Master

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Masters courseBioinformatics Data Analysis
and Tools

• Lecture 5 Markov models
• Centre for Integrative Bioinformatics

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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

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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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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

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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

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Markov Chain Models
Pr(cggt) Pr(c)Pr(gc)Pr(gg)Pr(tg)
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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

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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
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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

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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

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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)

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A Fifth Order Markov Chain
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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

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Inhomogenous Markov Chains
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A Fifth Order InhomogenousMarkov Chain
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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 __

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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

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Interpolated Markov Models

• The IMM idea manage this trade-off by
  interpolating among models of various orders
• Simple linear interpolation

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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

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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

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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

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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
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The GLIMMER System

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

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IMMs in GLIMMER

• How does GLIMMER determine the values?
• First, let us express the IMM probability
  calculation recursively

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

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

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Homology-derived Secondary Structure of Proteins
(HSSP) Sander Schneider, 1991
Its all about trying to push dont know region
down
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The Parameters of an HMM
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HMM for Eukaryotic Gene Finding
Figure from A. Krogh, An Introduction to Hidden
Markov Models for Biological Sequences
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A Simple HMM
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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

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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.

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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?

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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)

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How Likely is a Given Sequence?
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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
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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

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How Likely is a Given Sequence
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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
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Forward algorithm example

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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?

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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

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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

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Finding the Most Probable PathThe Viterbi
Algorithm
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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

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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).
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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

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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

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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

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The Expectation step
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The Expectation step
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The Expectation step
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The Expectation step
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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

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The Expectation step
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The Backward Algorithm
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The Expectation step
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The Expectation step
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The Expectation step
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The Maximization step
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The Maximization step
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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