HMMConverter A tool-box for hidden Markov models with two novel, memory efficient parameter training algorithms

Transcription

1 HMMConverter A tool-box for hidden Markov models with two novel, memory efficient parameter training algorithms by TIN YIN LAM B.Sc., The Chinese University of Hong Kong, 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) October, 2008 c TIN YIN LAM 2008

2 Abstract Hidden Markov models (HMMs) are powerful statistical tools for biological sequence analysis. Many recently developed Bioinformatics applications employ variants of HMMs to analyze diverse types of biological data. It is typically fairly easy to design the states and the topological structure of an HMM. However, it can be difficult to estimate parameter values which yield a good prediction performance. As many HMM-based applications employ similar algorithms for generating predictions, it is also time-consuming and error-prone to have to re-implement these algorithms whenever a new HMM-based application is to be designed. This thesis addresses these challenges by introducing a tool-box, called HMMConverter, which only requires an XML-input file to define an HMM and to use it for sequence decoding and parameter training. The package not only allows for rapid proto-typing of HMM-based applications, but also incorporates several algorithms for sequence decoding and parameter training, including two new, linear memory algorithms for parameter training. Using this software package, even users without programming knowledge can quickly set up sophisticated HMMs and pair-hmms and use them with efficient algorithms for parameter training and sequence analyses. We use HMMConverter to construct a new comparative gene prediction program, called Annotaid, which can predict pairs of orthologous genes by integrating prior information about each input sequence probabilistically into the gene prediction process and into parameter training. Annotaid can thus be readily adapted to predict orthologous gene pairs in newly sequenced genomes. ii

3 Table of Contents Abstract ii Table of Contents iii List of Figures vii Acknowledgements ix 1 HMMs in Bioinformatics applications Introduction Introduction to hidden Markov models Theoretical background Variants of HMMs for Bioinformatics applications Challenges in designing and implementing HMMs Discussion and conclusion Decoding and parameter training algorithms for HMMs Introduction and motivation Decoding algorithms The Viterbi algorithm The Hirschberg algorithm iii

4 Table of Contents 2.3 Parameter training The Baum-Welch algorithm The Viterbi training algorithm Discussion and conclusion HMMConverter Motivation Main features Input and output files Special features Sequence decoding algorithms Training algorithms Training using sampled state paths Sampling a state path from posterior distribution A new, linear memory algorithm for parameter training using sampled state paths A new linear memory algorithm for Viterbi training Pseudo-counts and pseudo-probabilities Training free parameters Availability Discussion and conclusion Specifying an HMM or pair-hmm with HMMConverter Introduction The XML-input file The <model> component of the XML file iv

5 Table of Contents The <sequence analysis> component of the XML file Other text input files The input sequence file The free emission parameter file The free transition parameter file The tube file The prior information file The output files of HMMConverter Examples Implementation of HMMConverter Introduction Converting XML into C++ code The model parameters class Other classes Decoding and training algorithms The Viterbi and Hirschberg algorithms The parameter training algorithms Annotaid Introduction and Motivation Main features of Annotaid The pair-hmm of Annotaid Banding and sequence decoding algorithms Integrating prior information along the input sequences Parameter training in Annotaid v

6 Table of Contents 6.3 Availability Discussion and Future Work Bibliography vi

7 List of Figures 1.1 A Markov model for the CpG-Island process An HMM for the CpG-Island A 5-state simple pair-hmm A simple pair-hmm for predicting related protein coding regions in two input DNA sequences A simple generalized pair-hmm for predicting related protein coding regions in two DNA sequences Illustration of the Viterbi algorithm Illustration of the Hirschberg algorithm (1) Illustration of the Hirschberg algorithm (2) The XML-input file for defining an HMM in HMMConverter Using Blast to generate a tube for HMMConverter Defining a tube for HMMConverter explicitly A state which carries labels from two different annotation label sets Illustration of the special emission feature in HMMConverter Using the Hirschberg algorithm inside a tube Illustration of the forward algorithm vii

8 List of Figures 3.8 Sampling a state path from the posterior distribution Illustration of the novel, linear memory algorithm for parameter training using sampled state paths A part of the pair-hmm of Doublescan Defining a state of an HMM or pair-hmm in HMMConverter A part of the pair-hmm of Doublescan Specifying a tube explicitly in a flat text file A simple 5-state pair-hmm Combing the Hirschberg algorithm with a tube Special emissions in Projector and Annotaid Combining the parameter training with a tube viii

9 Acknowledgements I would like to thank my supervisor, Prof. Irmtraud Meyer for her advice, support and encouragement during the course of this project. A special thank you goes to Prof. Anne Condon for agreeing to be the second reader of my thesis. I thank Nick Wiebe, Rodrigo Goya, Chris Thachuk and all other members of the BETA lab who have made my life on campus enjoyable as they are always being helpful and nice. ix

10 Chapter 1 HMMs in Bioinformatics applications 1.1 Introduction Hidden Markov models (HMMs) are flexible and powerful statistical tools for many different kinds of sequence analysis. HMMs were first introduced in a series of papers by Leonard E. Baum in the 1960s, and an early application of HMMs is speech recognition in the mid- 1970s. Today, HMM-based methods are widely used in many different fields such as speech recognition and speech synthesis [42, 45], natural language processing [32], hand-writing recognition [27] and Bioinformatics [11]. Among the above applications, HMMs are especially popular for sequence analysis in Bioinformatics. In Bioinformatics, HMMs are used for sequence annotation such as gene prediction [8, 23], sequence analysis such as transcription binding site recognition [17] and peptide identification [22], sequence alignment such as protein alignment [21, 24] and other applications such as protein-protein interaction prediction [41]. Many variations of HMMs have been developed in the recent two decades for different Bioinformatics applications. In addition, new algorithms for sequence decoding and parameter training have been introduced. Although HMMs are a powerful and popular concept, several practical challenges often have 1

11 1.1. Introduction to be overcome in practice. It is not easy to design an HMM, especially to assign values to its parameters. The time and memory requirements for analyzing long input sequences can be prohibitive. Finally, it can be very time consuming and error-prone to implement an HMM and the corresponding prediction and parameter training algorithms. This thesis presents an HMM generation tool called HMMConverter, which only requires an XML-file and simple text input files for defining an HMM and for using the HMM for predictions and for training its parameters. Users of HMMConverter are not required to have programming skills. The tool provides the most commonly used Viterbi algorithm [47], and the more memory efficient Hirschberg algorithm [16] for sequence decoding. For parameter training, HMMConverter offers not only a new, linear memory algorithm for Baum- Welch training, but also a novel linear memory variation of the Viterbi training algorithm as well as a new and efficient algorithm for training with sampled state paths. Furthermore, HMMConverter supports many special features for constructing various types of HMMs, such as higher order HMMs, pair-hmms as well as HMMs which take external information on the input sequences into account for parameter training and generating predictions. In this chapter, we give some theoretical background on HMMs, and describe variants of HMM and their applications in Bioinformatics. We also further discuss the challenges of designing and implementing an HMM. Existing sequence decoding and parameter training algorithms for HMM are introduced in chapter 2. Chapter 3 describes the main features of HMMConverter, and includes a detailed explanation of the novel parameter training algorithms. The specifications of an HMM within HMMConverter and the input and output format for the tool are described in chapter 4, some implementation details of HMMConverter are also provided in chapter 5. Finally, we used this tool to implement a novel comparative gene prediction pair-hmm called Annotaid which can take various types of prior information on each of the two input sequences into account in order to give more accu- 2

12 1.2. Introduction to hidden Markov models rate predictions. The features and applications of this new model are described in chapter Introduction to hidden Markov models Theoretical background A hidden Markov model (HMM) is a variation of a Markov model, where the state sequence of a Markov chain is hidden from the observation. We first explain what a Markov chain is and then use a simple biological example to introduce HMMs. Markov model A Markov chain is a random process with a Markov property. In brief, a random process is a process for which the next state is determined in a probabilistic way. The Markov property implies that given the current state in the random process, the future state is independent of the past states. Let X i be a random variable for all positive integers i, representing the state of the process at time i. The sequence X 1, X 2,, X i, is a Markov chain if: P (X i = x X i 1 = x i 1,, X 1 = x 1 ) = P (X i = x X i 1 = x i 1 ) A time-homogeneous Markov chain means that the next state of the process is independent of the time, but only depends on the current state: P (X i = x X i 1 = y) = P (X i 1 = x X i 2 = y) One can also define higher order Markov chains. A Markov chain of n-th order ( n) is a Markov chain for which depends on the current state and the previous n 1 states. Note 3

13 1.2. Introduction to hidden Markov models that when we say Markov chain, we refer to the simplest case, i.e. a 1st order Markov chain. A Markov chain can be represented by a finite state machine [11]. CpG-Island Now, we the use CpG-Island [4] example from the book Biological Sequence Analysis [11] to introduce hidden Markov models. In a genomic DNA sequence, the dinucleotide CG (i.e. a neighboring C and G on the same strand) appears more frequently around the transcription start sites of genes than in other regions of the genome. We call regions which have an increased frequency of CG pairs a CpG-Island. They are usually a few hundred to a few thousand bases long. Suppose we want to identify CpG-Islands in a long genomic sequence. We can do this by defining a Markov chain. Each state of the model in figure represents a DNA nucleotide in a CpG-Island or not in a CpG-Island. The Markov property holds for this model as the probability of going from a given state to the next state only depends on the current state, but not the previous states. Figure shows the Markov model of the CpG-Island process, but this model is lacking something as we cannot distinguish whether an observed nucleotide is from a CpG-Island(+) or not (-). We can modify this model a little, see figure In this new model, each observed nucleotide can be read by two states, one state with label + representing a nucleotide from a CpG-Island and one state with - representing a nucleotide outside a CpG-Island. This is a hidden Markov model for a CpG-Island process, where the hidden feature is the underlying + or - state associated to nucleotides in the observed sequence. The transition probability from the C+ state to the G+ state should be larger than the transition probability from the C- state to the G- state. The start state and the end state in the two models are both silent states which do not read any character from the input sequence. The model always start with the start state 4

14 1.2. Introduction to hidden Markov models Figure 1.1: A Markov model for the CpG-Island process This is a Markov model for the CpG-Island process, but it cannot solve the CpG-Island annotation problem. Figure 1.2: An HMM for the CpG-Island This figure shows an HMM which can model CpG-Islands. The states of the model are all connected to each other, i.e. it is a complete graph(k 8 ) where each state also has a transition back to itself. 5

15 1.2. Introduction to hidden Markov models before reading any character from the input sequence. On the other hand, every state path ends at the end state. In the above example, the start state is connected to all the other states (except the end state), and all the states except the start state are connected to the end state. Now we are ready to give a more formal definition of HMM, an HMM is unambiguously defined by the following components: A set of N states S = {0,, N 1}, where 0 is the start state and N 1 is the end state. An alphabet A which contains the symbols of observations of the HMM (i.e. {A,C,G,T} is the DNA alphabet for the CpG-Island HMM in the above example). A set of transition probabilities T = {t i,j i, j S}, where t i,j [0, 1] is the transition probability of going from state i in the model to state j and j S t i,j = 1 for all states i in S. A set of emission probabilities E = {e i (γ) i S, γ A}, where e i (γ) [0, 1] is the emission probability of state i for symbol γ and γ A e i(γ ) = 1 for all states i in S. An HMM as defined above corresponds to a probabilistic model. It can be used in a number of ways to generate predictions. In the following, let M denote an HMM, X = (x 1,, x L ) an input sequence of length L where x i is the i-th letter of the sequence and π = (π 0,, π l ) a state path of length l + 1 in the model M where π k is the state at the k-th position in the state path, i.e. π 0 is always the start state and π l is always the 6

16 1.2. Introduction to hidden Markov models end state. We can use M in a number of ways to generate predictions: Sequence decoding: For a given input sequence X and model M, we can for example calculate the state path with the highest overall probability in the model, i.e. the state path π which maximizes P (π X, M). In that case, model M analyzes the sequence X and we will, in all of the following, say that the states of model M read the symbols of observations of input sequence X. Parameter training: For a given set of training sequences X and a given model M, we can train the set of free parameters θ of M, for example by using training algorithms which maximize P (X θ, M). Generating sequences of observations: As any HMM defines a probability distribution over sequences of observations, we can use any given HMM M to generate sequences of observations. In that case, we say that the states of model M emit symbols of observations. In this thesis, this way of using an HMM is not employed. We will, in all of the following, therefore refer to a state as reading one or more symbols from an input sequence. The CpG-Island finding problem can be solved with a decoding algorithm, which computes the most probable state path π for an input DNA sequence X and model M. Before we can use an HMM for decoding, we first have have to determine good values for its parameters which may require parameter training. Algorithms for decoding and parameter training will be discussed in more detail in this chapter and in chapter 2. In the following section, we first describe variants of HMMs that have been developed recently for Bioinformatics applications. 7

17 1.2. Introduction to hidden Markov models Variants of HMMs for Bioinformatics applications Pair-HMMs and n-hmms A pair-hmm is a simple extension of an HMM that reads two rather than only one sequence at a time. One of the main applications of pair-hmms is to generate sequence alignments, so pair-hmms are particularly prevalent in Bioinformatics [39]. For example, a pair-hmm called WABA [21] can be used to identify pairs of protein coding exons in two orthologous input DNA sequences. This pair-hmm generates a global alignment between the two input sequences unlike TBlastX which generates local alignments [1]. Pair-HMMs are also used for comparative gene prediction [26, 34, 36]. These pair-hmms take a pair of homologous DNA sequences as input and predict pairs of homologous gene structures. Comparative gene prediction methods using pair-hmms have been shown [26, 36] to perform better than non-comparative approaches [8]. Figure shows a simple 5-state pair-hmm. In an n- HMM, each state can read letters from up to n input sequences. n-hmms can be used for multiple sequence alignment, but because of their high memory and time requirements, they are typically used only for short sequences, such as protein sequences. Generalized HMMs Generalized HMMs [8, 48] allow states to read more than one letter at a time from an input sequence. In Bioinformatics applications, a state of an HMM often corresponds to a biological entity, e.g. a codon consisting of 3 nucleotides, such a state would read 3 letters from the input sequence at a time. Figure shows a simple generalized pair-hmm for finding related protein-coding exons in two input DNA sequences. In particular, any HMM used for gene prediction should be a generalized HMM. 8

18 1.2. Introduction to hidden Markov models Figure 1.3: A 5-state simple pair-hmm This is a 5-state simple pair-hmm, where the Match state reads one character from each of the two input sequences and aligns them according to the emission probability of the combination of the two characters read by the state. The Emit X state reads one character from input sequence X and the Emit Y state does the same thing as the Emit X state, but it reads one character from the input sequence Y at a time. 9

19 1.2. Introduction to hidden Markov models Figure 1.4: A simple pair-hmm for predicting related protein coding regions in two input DNA sequences This simple pair-hmm can be used for predicting related protein coding regions in two input DNA sequences. There are 3 states to model the protein coding part and 3 states to model the non-coding part of the two input sequences. The states which model the protein coding part read one codon at a time, e.g. the Match Codon state reads 3 nucleotides from each input sequence at a time. 10

20 1.2. Introduction to hidden Markov models Explicit state duration HMMs Explicit state duration HMMs extend the concept of HMMs by keeping track of the state s duration, i.e. the time that the state path has stayed in the same state is also remembered in the decoding process. In this type of HMM, a set of parameters or functions for modeling the state duration is needed in addition to the set of transition and emission probabilities, and the probability of the next state is conditional on both the current state and the state duration. Genscan [8] and SNAP [25] are two ab initio gene prediction models which are based on an explicit state duration HMM. Evolutionary HMMs For many types of biological sequence analysis, it is important to know the evolutionary correlations between the sequences. [24] describe a method that uses evolutionary information to derive the transition probabilities of the insertion and deletion states of sequence alignment pair-hmms. Evogene [40] is an evolutionary HMM (EHMM), which incorporates the evolutionary relationship between the input sequences into the sequence decoding process. Evogene consists of an HMM and a set of evolutionary models. In the model, each state of the HMM consists of a set of alphabets over fixed alignment columns and a state-specific evolutionary model, and the emission probabilities of each state are calculated using the corresponding evolutionary model for each column in the input alignment. The Felsenstein algorithm [13] is used to calculate the likelihood of an alignment column given an evolutionary input tree and an evolutionary model. 11

21 1.2. Introduction to hidden Markov models HMMs which take external evidence into account Because of significant improvements in biological sequencing technology, large amounts of genome-wide data such as cdnas, ESTs and protein sequences [3, 18, 37, 38] are currently being generated. It is desirable to incorporate these types of additional evidence into computational sequence analysis methods to get a better performance. In recent years, many HMMs that can incorporate prior information on the input sequences into the prediction process have been developed. They are especially popular in gene prediction because of the highly available data. Twinscan [26] is one of the first HMMs that incorporates external information into the gene prediction process. It extends the classical gene prediction HMM Genscan [8] by incorporating matches to several homologous genomic sequences to the input DNA sequence into the predictions. Twinscan first generates a conserved sequence from the matches between homologous DNA sequences and the DNA sequence of interest. This conserved sequence is then used to bias the emission probabilities of the underlying Genscan HMM. Doublescan [34] is a comparative gene prediction pair-hmm which performs ab initio gene prediction and sequence alignment simultaneously by taking two un-annotated and unaligned homologous genomic sequences as input. It incorporates splice site information along each input sequence using the predictions of Stratasplice [29], which generates log-odd scores and posterior probabilities for potential splice site. These scores are then integrated into the prediction process, and greatly improve the prediction accuracy of splice sites. Projector [36] extends Doublescan by using the known genes in one of the two input sequences to predict the gene structures in the other un-annotated DNA input sequence. It uses so-called special emissions, which integrate the information on the annotated sequence into the prediction process by biasing the emission probabilities. In its published version, Projector is only capable of incorporating prior probabilities with discrete values (i.e. ei- 12

22 1.3. Challenges in designing and implementing HMMs ther 0 or 1). More recently, several gene prediction HMMs have been developed for integrating various types of prior information into the prediction process [7, 15, 19, 44]. These programs (they all employ HMMs but not pair-hmms) all allow taking prior information with different levels of confidence in account. For example, in Jigsaw [19], a feature vector is used to store prior information on different features from various sources in probabilities. Simple multiplication rules are then used to incorporate the vector into the nominal emission probabilities. ExonHunter [7] calls a piece of external evidence an advisor, different advisors are combined to form a superadvisor by quadratic programming (it is a problem of minimizing or maximizing a quadratic function of several variables subject to linear constraints on these variables), which is then incorporated into the prediction process. All of these HMMs are capable of integrating various prior information with different confidence scores into the prediction process, but they all have some limitations. For example, they cannot deal with contradicting pieces of information, and one state of the model can only incorporate one type of prior information. 1.3 Challenges in designing and implementing HMMs As discussed previously, HMMs are a widely used powerful concept in many applications. However, it is often not easy to design and implement an HMM. Various algorithms exist for generating predictions with the model and for training the model s parameters. The main challenges are: Long input sequences lead to large memory and time requirements for predictions and parameter training. It is difficult to set up the transition and emission parameters of the model that yield a good performance. 13

23 1.3. Challenges in designing and implementing HMMs It is time consuming and tricky to implement the model and to test several HMMs against each other ( Proto-typing ). Dealing with long input sequences In Bioinformatics, the length of input sequences for HMMs varies between applications. For example, the average gene size (before splicing) of eukaryotic genes varies from around 1kb (yeast) to 50kb (human) or longer. Before discussing the sequence decoding and parameter training algorithms in detail, we first specify the time and memory requirements of these algorithms to give readers a better impression of the challenges. For a pair-hmm with N states, the Viterbi algorithm [47], which is the most widely used algorithm for sequence decoding, requires O(N 2 L 2 ) time and O(NL 2 ) memory for analyzing a pair of sequences of length L each. For a gene prediction pair-hmm consisting of 10 states, it would thus require between 10 8 and time units and between 10 7 and 10 9 many units of memory to analyze typical input sequences which contain only a single gene. These constraints are even more serious for parameter training, as most training algorithms are iterative processes which have to consider many input sequences. These challenges for parameter estimation of HMMs are described below. New efficient algorithms are needed to make decoding and parameter training practical even for long input sequences. Parameter training The set of transition probabilities and emission probabilities constitutes the parameters of an HMM. There are additional parameters for explicit state duration HMMs and evolutionary HMMs. As the performance of an HMM critically depends on the choice of the parameter values, it is important to find good parameter values. As the states of an HMM closely reflect the biological problem that is being addressed, it is typically fairly easy to design the states of 14

24 1.3. Challenges in designing and implementing HMMs the model. However, it can be difficult to manually derive good parameter values. There are two common ways to define the parameters of an HMM: (1) Users can manually choose the parameters. This is time consuming and also requires a very good understanding of the data. Also, this strategy may not result in values that optimize the performance. (2) An other strategy is to use an automatic parameter training algorithm. This strategy works best if we have a large and diverse set of annotated training sequences. Being able to use automatic parameter training for a pair-hmm, for comparative gene prediction means that it can be readily trained to analyze different pairs of genomes. However, as we explain now, parameter training is not an easy task. There are two common types of parameter training strategies for HMMs, maximum likelihood methods (ML) and expectation maximization methods (EM) [10]. The corresponding training algorithms are usually iterative processes, whose convergence depends on the set of initial values and the amount of training data. Depending on the training algorithm, the outcome of the training may depend on the initial parameter values and may also not necessarily maximize the resulting prediction performance. Pseudo-counts for the parameters may be needed to avoid over-fitting, especially if the training data is sparse or biased. We have discussed the importance of parameter training for setting up an HMM. Because of the difficulties of implementing training algorithms that can be used efficiently on realistic data sets, parameter training algorithms are not implemented in most HMM applications (i.e. the parameters are set up manually for one particular data set). An exception are some HMMs for gene prediction [25, 30, 46]. This is because (1) the set of parameters are very important for the gene prediction performance, (2) parameters need to be adapted to predict genes for different organisms, (3) due to the great improvement in biological sequencing techniques, there are large enough sets of known genes to allow parameter training. For gene prediction methods, the main challenge is to implement efficient parameter training algorithms. The 15

25 1.4. Discussion and conclusion existing parameter training algorithms are described in chapter 2. Proto-typing Designing an HMM for a new Bioinformatics application is fairly easy given a good understandings of the underlying biological problem, whereas the implementation of several possible HMMs can be a tedious task. Proto-typing several HMMs and implementing the sequence decoding and parameter training algorithms is time-consuming and error-prone. Although there are many variations of HMMs, they all use the same prediction and training algorithms. It is particularly inefficient if the same algorithms have to be implemented repeatedly whenever a new HMM application is created. If there was a software tool for generating an HMM with the corresponding algorithms, then users would be able to focus on designing the HMM, i.e. the states of the model, which is the only task that requires human expertise and insight into the biological problem. 1.4 Discussion and conclusion HMMs are a very powerful statistical concept which allows many different kinds of sequence analysis applications. In this chapter, we have introduced the theoretical background of HMMs and some popular variants of HMMs and also described the popularity of these models in Bioinformatics. As we explained, it is typically not easy to design and implement an HMM in practice. We have discussed several challenges, such as long input sequences which lead to large memory and time requirements, difficulties with setting up the parameters of the model, and the time consuming and error-prone task of implementing efficient algorithms. Even though many different HMMs have been developed for different applications, they use similar prediction and parameter training algorithms. It is therefore not efficient if users 16

26 1.4. Discussion and conclusion spend most of their time implementing and re-implementing these algorithms for different HMMs, when they could be focusing on the design and parameterization of the HMM. We here propose an HMM-generating tool HMMConverter, which takes an XML input file for defining an HMM, its states and parameters and the algorithms to be used for generating predictions and training the parameters. Using this tool, a user is not required to have computer programming knowledge to set up an HMM and to use it for data analyses and parameter training. This tool provides several widely used sequence decoding and parameter training algorithms, including two novel parameter training algorithms. Furthermore, HMM- Converter supports many special features, such as incorporating external information on each input sequence with confidence scores, and several heuristic algorithms for reducing the run time and memory requirement greatly while keeping the performance essentially unchanged. These features are especially useful for Bioinformatics applications. In the following chapter, we introduce several widely used existing sequence decoding and parameter training algorithms. The HMM generating tool-hmmconverter is described in chapter 3, including details on all special features and novel algorithms. Chapter 4 explains how to specify an HMM using HMMConverter and chapter 5 describes some implementation details of HMMConverter. Chapter 6 describes a novel pair-hmm for comparative gene prediction called Annotaid, which is constructed using the HMMConverter framework. 17

27 Chapter 2 Decoding and parameter training algorithms for HMMs 2.1 Introduction and motivation In biological sequence analysis or biological sequence annotations, we are interested in finding good annotations for the raw biological sequences. This is the sequence decoding problem discussed in chapter 1. Every HMM assigns an overall probability to each state path which is equal to the product of the encountered transition and emission probabilities, and every state path corresponds to an annotation of the input sequence. For an HMM with well-chosen parameters, high-quality annotations should correspond to high-probability state paths, so the task for sequence decoding is to find a state path with high probability. The Viterbi algorithm [47] is one of the most widely used sequence decoding algorithms which calculates the most probable state path for a given input sequence and HMM. For many Bioinformatics applications, it is often the case that several state paths that yield the same annotation. For example, in a simple pair-hmm which generates global alignments of protein coding regions in two input DNA sequences (see figure 2.1), there are states Emit Codon x and Match Codon, which both assign the annotation label Codon to the letters they read from sequence X. In this pair-hmm, several different state paths that cor- 18

28 2.2. Decoding algorithms respond to different global alignments between the two sequences could yield the same gene annotation for the two input sequences. From this example, we can see that it is always possible to translate a state path in the model into an annotation of the input sequence. Figure 2.1: A simple generalized pair-hmm for predicting related protein coding regions in two DNA sequences This simple generalized pair-hmm has been presented in chapter 1. In particular, the Match Codon state, the Emit Codon X state and the Emit Codon Y state all assign the annotation label Codon to the letters they read from the input sequences. The three other states (except the start and the end state) assign the annotation label non coding. 2.2 Decoding algorithms In this section, we describe two existing sequence decoding algorithms for HMMs: The most commonly used Viterbi algorithm [47] and the Hirschberg algorithm [16]. The Hirschberg 19

29 2.2. Decoding algorithms algorithm can be regarded as a more memory efficient version of the Viterbi algorithm. For a given HMM and an input sequence, both algorithms derive the state path with the highest overall probability The Viterbi algorithm The Viterbi algorithm is one of the most widely used sequence decoding algorithm for HMMs. It first calculates the maximum probability that a state path in a given model M can have for a given input sequence, and then retrieves the corresponding optimal state path by a back-tracking process. In this section, we introduce the algorithm for an HMM, i.e. a single tape HMM where only the start and and end states are silent and all other states read one letter from the input sequence at a time. The algorithm consists of two parts: Part 1: Calculate the optimum probability of any state path using dynamic programming. Part 2: Retrieve the corresponding optimal state path by a back-tracking process. We first introduce some notations before explaining the algorithm with pseudo-code, some of these notations have already been introduced in chapter 1. X = (x 1,..., x L ) denotes an input sequence of length L, where x i is i-th letter of the sequence. S = {0,..., N 1} denotes the set of N states in the HMM, where 0 is the start state and N 1 is the end state. T = {t i,j i, j S}, where t i,j is the transition probability from state i to state j 20

30 2.2. Decoding algorithms E = {e i (γ) i S, γ A}, where e i (γ) is the emission probability of state i to read symbol γ from alphabet A v denotes the two-dimensional Viterbi matrix, where v s (i) is the probability of the most probable path that ends in state s and that has read the input sequence up to and including position i. ptr denotes the two-dimensional pointer matrix, where ptr i (s) is the previous state from which the maximum probability at state s and sequence position i was derived. π = (π 0,, π L+1 ) represents a state path in the model for a given input sequence of length L, where π i is the i-th state of the state path. In particular, π 0 is the start state and π L+1 is the end state. π denotes the optimal state path, it is the state path with the highest overall probability on sequence X, i.e. π = argmax π P (π X, M). We also call this state path the Viterbi path. In order to calculate the optimum probability among all state paths for a given HMM and a given input sequence, the Viterbi algorithm uses the following dynamic programming approach: Part 1: Initialization before reading the first letter from the input sequence v s (0) = 1, if s = start 0, if s start Recursion For i = 1,, L 21

31 2.2. Decoding algorithms For s = 1,, N 2 v s (i) = e s (i)max s S {v s (i 1)t s,s } (2.1) ptr i (s ) = argmax s S {v s (i 1)t s,s } (2.2) Termination at the end of the sequence and in the end state v N 1 (L) = max s S {v s (L)t s,end } ptr L (N 1) = argmax s S {v s (L)t s,end } From the definition of v, we know that the probability of the optimal state path P (x, π ) = v N 1 (L). The corresponding optimal state path can be obtained by a back-tracking procedure. Part 2: Back tracking starts at the end of the sequence in the end state and finishes at the start of the sequence in the start state. It is defined by the recursion: π i 1 = ptr i (π i ) Figure illustrates the first part of the Viterbi algorithm. The above algorithm can be easily generalized for pair-hmms by using a three dimensional matrix and looping over both X and Y dimensions in the recursion. For a generalized HMMs where states can read more than one character at a time from the input sequence, the entries of the Viterbi matrix are calculated by: v s (i) = e s (i)max s (v s (i x (s ))t s,s ), where x (s ) is the number of characters read by state s. 22

32 2.2. Decoding algorithms Figure 2.2: Illustration of the Viterbi algorithm The left part shows how the Viterbi algorithm calculates the Viterbi matrix from left to right in the recursion. The algorithm first loops along the input sequence X and then loops over the states in the model. It terminates in the top right corner of the matrix at the end of the input sequence and in the end state. The right part demonstrates the calculation of entry v s (i), where the matrix element for state s and sequence position i is derived from state s at sequence position (i 1), i.e. v s (i) = e s (i) v s (i 1)t s,s = e s (i) max s S {v s (i 1)t s,s }. Let T max be the maximum number of transitions into any state in the model. The Viterbi algorithm takes O(NT max L n ) operations and O(NL n ) memory for an n-hmms which reads n input sequences at a time. These time and memory requirements impose serious constraints when employing this algorithm with n-hmm and long input sequences The Hirschberg algorithm From equation 2.1, we can see that the Viterbi algorithm can be continued if we keep the Viterbi elements for the previous sequence position in memory. This means that only O(NL n 1 ) memory is required for an n-hmm with N states and n input sequences of identi- 23

33 2.2. Decoding algorithms cal length L if we only want to calculate the probability of the Viterbi path. However, in order to obtain the Viterbi path, O(NL n ) memory is required for storing the back-tracking pointers (see equation 2.2) for the back-tracking process. The memory needed for the back-tracking pointers thus dominates the memory requirements for the Viterbi algorithm. The Hirschberg algorithm is a more memory efficient version of the Viterbi algorithm. It requires O(NL n 1 ) memory instead of O(NL n ) for an n-hmm. Furthermore, the time requirement of the Hirschberg algorithm is of the same order as for the Viterbi algorithm. It uses the same recurrences as the Viterbi algorithm, but searches the Viterbi matrix in a smarter way. We now describe the Hirschberg algorithm for a pair-hmm. Instead of filling the Viterbi matrix in a single direction along one of the two input sequences, the Hirschberg algorithm traverses the matrix from both ends using two Hirschberg strips of O(NL) memory each (see figure 2.2.2). We call them the forward strip and the backward strip, respectively. In order to perform the backward traversal, a mirror model of the original HMM has to be set up by reversing the transitions and exchanging the end state and start state with respect to the original HMM. This mirror model does not necessarily define an HMM because the sum of the transition probabilities from a state may no longer be 1. The forward strip calculated the 3-dimensional matrix in the same direction as the Viterbi algorithm, while the backward strip uses the mirror model to move in the opposite direction, i.e. it starts at the end state and at the end of one sequence. The matrix element (i, j, s) in the backward strip stores the probability of the optimal state path that has read the input sequences from their ends up to and including position i of sequence X and position j of sequence Y and that finishes in state s. Figure shows how the two Hirschberg strips move in opposite directions and meet at the middle position of one of two input sequences (i.e. sequence X in 24

34 2.2. Decoding algorithms this example). Figure 2.3: Illustration of the Hirschberg algorithm (1) This figure shows the projection of the 3-dimensional Viterbi matrix onto the X Y plane. The two Hirschberg strips traverse the matrix from both ends of sequence X and meet at the middle of the sequence X. For a generalized HMM, the two strips have a width of max s S ( x (s))+1, so they will overlap instead of touching each other. The additional columns of the strip are used to store the values calculated for the previous sequence position which are needed in order to continue the calculation. Suppose the two strips meet at position i of sequence X, where the forward strip stores the Viterbi matrix elements for position i of sequence X and the backward strip stores the corresponding probabilities for position i + 1 of sequence X. At that sequence position in X, the probability of the optimal state path P (X, Y π ) can be calculated by finding the transition probabilities from states in the forward strip to states in the backward strip, such that the product of this transition probability and the two entries in the strip is maximum. Using this criterion, we can identify the Viterbi matrix element (i, j, s) through which the optimal state path π goes. 25

35 2.2. Decoding algorithms In summary, when the two Hirschberg strips meet, the optimum probability and a coordinate on the optimal state path can be determined. The process only continues on the left sub-matrix and the right sub-matrix which corresponds to only half of the search space of the previous iteration (see figure 2.2.2). From the above example, the position i of sequence X defines the right boundary of the left sub-matrix, and the coordinate (i, j, s) defines the end position, this coordinate also defines the start position of the right sub-matrix. Figure Figure 2.4: Illustration of the Hirschberg algorithm (2) The coordinate (i,j,s) on the optimal state path was determined in the first iteration of the Hirschberg algorithm (see figure 2.2.2). The matrix is then split into left and right submatrices and the next round of iterations is started for each of the two sub-matrices. This recursive procedure continues, until the complete state path is determined shows that two smaller Hirschberg strips search in each of the two sub-matrices. The recursive process continues until the optimal state path is completely determined. The figure also shows that in each iteration, the total search space for the Hirschberg strips is reduced by a factor of two. So the number of operations performed by the Hirschberg algorithm is given by: t+ t 2 + t 4 + 2t, where t denotes the number of operations required for searching 26

36 2.3. Parameter training the entire matrix, i.e. the number of operations in the Viterbi algorithm. 2.3 Parameter training In order to calculate the most probable state path for a given input sequence or sequences, a set of parameters must be provided. The parameters are the set of transition probabilities and emission probabilities. HMMs of the same topology but different parameter values can be used for different applications. For example, the average length of an exonic stretch in human genes is different from that in mouse genes, so the transition probabilities from the exon state to intergenic state in the HMM should not be the same. The values of these parameters determine the prediction result and the performance of the HMM. In general, the set of parameters of the HMM should be adjusted according to applications, so the parameter estimation is very crucial. However, it is often a difficult task to obtain good parameter estimation, because the parameter training process is expensive to run and training data is usually inadequate. Parameter training is essential in order to set up HMMs for novel biological data sets and to optimize their prediction performance. First of all, a set of training sequences is required for the training process. If the state paths for all the training sequences are known, then the parameter estimation becomes a simple counting process where the performance is optimized via a maximum likelihood approach. Suppose there are M training sequences. Let: X = {X 1,..., X M } denote the set of M training sequences. θ denote the set of parameters of the HMM. This set refers to the set of transition and emission probabilities, i.e. θ = {T, E}. 27

37 2.3. Parameter training r s,s the pseudo-count for the number of times the transition s s is used. r s (γ) the pseudo-count for the number of times states s reads symbol γ. We can then derive the parameters that maximize the likelihood, P (X θ), by setting: t s,s = T s,s s T, where (2.3) s, s T s,s = number of transitions from s to s in all the state paths including any pseudo-counts r s,s. Similarly, e s (γ) = E s(γ) γ E s(γ, where (2.4) ) E s (γ) = number of times state s reads letter γ in all the state paths including any pseudocounts r s (γ). Often, the state paths that correspond to a known annotation are not known or too numerous. In that case, parameter estimation becomes more difficult. In this situation, we can employ two frequently used parameter training algorithms: the Baum-Welch algorithm [2] and the Viterbi training [11] The Baum-Welch algorithm The Baum-Welch algorithm is a special case of an expectation maximization (EM) algorithm, which is a general method for probabilistic parameter estimation. The Baum-Welch algorithm iteratively calculates new estimates for all parameters and the overall log likelihood of the HMM can be shown to converge to a local maximum. We introduce a few more notations, 28