HMM

Hidden Markov Model

CpG islands

In human genome, CG dinucleotides are relatively rare CpG pairs undergo a process called

methylation that modifies the C nucleotide

A methylated C mutates (with relatively high chance) to a T

Promotor regions are CG rich These regions are not methylated, and

thus mutate less often These are called CG (aka CpG) islands

Initiation of Transcription from a Promoter

Properties of CpG Islands

Finding CpG islands

Simple approach: Pick a window of size N

(N = 100, for example) Compute log-ratio for the sequence in the

window, and classify based on that

Problems: How do we select N ? What do we do when the window

intersects the boundary of a CpG island?

Slot machine

Fair Bet Casino The game is to flip coins, which results in

only two possible outcomes: Head or Tail. The Fair coin will give Heads and Tails

with same probability ½. The Biased coin will give Heads with

prob. ¾.

Fair coin: ½ ½

Biased coin: ¾ ¼

The “Fair Bet Casino” (cont’d)

Thus, we define the probabilities: P(H|F) = P(T|F) = ½ P(H|B) = ¾, P(T|B) = ¼ The crooked dealer changes

between Fair and Biased coins with probability 10%

Game start:

T H H H H T T T T H T

F FF F F F FBBB B

The Fair Bet Casino Problem

Input: A sequence x = x1x2x3…xn of coin tosses made by two possible coins (F or B).

Output: A sequence π = π1 π2 π3…

πn, with each πi being either F or B indicating that xi is the result of tossing the Fair or Biased coin respectively.

Log-odds Ratio

We define log-odds ratio as follows:

log2(P(x|fair coin) / P(x|biased coin)) = Σk

i=1 log2(p+(xi) / p-

(xi))

= n – k log23

Log-odds Ratio in Sliding Windows

x1x2x3x4x5x6x7x8…xn

Consider a sliding window of the outcome sequence. Find the log-odds for this short window.

Log-odds value

0

Fair coin most likely used

Biased coin most likely used

Disadvantages:- the length of Fair coin seq is not known in advance- different windows may classify the same position differently

HMM

Markov Process & Transition Matrix

A stochastic process for which the probability of entering a certain state depends only on the last state occupied is called a Markov process. The process is governed by a transition matrix.

Ex. Suppose that the 1995 state of land use in a city of 50 square miles of area is

Assuming that the transition matrix for 5-year intervals are given by

The 2000 state: I (Residential) = ?

I (residential used) 30%II (commercially used) 20%III (industrially used) 50%

To I To II To III

From I 0.8 0.1 0.1

From II

0.1 0.7 0.2

From III

0 0.1 0.9

.8*30+.1*20+0*50 = 26%

Markov ModelA stochastic process for which the

probability of entering a certain state depends only on the last state occupied is called a Markov process.

Residentiallyused

Commerciallyused

Industriallyused

0.8

0.1 0.9

0.0

0.1

0.2

0.7

0.1 0.1

RIICCRRRCCII

Markov Chain

Basic Mathematics Markov chains: probability of a

sequence: S=a1a2...an

P(S)=P(a1)P(a2|a1)P(a3|a1 a2)...P(an|a1... an-1) P(S)=P(a1)P(a2|a1)P(a3|a2)...P(an|an-1) P(S)=P(a1) P P(ai|ai-1)

ExerciseProbability of sequence CATG

Bayes’ theorem

A T G C

A 0.2 0.35 0.15 0.3

T 0.25 0.15 0.4 0.2

G 0.25 0.25 0.25 0.25

C 0.3 0.25 0.2 0.25

Hidden Markov Model1 2 4 3 1 4 5 2 3 4 6 1 2 2 1 …Which dice was used is hidden.

HMM The state sequence is hidden The process is governed by a transition

matrix

akl = P (i=l | i-1=k) and emission probabilities

ek(b)= P (xi=b | i=k) HMMs: prob. depends on states passed

thru if known: (states = s1 s2 ... sn)

P(Seq)=P(a1|s1)P(s2|s1)P(a2|s2)P(s3|s2)...P(sn|sn-

1)P(an|sn) if unknown, sum over all possible paths find the

sequence that maximize P(S)?

Hidden Markov Model (HMM)

Can be viewed as an abstract machine with k hidden states that emits symbols from an alphabet Σ.

Each state has its own probability distribution, and the machine switches between states according to this probability distribution.

While in a certain state, the machine makes 2 decisions: What state should I move to next? What symbol - from the alphabet Σ -

should I emit?

Why “Hidden”?

Observers can see the emitted symbols of an HMM but have no ability to know which state the HMM is currently in.

Thus, the goal is to infer the most likely hidden states of an HMM based on the given sequence of emitted symbols.

Fair Bet Casino Problem

Fair Bet Casino ProblemAny observed outcome of coin tosses could have been generated by any sequence of states!

Need to incorporate a way to grade different sequences differently.

Decoding Problem

HMM Parameters

Σ: set of emission characters.Ex.: Σ = {H, T} for coin tossing

Σ = {1, 2, 3, 4, 5, 6} for dice tossing Σ = {A, C, G, T} for a DNA seq

Q: set of hidden states, each emitting symbols from Σ.

Q={F,B} for coin tossing Q={intron, exon} for a gene

HMM Parameters (cont’d)

A = (akl): a |Q| |Q| matrix of probability of changing from state k to state l.

aFF = 0.9 aFB = 0.1

aBF = 0.1 aBB = 0.9

E = (ek(b)): a |Q| |Σ| matrix of probability of emitting symbol b while being in state k.

eF(0) = ½ eF(1) = ½

eB(0) = ¼ eB(1) = ¾

HMM for Fair Bet Casino The Fair Bet Casino in HMM terms:

Σ = {0, 1} (0 for Tails and 1 Heads)Q = {F,B} – F for Fair & B for Biased coin.

Transition Probabilities A *** Emission Probabilities EFair Biased

Fair aFF = 0.9

aFB = 0.1

Biased

aBF = 0.1

aBB = 0.9

Tails(0) Heads(1)

Fair eF(0) = ½

eF(1) = ½

Biased

eB(0) = ¼

eB(1) = ¾

HMM for Fair Bet Casino (cont’d)

HMM model for the Fair Bet Casino Problem

Hidden Paths A path π = π1… πn in the HMM is defined

as a sequence of states. Consider path π = FFFBBBBBFFF and

sequence x = 01011101001 (0=T, 1=H)

x 0 1 0 1 1 1 0 1 0 0 1

π = F F F B B B B B F F FP(xi|πi) ½ ½ ½ ¾ ¾ ¾ ¼ ¾ ½ ½ ½

P(πi-1 πi) ½ 9/10 9/10 1/10

9/10 9/10

9/10 9/10

1/10 9/10

9/10 Transition probability from state πi-1 to state πi

Probability that xi was emitted from state πi

Decoding Problem

Goal: Find an optimal (most probable) hidden path of states given observations.

Input: Sequence of observations x = x1…xn generated by an HMM M(Σ, Q, A, E)

Output: A path that maximizes P(x, π) over all possible paths π.

Decoding Problem

T H H H H T T T TF

B

P(x, π) Calculation

P(x, π): Probability that sequence x was generated by the path π: n

P(x, π) = P(π0→ π1) · Π P(xi|πi) · P(πi → πi+1)

i=1

= a π0, π1 · Π e πi (xi) · a πi,

πi+1

= Π e πi+1 (xi+1) · a

πi, πi+1

if we count from i=0 instead of i=1 to

i=n

Number of possible paths?

Building Manhattan for Decoding Problem

Andrew Viterbi used the Manhattan grid model to solve the Decoding Problem.

Every choice of π = π1… πn

corresponds to a path in the graph. The only valid direction in the graph

is eastward. This graph has |Q|2(n-1) edges.

?

Graph for Decoding Problem

Decoding Problem vs. Alignment Problem

Valid directions in the alignment problem.

Valid directions in the decoding problem.

Decoding Problem as Finding a Longest Path

in a DAG The Decoding Problem is reduced to

finding a longest path in the directed acyclic graph (DAG) above.

Notes: the length of the path is defined as the product of its edges’ weights, not the sum.

Decoding Problem: weights of edges

The weight w = el(xi+1) . akl

w?

(k, i) (l, i+1)

T H H H H T T T TF

B

Decoding Problem n

Maximize: P(x, π) = Π e πi+1 (xi+1) . a πi, πi+1

i=0

Decoding Problem (cont’d)

Every path in the graph has the probability P(x,π) (= length of the path).

The Viterbi algorithm finds the path that maximizes P(x, π) among all possible paths.

The Viterbi algorithm runs in O(n|Q|2) time.

?

Decoding Problem and Dynamic Programming

sl,i+1=maxk Q {sk,i · weight of (k,i) (l,i+1)}

=maxk Q {sk,i · akl · el (xi+1)}

=el (xi+1) · maxk Q {sk,i · akl}

lk

i i+1

Decoding Problem (cont’d)

Initialization: sbegin,0 = 1

sk,0 = 0 for k ≠ begin.

Let π* be the optimal path. Then,

P(x, π*) = maxk Є Q {sk,n . ak,end}

Most probable path: Viterbi alg

Dynamic programming define pi,j as the prob of the most probable path

ending in state j after emitting element i Define solution recursively:

suppose we know pi-1,j states up to previous char

update: pi,k=P(ai, sk) * maxj(pi-1,j*P(sk, sj)) traceback

keep table of state probs, start with 1st char, assign prob to each state, iterate updates...

a1 a2 a3 a4s1 0.6 0.9 0.9 0.2s2 0.1 0.1 0.05 0.8s3 0.3 0 0.05 0

Most probable path: Viterbi alg

Dynamic programming

a1 a2 a3 a4

s1 0.6 0.9 0.9 0.2

s2 0.1 0.1 0.05 0.8

s3 0.3 0 0.05 0

Problem with Viterbi Algorithm

The value of the product can become extremely small, which leads to under-flowing.

To avoid overflowing, use log value instead.

sk,i+1= log el(xi+1) + max k Є Q {sk,i + log(akl)}

Exercise

Consider a hidden Markov model with the following transition and emission matrices:

What is the most probable sequence of states for a given DNA sequence ACGG?

exon intron

exon 0.9 0.1

intron 0.05 0.95

purine pyrimidine

exon 0.3 0.7

intron 0.5 0.5

Forward-Backward Problem

Given: a sequence of coin tosses generated by an HMM.

Goal: find the probability that the dealer was using a biased coin at a particular time i.

T H H H H T T T T H T

Forward-Backward Problem

T H H H H T T T TF

Bi

Forward Algorithm Define fk,i (forward probability) as

the probability of emitting the prefix x1…xi and reaching the state πi = k.

The recurrence for the forward algorithm:

fk,i = ek(xi) Σ fl,i-1 alk l Є Q

Similar to Viterbi, except replace ‘max’ with probabilistic ‘sum’!

kl

i1 i

Dishonest Casino Computing posterior probabilities

for “fair” at each point in a long sequence:

Backward Algorithm

However, forward probability is not the only quantity that provides info to P(πi = k|x).

The sequence of transitions and emissions that the HMM undergoes between πi+1 and πn also affect P(πi = k|x).

forward xi backward

Define backward probability bk,i as the probability of being in state πi = k and emitting the suffix xi+1…xn.

The recurrence for the backward algorithm:

bk,i = Σ akl el(xi+1) bl,i+1

l Є Q

Backward Algorithm (cont’d)

k l

i i+1

The probability that the dealer used a biased coin at any moment i:

P(x, πi = k) fk(i) . bk(i)

P(πi = k|x) = _______________ = ____________

P(x) P(x)

Backward-Forward Algorithm

P(x) is the sum of P(x, πi = k) over all k

Markov chain for CpG Islands

Construct a Markov chain for CpG rich and another for CpG poor regions

Using maximum likelihood estimates from 60K nucleotide, the two models:

Ratio Test for CpC islands

Given a sequence X1,…,Xn , compute the likelihood ratio

1

1

1

11

1

( , , | )( , , ) log

( , , | )

log i i

i i

i i

nn

n

X X

i X X

X Xi

P X XS X X

P X X

A

A

2

2

HMM Approach Build one model that include “+” states

and “-” states

A state “remembers” last nucleotide and the type of region

A transition from a state to a + corresponds to the start of a CpG island

p? q?