Hidden markov models

Bài giảng về kỹ thuật Hidden markov models trong xử lý ảnh (chương trình cao học)

Hidden Markov Models

Ankur Jain Y7073

What is Covered

• Observable Markov Model

• Hidden Markov Model

• Evaluation problem

• Decoding Problem

• The output of the process is the set of states at each instant of time

Markov Models

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Calculation of sequence probability

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Rain Dry

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• Two states : ‘Rain’ and ‘Dry’

• Suppose we want to calculate a probability of a sequence of states in our example, {‘Dry’,’Dry’,’Rain’,Rain’}

P({‘Dry’,’Dry’,’Rain’,Rain’} ) = ??

Example of Markov Model

Hidden Markov models.

• The observation is turned to be a probabilistic function (discrete

or continuous) of a state instead of an one-to-one correspondence

of a state

•Each state randomly generates one of M observations (or visible states)

• To define hidden Markov model, the following probabilities

aij= P(si| sj) , matrix of observation probabilities B=(bi (vm )),

bi(vm ) = P(vm| si) and a vector of initial probabilities =(i), i = P(si) Model is represented by M=(A, B, ).

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HMM Assumptions

• Markov assumption: the state transition depends only on

the origin and destination

• Output-independent assumption: all observation frames

are dependent on the state that generated them, not on

neighbouring observation frames

Low High

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DryRain

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Example of Hidden Markov Model

• Two states : ‘Low’ and ‘High’ atmospheric pressure.

• Two observations : ‘Rain’ and ‘Dry’

Example of Hidden Markov Model

•Suppose we want to calculate a probability of a sequence of

observations in our example, {‘Dry’,’Rain’}

•Consider all possible hidden state sequences:

where first term is :

Calculation of observation sequence probability

Evaluation problem Given the HMM M=(A, B, ) and the

O

• Learning problem Given some training observation sequences

O=o1 o2 oK and general structure of HMM (numbers of hidden

probability

O=o1 oK denotes a sequence of observations ok{v1,…,vM }.

Main issues using HMMs :

• Typed word recognition, assume all characters are separated.

• Character recognizer outputs probability of the image being particular character, P(image|character)

0.5 0.03 0.005

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Word recognition example(1).

Hidden state Observation

• Hidden states of HMM = characters.

• Observations = typed images of characters segmented from the image Note that there is an infinite number of

observations

• Observation probabilities = character recognizer scores

•Transition probabilities will be defined differently in two

• If lexicon is given, we can construct separate HMM models for each lexicon word.

0.5 0.03

• Here recognition of word image is equivalent to the problem

of evaluating few HMM models

•This is an application of Evaluation problem.

Word recognition example(3).

0.4 0.6

• We can construct a single HMM for all words.

• Hidden states = all characters in the alphabet

• Transition probabilities and initial probabilities are calculated from language model

• Observations and observation probabilities are as before

• This is an application of Decoding problem.

Word recognition example(4).

•Evaluation problem Given the HMM M=(A, B, ) and the

• Direct Evaluation :Trying to find probability of observations

O=o1 o2 oT by means of considering all hidden state sequences

•P(o1 o2 oT ) = P(o1 o2 oT , S ) {S is state sequence}

•P(o1 o2 oT ) = P(o1 o2 oT /S ) P(S)

•P(S) = {Markov property}

Evaluation Problem.

• NT hidden state sequences - exponential complexity.

• Use Forward-Backward HMM algorithms for efficient

calculations

of the partial observation sequence o1 o2 ok and that the hidden state at time k is si : k(i)= P(o1 o2 ok , qk= si )

• Define the backward variable k(i) as the joint probability of the

state at time k is si : k(i)= P(ok+1 ok+2 oK |qk= si)

k(i) k(i) = P(o1 o2 oK , qk= si)

• P(o1 o2 oK) = i k(i) k(i)

•We want to find the state sequence Q= q1…qK which maximizes

P(Q | o1 o2 oK ) , or equivalently P(Q , o1 o2 oK )

• Brute force consideration of all paths takes exponential time Use efficient Viterbi algorithm instead.

state sequence q1… qk-1 and getting into qk= si

k(i) = max P(q1… qk-1 , qk= si , o1 o2 ok)

where max is taken over all possible paths q1… qk-1

Decoding problem

• General idea:

if best path ending in qk= sj goes through qk-1= si then it

•Termination: choose best path ending at time K

maxi [ K(i) ]

• Backtrack best path

This algorithm is similar to the forward recursion of evaluation

problem, with  replaced by max and additional backtracking.

Viterbi algorithm (2)

Learning problem

• The most difficult of the three problems, because there is

no known analytical method that maximizes the joint

probability of the training data

• Solved by the Baum-Welch (known as forward backward) algorithm and EM (Expectation maximization) algorithm

HMM & Neural network

• Predicting the Protein Structures

• Fault detection in ground antenna

• What is hidden in HMM??

• How many states do I think my model needs?

• How many possible observations do I think there are?

• What kind of topology do I think my state transition graph should have?

Thank You