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Title: Master


1
Masters courseBioinformatics 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
A widely used machine learning approach Markov
models
  • Contents
  • 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 Xi depends only on Xi-1

7
Markov Chain Models
Pr(cggt) Pr(c)Pr(gc)Pr(gg)Pr(tg)
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 Similarly we can denote the probability of
a sequence x as Where aBxi 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 ? to maximize
  • Pr(D ?)
  • 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 were given the sequences
  • accgcgctta
  • gcttagtgac
  • tagccgttac
  • Then the maximum likelihood estimates are
  • Pr(a) 6/30 0.2 Pr(g) 7/30 0.233
  • Pr(c) 9/30 0.3 Pr(t) 8/30 0.267

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These data are derived from genome sequences
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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 InhomogenousMarkov Chain
26
Selecting the Order of aMarkov 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 aMarkov 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 gt 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 /1000
  • ----------------------
  • Gly GGG 1.89
  • Gly GGA 0.44
  • Gly GGU 52.99
  • Gly GGC 34.55
  • Glu GAG 15.68
  • Glu GAA 57.20
  • Asp GAU 21.63
  • 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
fixed- sized 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 havent seen xi-1 xi-n more than 400
    times, then compare the counts for the following

Use a statistical test ( ?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
?2 score when comparing nth-order with
n-1th-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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Plot sensitivity over 1-specificity
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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 mmatch state, I-insert state,
ddelete state go from left to right. I and m
states output amino acids d states are silent.
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
Its all about trying to push dont 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 SequenceThe Forward
Algorithm
  • Define fk(i) to be the probability of being in
    state k
  • Having observed the first i characters of x we
    want to compute fN(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
The forward algorithm
  • Initialisation
  • f0(0) 1 (start),
  • fk(0) 0 (other silent states k)
  • Recursion fl(i) el(i)?k fk(i-1)akl
    (emitting states),
  • fl(i) ?k fk(i)akl (silent states)
  • Termination
  • Pr(x) Pr(x1xL) f N(L) ?k fk(L)akN

probability that were in start state and have
observed 0 characters from the sequence
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 PathThe Viterbi
Algorithm
  • Define vk(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 vN(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 vN(L) efficiently

64
Finding the Most Probable PathThe Viterbi
Algorithm
  • Initialisation
  • v0(0) 1 (start), vk(0) 0 (non-silent states)
  • Recursion for emitting states (i 1L)
  • Recursion for silent states

65
Finding the Most Probable PathThe 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 havent 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 dont 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 ParametersThe 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( ?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
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