms that the performance bottleneck comes from the evaluations of p(Ytp |Ztp = k, θ) and Rt,g [k]. With the exception of the possible random initialization, the algorithm is deterministic so there are no hidden costs from random number generation. It is interesting to note that, due to the hard patients assignments in hmmmix-hard, that algorithm can do certain things in O P T K 2 that would take hmmmix-soft O GP T K 2 operations to do. The overall cost of hmmmix-hard is still O GP T K 2 because patient assignments are done by checking how well every patient fits every cluster.  4.8  Scalability  As explained in section 4.7, the hmmmix-soft algorithm has an asymptotic computational cost of O(GP T K 2 ). Our basic implementation has memory requirements proportional to O(GT K 2 + P T K) simply because of the size of the quantities found in the formulas of section 3.3. The algorithm scales linearly in terms of increasing the number of groups, patients or 38  length of the sequences. It would be hard to expect better, but the memory requirements can still be a problem when dealing with very large datasets.  4.8.1  Hardware issues  We ran hmmmix-soft on a dual-core Mac Mini with 2 gigabytes of RAM for various sequence lengths ranging from T = 200 to T = 1, 000, 000. We used artificial data with G = 10 groups and P = 100 patients. In figure 4.4(a) and 4.4(b) we show with the dotted blue curve the execution time for one iteration of the main loop. We can see in the second plot 4.4(b) that the line breaks at T = 80, 000 when Matlab runs out of memory. We ran more tests on a PC with a dual-core AMD 64-bit processor with 4 gigabytes of RAM and 12 gigabytes of swap memory. The goal was to see if the swap could be used to allow the algorithm to handle larger datasets. When running hmmmix-soft on a large dataset with G = 10, P = 100, T = 250 000, the available memory wasn’t enough to satisfy the algorithm and there were a lot of page misses. In fact, the processor was reported to be only running at around 20% efficiency because most of the execution time was spent transferring variables to and from the swap (the swap being on a regular hard drive). One time, Matlab issued an out of memory error while processing a dataset with T = 100 000 even though the swap could easily have handled this. This is clearly unacceptable and the swap memory solution is a temporary fix that will break if we multiply the number of patients or do anything else to increase the datasets by an order of magnitude. After accepting the fact that, by using the swap memory, we were going to do reads and writes to the hard drive during the execution, we wrote a version of the algorithm that keeps nothing in memory except what it needs at a given moment. After splitting the initial dataset of size (P, T ) into P sequences stored on the hard drive, the total memory usage for the algorithm falls to O(T K 2 ). We decided to draw the line there, but we could have gone further and stored the chromosomes separately. To compare the two approaches, we ran the low-memory implementation on the same Mac Mini. The results are again show in figure 4.4 with the solid black curve. We can see in 4.4(a) how the disk I/O operations slow down the execution significantly for short sequences. It becomes more acceptable in 4.4(b) in the context of larger datasets, and it is absolutely necessary in 4.4(c) with huge datasets. The low-memory implementation also reports that the Mac Mini’s two cores are both used to their maximum capacity, unlike the 20% efficiency that we got earlier on by delegating the task of memory management to the kernel. The issue might be more complicated and there might be a way to help Matlab manage its memory better. In any case, we recommend the low-memory implementation for larger datasets. 39  execution time for one loop, in seconds  60 normal memory saver  50 40 30 20 10 0 0  2000  4000 6000 8000 T, length of sequence  10000  execution time for one loop, in seconds  (a) short sequences 600 normal memory saver  500 400 300 200 100 0 0  2  4 6 T, length of sequence  8  10 4  x 10  execution time for one loop, in seconds  (b) T=10,000 to 100,000 7000 memory saver 6000 5000 4000 3000 2000 1000 0 0  2  4 6 T, length of sequence  8  10 5  x 10  (c) T=100,000 to 1,000,000  Figure 4.4: The execution time for the basic implementation of hmmmix-soft (dotted blue curve) compared to a low-memory implementation (solid black curve). 40  With figure 4.4 we can also confirm our asymptotic analysis claiming that the algorithm scales linearly with T . Discussions about hard drive seek times, contiguous memory on the disk, flash hard drives and network file sharing access speeds fall outside of our domain of expertise.  4.8.2  Parallelization  The hmmmix-soft algorithm can be parallelized by at least a factor of G. The updates to the P patients assignments Cgp are independent and so are the updates to the G chains. In theory, the only part of the algorithm that doesn’t lend itself naturally to parallelization is the forwards-backwards algorithm that runs on sequences of length T . Although it seems sequential in nature, it’s still possible to do something to parallelize it when the number of states K is small compared to the number of processors by adapting the approach from [BMR97]. We don’t think it’s worth the trouble, though, considering that the bottleneck in the algorithm is elsewhere and that the sequences are already separated into multiple chromosomes that can be processed in parallel. From a more practical standpoint, the main obstacle to parallelization is the large quantities of data that needs to be passed around. This makes it hard for us to implement a simple parallelization scheme that doesn’t depend on the particular hardware available. Matlab has problems dealing with non-vectorized operations on arrays and it’s impossible to vectorize trivially the forwards-backwards algorithm. By rewriting that part of Sohrab Shah’s code, we gained a 100-fold increase in speed in forwards-backwards, resulting in a ≈ 20-fold increase for hmmmix-hard. The exact factor depends on the size of the data because it determines the characteristics of the new bottleneck. There are various places in the algorithm (e.g. during patient assignments) where we could theoretically speed things up by using only a subset of the full sequence t = 1, . . . , T of data. If we were to do that, though, it would make more sense to incorporate it into the pre-processing step where experts with domain-specific knowledge can point to certain regions where we should draw more samples. To use sequential data with varying step sizes we need to use non-stationary transition matrices in the forward-backwards algorithm. Our implementation of forwards-backwards supports this, but it hasn’t been incorporated into the hmmmix-soft algorithm.  4.9  Free energy variant  We haven’t explained in section 3.3.3 how to choose the constant τ in the Dirichlet (τ, . . . , τ ) prior on the parameters defining the distributions f (Gp ). It really depends on the nature of 41  the data and on the objectives. We find in [Ros98] a very interesting discussion about the use of an entropy term scaled by a factor S −1 to control the formation and fusion of clusters. This is similar to the annealing process in metallurgy where we heat metals just above the re-crystallization temperature and then let the metal cool down in a controlled way. This gets rid of the crystal defects and the internal stresses which they cause [Wik09]. In our case, our value of τ in the prior corresponds exactly to a scaling of S −1 = τ to the entropy terms H (Gp ). Justifying that scaling term as a prior hyperparameter is easy within a Bayesian framework, but tinkering with the entropy term in variational methods is a bit harder to justify as it affects the maximum (in terms of θ) that the original likelihood had. A “safe” thing to try is to start at equilibrium with τ = 1, let τ grow slowly, have the clusters move around and then come back to τ = 1 hoping to get better final clusterings. That isn’t to say that it is justified to change priors as we iterate in EM, but changing the scaling factor of the entropy terms can smooth out the objective function and allow us to escape from local maxima. A compelling example of this is presented in [UN98]. They start with a small value of S < 1 and they let S increase monotonically. At S = 1 they obtain the original objective function, but the final variational parameters obtained are closer to the global optimum than they would have been otherwise. Empirically, in many of the cases where we used the prior τ = 1, the algorithm took less than 10 iterations to converge, assigning many patients to one chain each. Selecting a higher value of τ slowed the convergence but the same final hard assignments often resulted. We didn’t change the value of τ while running hmmmix-soft and preferred to view τ as part of the prior instead of a modified entropy term. If it wasn’t for the success of [UN98] and [Ros98], though, we might not have defined a prior on G1:P as we did in section 3.3.3. Although we haven’t defined a Dirichlet prior on the states of the chains Mt1:G , we could g proceed by adding a scaling factor of S −1 to the entropy terms H M1:T in (3.14). This would have the effect of scaling the Rt,g [k] terms by S as well as raising the transition matrices Ag and initial states priors π g to the power S (renormalizing appropriately). It is also possible to play with the definition of the model to limit this scaling effect to only the Rt,g [Mtg ] terms (without affecting the transition matrices). Smaller values of S result in the chains being less affected by evidence when tuned by forwards-backwards. If we rewrite the complete data 1:G , we get log-likelihood (3.3) with scaled entropy terms on both G1:P and M1:T L(h) =  g 1:G h(M1:T )h(G1:P ) log f (Y, M, G, A, π, θ) + SM H[h(M1:T )] + SG H[h(G1:P )]  (4.3)  One of the approaches tried (to reduce the influence of bad initial clustering) involved having one cluster for every patient and then letting the clusters merge by running hmmmix-soft. 42  The clusters would only merge if given enough incentives by way of scaling the entropy terms sufficiently. In one particular experiment, we observed that smaller values of S (i.e. large τ ) led to better results by having the clusters merge progressively instead of suddenly collapsing from 100 clusters to 13. The conclusions of many experiments were heuristic rules like this that were not really generalizable. If we choose to scale the entropy terms by such factors, to select τ we have to take into consideration the length T of the sequences. We can see in (3.24) how the τ −1 factor is fighting against the sum Tt=1 . A good value of τ when T = 105 might not be good when T = 108 . The same applies to the pseudocounts on the transition matrices. They should be chosen based on how many transitions we expect in the whole sequence.  43  Chapter 5  Conclusion and future work 5.1  Summary  In this thesis we showed how the hmmmix model from [Sha08] can be trained using variational methods. This gives the model more flexibility in terms of allowing partial assignments of patients to clusters and controlling the weight that evidence has on the training process. We studied the effects of using different numbers of clusters, compared k-medoids cluster initialization to random cluster initialization, and tried a few methods to determine the best number of clusters to use. We compared the performance of our approach to that of [Sha08] on synthetic data using their benchmark test and with data sampled from the model. We showed that our method generally leads to better results, with bigger improvements in cases where k-medoids fails to produce good cluster initializations. We also compared the two methods on real data in terms of the visual representation of the segments where mutations occur. Finally, we analyzed the computational complexity and memory usage of our algorithm and showed how a low-memory implementation can be used to handle very large datasets without paying an unreasonable price on performance. We briefly addressed the issue of parallelization and other optimization techniques.  5.2  Future work  Our initial goal was to have a method that can completely do away with the need for good clustering initialization. In most experiments, the results were still better using k-medoids instead of a random initialization. If we had a real dataset on which k-medoids was useless, we could check if hmmmix-soft can get good results. We briefly mentioned the possibility of selecting special subsets of the gene sequences to focus on more promising segments. This is one of the next natural steps and some portions of our code already support non-stationary transition matrices for this. An experiment could be 44  done to quantify the performance losses incurred when using significantly less observations than are available. It would be easier in that case to control the trade-off between the quality of the results and the execution time as determined by the size of the datasets. There is also work to be done to implement a more efficient version of hmmmix-soft that exploits parallelism. Multicore processors are now the norm and one could imagine how hmmmix-soft could be made to run on Amazon’s Elastic Compute Cloud. It would be interesting to know what is actually worth doing in practice given specific hardware. As mentioned in [RIF+ 06], the deletion mutations are on average three times shorter than the higher copy number mutations. It could be possible to pick our model parameters taking that fact into consideration. It would have been possible also to study what set of values for hyperparameters generate data that comes closest to real experimental data. Maybe it would have been possible to use in some way the observation from [RIF+ 06] that the proportion of chromosomes susceptible to CNVs varies from 6% to 19%. These are just examples of possible improvements to the model based on domain knowledge. There are probably many others, and we don’t really know if we’d get significantly better results by complicating the model. One of the good features of the original hmmmix model was that it could discover interesting patterns in an unsupervised way without much assumptions. 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[SR03]  Ruslan Salakhutdinov and Sam Roweis, Adaptive overrelaxed bound optimization methods, Proceedings of the International Conference on Machine Learning, vol. 20, 2003, pp. 664–671. 47  [SRG03]  Ruslan Salakhutdinov, Sam T. Roweis, and Zoubin Ghahramani, Optimization with EM and Expectation-Conjugate-Gradient, 2003.  [TSK05]  P.N. Tan, M. Steinbach, and V. Kumar, Introduction to data mining, AddisonWesley Longman Publishing Co., Inc. Boston, MA, USA, 2005.  [TWH01]  R. Tibshirani, G. Walther, and T. Hastie, Estimating the number of clusters in a data set via the gap statistic, Journal of the Royal Statistical Society. Series B, Statistical Methodology (2001), 411–423.  [UN98]  N. Ueda and R. Nakano, Deterministic annealing EM algorithm, Neural Networks 11 (1998), 271–282.  [Wik09]  Wikipedia, Annealing (metallurgy) — Wikipedia, the free encyclopedia, 2009, [Online; accessed 15-May-2009].  48  Appendix A  Derivation of update formulas for parameters 1:P We show in this section the derivations for the update formulas of the parameters µ1:P 1:K , λ1:K found in section 3.3.4. We start from 1:G 1:P 1:G h(M1:T )h(G1:P ) log f (Y1:T |M1:T , G1:P , θ)f (θ)  L(θ) = = ≥  1:G h(M1:T )h(G1:P ) log  1:P 1:P 1:P 1:G 1:P f (Y1:T |Z1:T , θ)f (Z1:T |M1:T , G1:P )dZ1:T + log f (θ)  1:G 1:P 1:G 1:G h(M1:T )h(G1:P )f (Z1:T |M1:T , G1:P )dM1:T dG1:P  (A.1)  1:P 1:P 1:P log f (Y1:T |Z1:T , θ)f (θ) dZ1:T  (A.2) where we want to maximize (A.2) with respect to  1:P µ1:P 1:K , λ1:K .  For the rest of this section, we will be dealing with the distribution 1:P q(Z1:T )=  1:G 1:P 1:G 1:G h(M1:T )h(G1:P )f (Z1:T |M1:T , G1:P )dM1:T dG1:P  (A.3)  from equation (A.2). We can rewrite (A.2) as an expectation 1:P 1:P |Z1:T , θ)f (θ) . Eq(Z 1:P ) log f (Y1:T 1:T  p 1:P ) = However, as q(Z1:T t p q(Zt ), it will be more convenient to define a special notation for the quantities q(Ztp = k) and use them to write the formulas. Thus, we define :  ρpt (k) :=  1:G 1:G 1:G h(M1:T )h(G1:P )f (Ztp = k|M1:T , G1:P )dM1:T dG1:P  h(Mt1:G )h(G1:P )f (Ztp = k|Mt1:G , G1:P )dMt1:G dG1:P  = G  K  Cgp  = g=1  h(Mtg = j)f (Ztp = k|Mtg = j, Gp = g)  (A.4) (A.5) (A.6)  j=1  As illustrated at line (A.5), these quantities are easily computed from the smoothed marginals 1:G ). We need the partial patients h(Mtg ) because the Ztp are independent given the chains h(M1:T 49  assignments Cgp and the parameters αL , αN , αG that govern f (Ztp = k|Mtg = j, Gp = g) to evaluate ρpt (k). At line (A.6) we find a more “algorithmic” expression. Note that the values of f (Ztp = k|Mtg = j, Gp = g) defined by equation (1.1) don’t depend on the chain, the patient or the time so we can define a K-by-K matrix S(j, k) := f (Ztp = k|Mtg = j, Gp = g) to precompute and cache the values in (A.6). We rewrite our lower bound (A.2) as ρpt (k) log f (Ytp |Ztp = k, µpk , λpk ) + p  k  t  log f (µpk , λpk ) k  (A.7)  p  1:P that we now want to maximize with respect to θ = (µ1:P 1:K , λ1:K ). We will temporarily refer to the first term of (A.7) as A1 and to the second as A2 for convenience. Note that, among all 1:P , η 1:P , m1:P , γ 1:P , S 1:P ), A depends on ν 1:P and A depends on the hyperparameters (ν1:K 1 2 1:K 1:K 1:K 1:K 1:K 1:P , m1:P , γ 1:P ,S 1:P . η1:K 1:K 1:K 1:K  As Archambeau explains in [Arc05], there is no closed form solution for estimating the parameters of a Student-t, but we can always use a EM trick from Liu and Rubin [LR95] involving a latent variable u to estimate µ, λ. We follow that procedure, adopting the notation from [Arc05]. This essentially amounts to decomposing A1 as ρpt (k) log p  k  N  Ytp |µpk , uλpk  t  νkp νkp Gamma u| , 2 2  du  (A.8)  and maximizing instead the quantity ¯pt (k)| ρpt (k) log N Ytp |µpk , u ¯pt (k)λpk + log Gamma u  A∗1 = p  k  t  νkp νkp , 2 2  .  (A.9)  We use u ¯pt (k) to denote the fact that we have such a latent variable u for all patients 1:P , all time steps 1:T and all states 1:K. The prior A2 was defined in section 2.1 to be log N µpk |mpk , precision = λpk ηkp + log Gamma λpk |γkp , Skp .  A2 = k  (A.10)  p  We are now looking for a set of values u ¯pt (k), µpk , λpk for which the derivatives of A∗1 + A2 with respect to all these variables are equal to zero. There are no constraints other than λ1:P 1:K > 0 p p p because the precision is a positive quantity. Dropping all constants not involving µk , λk , u ¯t (k),  50  we get that A∗1 + A2 is equal to ρpt (k) k  t  p  u ¯p (k)λpk 1 log u ¯pt (k)λpk − t Ytp − µpk 2 2  νkp νp p − 1 log u ¯pt (k) − k u ¯ (k) 2 2 t  +  η p λp 1 log λpk − k k µpk − mpk 2 2  + p  k  2  2  + γkp − 1 log λpk − Skp λpk . (A.11)  We now set all the derivatives to zero and see what equalities must hold for us to have a ∂(A∗ +A ) maximum. First, we set ∂ u¯1p (k)2 = 0 : t  p 1 p −1 λk u ¯t (k) − Ytp − µpk 2 2  0 = ρpt (k)  u ¯pt (k)  Then, we set  ∂(A∗1 +A2 ) ∂µpk  =  λpk 2  1 2  +  νkp 2  Ytp −  t  2 2 νp p 2 µk + 2k  −  t p ¯pt (k)λpk Ytp t ρt (k) u p ¯pt (k)λpk t ρt (k) u  µpk = Finally, we set  νkp νp −1 −1 u ¯pt (k) − k 2 2  λpk Ytp − µpk 2  2  νkp − 1  =  λpk Ytp − µpk  +  2  νkp 2  (A.12) (A.13)  + νkp  2ηkp λpk p 2 p p p p ¯t (k)λk Yt − µk (−1) − µk − mpk − u 2 2  ρpt (k) u ¯pt (k)λpk Ytp + ηkp λpk mpk +  0=  =  +  (A.14)  =0: ρpt (k)  0=  νkp −1 2  1 + 2  −1  u ¯pt (k)  2  ∂(A∗1 +A2 ) ∂λpk  ρpt (k) u ¯pt (k)λpk (−1)µpk − ηkp λpk µpk  t p p p + ηk λk mk + ηkp λpk  =  p ¯pt (k)Ytp + ηkp mpk t ρt (k) u p ¯pt (k) + ηkp t ρt (k) u  u ¯pt (k) 2  Ytp − µpk  p t ρt (k)  + 21 (λpk )−1 −  ηkp 2  1 p −1 2 (λk )  µpk − mpk  − 2  (λpk )−1  (A.17)  p p 1 1 t ρt (k) + 2 + (γk 2 p η 2 p 1 p ut (k) Ytp − µpk − 2k t ρt (k) (− 2 )¯  =  p ¯pt (k) t ρt (k) u  2  + γkp − 1 (λpk )−1 − Skp  (λpk )−1  −  (A.16)  = 0 to have :  0=  0=  (A.15)  2  (A.18)  − 1) 2  µpk − mpk  Ytp − µpk + ηkp µpk − mpk p p t ρt (k) + 2γk − 1  2  − Skp  (A.19)  + 2Skp  (A.20)  We iterate a few times updating the quantities in the order (A.14), (A.17), (A.20) until they stabilize. We don’t need to update the values for ρpt (k) as we iterate as they don’t depend on θ. 51  Appendix B  Measures of performance for clusterings B.1  Silhouette coefficient  The silhouette coefficient is a measure of clustering quality. By performing clustering with different numbers of clusters and evaluating the silhouette coefficients, we can pick the best number of clusters as the one that maximizes the silhouette coefficient. For each data point, we compute its silhouette coefficient as si =  bi − ai max(ai , bi )  (B.1)  where ai is the average distance to points in i’s cluster and bi is the minimum average distance to points in another cluster. The silhouette coefficient for a given clustering is the average of the silhouette coefficients for the points : S=  1 N  N  si .  (B.2)  i=1  Values for silhouette coefficients are always found in the interval [−1, 1] and with higher values corresponding to more desirable clustering configurations.  B.2  Jaccard coefficient  The Jaccard coefficient is a measure of clustering quality that requires knowledge of the true assignments. Its value is in the interval [0, 1] and a value of 1 can be achieved iff we learned the true assignments. Following the notation of [TSK05], we take the true clustering and we let C(i, j) be 1 if data points i, j are in the same cluster and 0 otherwise. For the predicted clustering, we let P (i, j) be 1 if data points i, j are in the same cluster and 0 otherwise. We define 52  • f00 : the number of pairs of points (i, j) for which C(i, j) = 0 and P (i, j) = 0 • f01 : the number of pairs of points (i, j) for which C(i, j) = 0 and P (i, j) = 1 • f10 : the number of pairs of points (i, j) for which C(i, j) = 1 and P (i, j) = 0 • f11 : the number of pairs of points (i, j) for which C(i, j) = 1 and P (i, j) = 1. The Jaccard coefficient is defined as f11 . f01 + f10 + f11  B.3  (B.3)  Gap statistic  This section of the appendix is not meant to be an explanation of the theory behind the gap statistic. It is only a description of the method that we followed to get the results from section 4.2. We explain how the gap statistic can be used to estimate the true number of clusters for a given dataset D. The experiment has to be repeated for a number of datasets in order for us to reach any reliable conclusion about the use of the gap statistic, but we will turn to that after explaining the method on a dataset D. For a given clustering E = (E1 , . . . , EC ) of the data D, we define the within cluster spread as spread(Ei ) =  1 2|Ei |  dist(x, y).  (B.4)  x,y∈Ei  for some choice of distance function. We use here the square Euclidean distance dist(x, y) = x − y 22 . We also define the spread for the set of clusters E as C  spread(E) =  spread(Ei ).  (B.5)  i=1  We need a “null set” to serve as baseline measure so we generate a series of datasets D11 , D12 , . . . , D1M having a single cluster. For C = 1, . . . , CMAX , we perform our clustering (using hmmmixsoft with random initializations, in this particular case). For every dataset D1i , we let i = spread(E i , . . . , E i ) where E i , . . . , E i are the resulting clusters from dataset D i with BC 1 1 1 C C the method using C initial clusters. These values will come into play later in the role of M 1 i i=1 log BC . M Now we go back to our original dataset D having G clusters and we use our clustering method again using C = 1, . . . , CMAX clusters, repeating this process N times for every value of C. 53  From this we define quantities AiC as the spread of the ith set of clusters obtained from the method with C clusters. We are interested in the estimated expections A˜C = N1 of the standard deviations of those quantities defined as sC =  1 N +1  N i i=1 log AC  N  log AiC − A˜C  2  .  as well as the estimates  (B.6)  i=1  The gap statistic is defined as 1 Gap(C) = N  N  log AiC  1 − M  M i log BC  (B.7)  min {C|Gap(C) ≥ Gap(C + 1) − sC+1 } .  (B.8)  i=1  i=1  and the estimate for the number of clusters is given by C  This procedure is for a single dataset D. To get a good idea of the reliability of the gap statistic, we have to do this multiple times by generating other datasets with G clusters. The M 1 i estimates M i=1 log BC can be reused throughout the trials, and since their role is important it makes sense to use a large value of M .  54  

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