Lecture 5: Markov models

Transcription

1 Master s course Bioinformatics Data Analysis and Tools Lecture 5: Markov models Centre for Integrative Bioinformatics

2 Problem in biology Data and patterns are often not clear cut When we want to make a method to recognise a pattern (e.g. a sequence motif), we have to learn from the data (e.g. maybe there are other differences between sequences that have the pattern and those that do not) This leads to Data mining and Machine learning

3 Contents: A widely used machine learning approach: Markov models Markov chain models (1st order, higher order and inhomogeneous models; parameter estimation; classification) Interpolated Markov models (and back-off models) Hidden Markov models (forward, backward and Baum- Welch algorithms; model topologies; applications to gene finding and protein family modeling

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5 Markov Chain Models a Markov chain model is defined by: a set of states some states emit symbols other states (e.g. the begin state) are silent a set of transitions with associated probabilities the transitions emanating from a given state define a distribution over the possible next states

6 Markov Chain Models given some sequence x of length L, we can ask how probable the sequence is given our model for any probabilistic model of sequences, we can write this probability as key property of a (1st order) Markov chain: the probability of each X i depends only on X i-1

7 Markov Chain Models Pr(cggt) = Pr(c)Pr(g c)pr(g g)pr(t g)

8 Markov Chain Models Can also have an end state, allowing the model to represent: Sequences of different lengths Preferences for sequences ending with particular symbols

9 Markov Chain Models The transition parameters can be denoted by where a x i = 1xi Pr( x x ) i i 1 Similarly we can denote the probability of a sequence x as ax i 1 x i Where a Bxi represents the transition from the begin state

10 Example Application CpG islands CGdinucleotides are rarer in eukaryotic genomes than expected given the independent probabilities of C, G but the regions upstream of genes are richer in CG dinucleotides than elsewhere CpG islands useful evidence for finding genes Could predict CpG islands with Markov chains one to represent CpG islands one to represent the rest of the genome Example includes using Maximum likelihood and Bayes statistical data and feeding it to a HM model

11 Estimating the Model Parameters Given some data (e.g. a set of sequences from CpG islands), how can we determine the probability parameters of our model? One approach: maximum likelihood estimation given a set of data D set the parameters θ to maximize Pr(D θ) i.e. make the data D look likely under the model

12 Maximum Likelihood Estimation Suppose we want to estimate the parameters Pr(a), Pr(c), Pr(g), Pr(t) And we re given the sequences: accgcgctta gcttagtgac tagccgttac Then the maximum likelihood estimates are: Pr(a) = 6/30 = 0.2 Pr(g) = 7/30 = Pr(c) = 9/30 = 0.3 Pr(t) = 8/30 = 0.267

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17 These data are derived from genome sequences

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21 Higher Order Markov Chains An nth order Markov chain over some alphabet is equivalent to a first order Markov chain over the alphabet of n-tuples Example: a 2nd order Markov model for DNA can be treated as a 1st order Markov model over alphabet: AA, AC, AG, AT, CA, CC, CG, CT, GA, GC, GG, GT, TA, TC, TG, and TT (i.e. all possible dipeptides)

22 A Fifth Order Markov Chain

23 Inhomogenous Markov Chains In the Markov chain models we have considered so far, the probabilities do not depend on where we are in a given sequence In an inhomogeneous Markov model, we can have different distributions at different positions in the sequence Consider modeling codons in protein coding regions

24 Inhomogenous Markov Chains

25 A Fifth Order Inhomogenous Markov Chain

26 Selecting the Order of a Markov Chain Model Higher order models remember more history Additional history can have predictive value Example: predict the next word in this sentence fragment finish (up, it, first, last,?) now predict it given more history Fast guys finish

27 Selecting the Order of a Markov Chain Model However, the number of parameters we need to estimate grows exponentially with the order for modeling DNA we need parameters for an nth order model, with n 5 normally The higher the order, the less reliable we can expect our parameter estimates to be estimating the parameters of a 2nd order homogenous Markov chain from the complete genome of E. Coli, we would see each word > 72,000 times on average estimating the parameters of an 8th order chain, we would see each word ~ 5 times on average

28 Interpolated Markov Models The IMM idea: manage this trade-off by interpolating among models of various orders Simple linear interpolation:

29 Interpolated Markov Models We can make the weights depend on the history for a given order, we may have significantly more data to estimate some words than others General linear interpolation

30 Gene Finding: Search by Content Encoding a protein affects the statistical properties of a DNA sequence some amino acids are used more frequently than others (Leu more popular than Trp) different numbers of codons for different amino acids (Leu has 6, Trp has 1) for a given amino acid, usually one codon is used more frequently than others This is termed codon preference Codon preferences vary by species

31 Codon Preference in E. Coli AA codon / Gly GGG 1.89 Gly GGA 0.44 Gly GGU Gly GGC Glu GAG Glu GAA Asp GAU Asp GAC 43.26

32 Search by Content Common way to search by content build Markov models of coding & noncoding regions apply models to ORFs (Open Reading Frames) or fixedsized windows of sequence GeneMark [Borodovsky et al.] popular system for identifying genes in bacterial genomes uses 5th order inhomogenous Markov chain models

33 The GLIMMER System Salzberg et al., 1998 System for identifying genes in bacterial genomes Uses 8th order, inhomogeneous, interpolated Markov chain models

34 IMMs in GLIMMER How does GLIMMER determine the values? First, let us express the IMM probability calculation recursively:

35 IMMs in GLIMMER If we haven t seen x i-1 x i-n more than 400 times, then compare the counts for the following: Use a statistical test ( χ 2 ) to get a value d indicating our confidence that the distributions represented by the two sets of counts are different

36 IMMs in GLIMMER χ 2 score when comparing n th -order with n-1 th -order Markov model (preceding slide)

37 The GLIMMER method 8th order IMM vs. 5th order Markov model Trained on 1168 genes (ORFs really) Tested on 1717 annotated (more or less known) genes

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40 Plot sensitivity over 1-specificity

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43 Hidden Markov models (HMMs) Given say a T in our input sequence, which state emitted it?

44 Hidden Markov models (HMMs) Hidden State We will distinguish between the observed parts of a problem and the hidden parts In the Markov models we have considered previously, it is clear which state accounts for each part of the observed sequence In the model above (preceding slide), there are multiple states that could account for each part of the observed sequence this is the hidden part of the problem states are decoupled from sequence symbols

45 HMM-based homology searching HMM for ungapped alignment Transition probabilities and Emission probabilities Gapped HMMs also have insertion and deletion states (next slide)

46 Profile HMM: m=match state, I-insert state, d=delete state; go from left to right. I and m states output amino acids; d states are silent. d 1 d 2 d 3 d 4 I 0 I 1 I 2 I 3 I 4 m 0 m 1 m 2 m 3 m 4 m 5 Start End Model for alignment with insertions and deletions

47 HMM-based homology searching Most widely used HMM-based profile searching tools currently are SAM-T99 (Karplus et al., 1998) and HMMER2 (Eddy, 1998) formal probabilistic basis and consistent theory behind gap and insertion scores HMMs good for profile searches, bad for alignment (due to parametrisation of the models) HMMs are slow

48 Homology-derived Secondary Structure of Proteins (HSSP) Sander & Schneider, 1991 It s all about trying to push don t know region down

49 The Parameters of an HMM

50 HMM for Eukaryotic Gene Finding Figure from A. Krogh, An Introduction to Hidden Markov Models for Biological Sequences

51 A Simple HMM

52 Three Important Questions How likely is a given sequence? the Forward algorithm What is the most probable path for generating a given sequence? the Viterbi algorithm How can we learn the HMM parameters given a set of sequences? the Forward-Backward (Baum-Welch) algorithm

53 Three basic problems of HMMs Once we have an HMM, there are three problems of interest. (1) The Evaluation Problem Given an HMM and a sequence of observations, what is the probability that the observations are generated by the model? (2) The Decoding Problem Given a model and a sequence of observations, what is the most likely state sequence in the model that produced the observations? (3) The Learning Problem Given a model and a sequence of observations, how should we adjust the model parameters in order to maximize Evaluation problem can be used for isolated (word) recognition. Decoding problem is related to the continuous recognition as well as to the segmentation. Learning problem must be solved, if we want to train an HMM for the subsequent use of recognition tasks.

54 Three Important Questions How likely is a given sequence? Forward algorithm What is the most probable path for generating a given sequence? How can we learn the HMM parameters given a set of sequences?

55 How Likely is a Given Sequence? The probability that the path is taken and the sequence is generated: (assuming begin/end are the only silent states on path)

56 How Likely is a Given Sequence?

57 How Likely is a Given Sequence? The probability over all paths is: but the number of paths can be exponential in the length of the sequence... the Forward algorithm enables us to compute this efficiently

58 How Likely is a Given Sequence: The Forward Algorithm Define f k (i) to be the probability of being in state k Having observed the first i characters of x we want to compute f N (L), the probability of being in the end state having observed all of x We can define this recursively

59 How Likely is a Given Sequence:

60 Initialisation: The forward algorithm f 0 (0) = 1 (start), f k (0) = 0 (other silent states k) probability that we re in start state and have observed 0 characters from the sequence Recursion: f l (i) = e l (i)σ k f k (i-1)a kl (emitting states), f l (i) = Σ k f k (i)a kl (silent states) Termination: Pr(x) = Pr(x 1 x L ) = f N (L) = Σ k f k (L)a kn probability that we are in the end state and have observed the entire sequence

61 Forward algorithm example

62 Three Important Questions How likely is a given sequence? What is the most probable path for generating a given sequence? Viterbi algorithm How can we learn the HMM parameters given a set of sequences?

63 Finding the Most Probable Path: The Viterbi Algorithm Define v k (i) to be the probability of the most probable path accounting for the first i characters of x and ending in state k We want to compute v N (L), the probability of the most probable path accounting for all of the sequence and ending in the end state Can be defined recursively Can use DP to find v N (L) efficiently

64 Finding the Most Probable Path: Initialisation: The Viterbi Algorithm v 0 (0) = 1 (start), v k (0) = 0 (non-silent states) Recursion for emitting states (i =1 L): Recursion for silent states:

65 Finding the Most Probable Path: The Viterbi Algorithm

66 Three Important Questions How likely is a given sequence? (clustering) What is the most probable path for generating a given sequence? (alignment) How can we learn the HMM parameters given a set of sequences? The Baum-Welch Algorithm

67 The Learning Problem Generally, the learning problem is how to adjust the HMM parameters, so that the given set of observations (called the training set) is represented by the model in the best way for the intended application. Thus it would be clear that the ``quantity'' we wish to optimize during the learning process can be different from application to application. In other words there may be several optimization criteria for learning, out of which a suitable one is selected depending on the application. There are two main optimization criteria found in the literature; Maximum Likelihood (ML) and Maximum Mutual Information (MMI).

68 The Learning Task Given: a model a set of sequences (the training set) Do: find the most likely parameters to explain the training sequences The goal is find a model that generalizes well to sequences we haven t seen before

69 Learning Parameters If we know the state path for each training sequence, learning the model parameters is simple no hidden state during training count how often each parameter is used normalize/smooth to get probabilities process just like it was for Markov chain models If we don t know the path for each training sequence, how can we determine the counts? key insight: estimate the counts by considering every path weighted by its probability

70 Learning Parameters: The Baum-Welch Algorithm An EM (expectation maximization) approach, a forward-backward algorithm Algorithm sketch: initialize parameters of model iterate until convergence Calculate the expected number of times each transition or emission is used Adjust the parameters to maximize the likelihood of these expected values Baum-Welch has as important feature that it always converges

71 The Expectation step

72 The Expectation step

73 The Expectation step

74 The Expectation step

75 The Expectation step First, we need to know the probability of the i th symbol being produced by state q, given sequence x: Pr( π i =k x) Given this we can compute our expected counts for state transitions, character emissions

76 The Expectation step

77 The Backward Algorithm

78 The Expectation step

79 The Expectation step

80 The Expectation step

81 The Maximization step

82 The Maximization step

83 The Baum-Welch Algorithm Initialize parameters of model Iterate until convergence calculate the expected number of times each transition or emission is used adjust the parameters to maximize the likelihood of these expected values This algorithm will converge to a local maximum (in the likelihood of the data given the model) Usually in a fairly small number of iterations