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Hidden Markov Models

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A Hidden Markov Model consists of 1.A sequence of states {X t |t T } = {X 1, X 2,..., X T }, and 2.A sequence of observations {Y t |t T } = {Y 1, Y 2,..., Y T }

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The sequence of states {X 1, X 2,..., X T } form a Markov chain moving amongst the M states {1, 2, …, M}. The observation Y t comes from a distribution that is determined by the current state of the process X t. (or possibly past observations and past states). The states, {X 1, X 2,..., X T }, are unobserved (hence hidden).

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X1X1 X2X2 X3X3 XTXT Y1Y1 Y2Y2 Y3Y3 YTYT The Hidden Markov Model …

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Some basic problems: from the observations {Y 1, Y 2,..., Y T } 1.Determine the sequence of states {X 1, X 2,..., X T }. 2.Determine (or estimate) the parameters of the stochastic process that is generating the states and the observations.;

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Examples

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Example 1 A person is rolling two sets of dice (one is balanced, the other is unbalanced). He switches between the two sets of dice using a Markov transition matrix. The states are the dice. The observations are the numbers rolled each time.

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Balanced Dice

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Unbalanced Dice

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Example 2 The Markov chain is two state. The observations (given the states) are independent Normal. Both mean and variance dependent on state. HMM AR.xls

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Example 3 –Dow Jones

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Daily Changes Dow Jones

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Hidden Markov Model??

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Bear and Bull Market?

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Speech Recognition When a word is spoken the vocalization process goes through a sequence of states. The sound produced is relatively constant when the process remains in the same state. Recognizing the sequence of states and the duration of each state allows one to recognize the word being spoken.

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The interval of time when the word is spoken is broken into small (possibly overlapping) subintervals. In each subinterval one measures the amplitudes of various frequencies in the sound. (Using Fourier analysis). The vector of amplitudes Y t is assumed to have a multivariate normal distribution in each state with the mean vector and covariance matrix being state dependent.

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Hidden Markov Models for Biological Sequence Consider the Motif: [AT][CG][AC][ACGT]*A[TG][GC] Some realizations: ACA---ATG TCAACTATC ACAC--AGC AGA---ATC ACCG--ATC

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A.8 C G T.2 A C.8 G.2 T A.8 C.2 G T A C1.0 G T A C G.2 T.8 A C.8 G.2 T A.2 C.4 G.2 T.2.4 1.0.6.4 Hidden Markov model of the same motif : [AT][CG][AC][ACGT]*A[TG][GC]

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Profile HMMs Begin End

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Computing Likelihood Let ij = P[X t+1 = j|X t = i] and = ( ij ) = the M M transition matrix. Let = P[X 1 = i] and = the initial distribution over the states.

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Now assume that P[Y t = y t |X 1 = i 1, X 2 = i 2,..., X t = i t ] = P[Y t = y t | X t = i t ] = p(y t | ) = Then P[X 1 = i 1,X 2 = i 2..,X T = i T, Y 1 = y 1, Y 2 = y 2,..., Y T = y T ] = P[X = i, Y = y] =

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Therefore P[Y 1 = y 1, Y 2 = y 2,..., Y T = y T ] = P[Y = y]

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In the case when Y 1, Y 2,..., Y T are continuous random variables or continuous random vectors, Let f(y| ) denote the conditional distribution of Y t given X t = i. Then the joint density of Y 1, Y 2,..., Y T is given by = f(y 1, y 2,..., y T ) = f(y) where = f(y t | )

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Efficient Methods for computing Likelihood The Forward Method Consider

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The Backward Procedure

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Prediction of states from the observations and the model:

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The Viterbi Algorithm (Viterbi Paths) Suppose that we know the parameters of the Hidden Markov Model. Suppose in addition suppose that we have observed the sequence of observations Y 1, Y 2,..., Y T. Now consider determining the sequence of States X 1, X 2,..., X T.

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Recall that P[X 1 = i 1,..., X T = i T, Y 1 = y 1,..., Y T = y T ] = P[X = i, Y = y] = Consider the problem of determining the sequence of states, i 1, i 2,..., i T, that maximizes the above probability. This is equivalent to maximizing P[X = i|Y = y] = P[X = i,Y = y] / P[Y = y]

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The Viterbi Algorithm We want to maximize P[X = i, Y = y] = Equivalently we want to minimize U(i 1, i 2,..., i T ) Where U(i 1, i 2,..., i T ) = -ln (P[X = i, Y = y]) =

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Minimization of U(i 1, i 2,..., i T ) can be achieved by Dynamic Programming. This can be thought of as finding the shortest distance through the following grid of points. By starting at the unique point in stage 0 and moving from a point in stage t to a point in stage t+1 in an optimal way. The distances between points in stage t and points in stage t+1 are equal to:

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Stage 0Stage 1Stage 2Stage T-1Stage T... Dynamic Programming

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By starting at the unique point in stage 0 and moving from a point in stage t to a point in stage t+1 in an optimal way. The distances between points in stage t and points in stage t+1 are equal to:

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Stage 0Stage 1Stage 2Stage T-1Stage T... Dynamic Programming

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Stage 0Stage 1Stage 2Stage T-1Stage T... Dynamic Programming

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Let Then and i 1 = 1, 2, …, M i t+1 = 1, 2, …, M; t = 1,…, T-2

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Finally

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Summary of calculations of Viterbi Path 1. i 1 = 1, 2, …, M 2. i t+1 = 1, 2, …, M; t = 1,…, T-2 3.

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An alternative approach to prediction of states from the observations and the model: It can be shown that:

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Forward Probabilities 1. 2.

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Backward Probabilities 1. 2. HMM generator (normal).xls

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Estimation of Parameters of a Hidden Markov Model If both the sequence of observations Y 1, Y 2,..., Y T and the sequence of States X 1, X 2,..., X T is observed Y 1 = y 1, Y 2 = y 2,..., Y T = y T, X 1 = i 1, X 2 = i 2,..., X T = i T, then the Likelihood is given by:

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the log-Likelihood is given by:

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In this case the Maximum Likelihood estimates are: = the MLE of i computed from the observations yt where X t = i.

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MLE (states unknown) If only the sequence of observations Y 1 = y 1, Y 2 = y 2,..., Y T = y T are observed then the Likelihood is given by:

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It is difficult to find the Maximum Likelihood Estimates directly from the Likelihood function. The Techniques that are used are 1. The Segmental K-means Algorithm 2. The Baum-Welch (E-M) Algorithm

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The Segmental K-means Algorithm In this method the parameters are adjusted to maximize where is the Viterbi path

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Consider this with the special case Case: The observations {Y 1, Y 2,..., Y T } are continuous Multivariate Normal with mean vector and covariance matrix when, i.e.

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1.Pick arbitrarily M centroids a 1, a 2, … a M. Assign each of the T observations y t (kT if multiple realizations are observed) to a state i t by determining : 2.Then

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3. And 4.Calculate the Viterbi path (i 1, i 2, …, i T ) based on the parameters of step 2 and 3. 5.If there is a change in the sequence (i 1, i 2, …, i T ) repeat steps 2 to 4.

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The Baum-Welch (E-M) Algorithm The E-M algorithm was designed originally to handle “Missing observations”. In this case the missing observations are the states {X 1, X 2,..., X T }. Assuming a model, the states are estimated by finding their expected values under this model. (The E part of the E-M algorithm).

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With these values the model is estimated by Maximum Likelihood Estimation (The M part of the E-M algorithm). The process is repeated until the estimated model converges.

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The E-M Algorithm Let denote the joint distribution of Y,X. Consider the function: Starting with an initial estimate of. A sequence of estimates are formed by finding to maximize with respect to.

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The sequence of estimates converge to a local maximum of the likelihood.

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Example: Sampling from Mixtures Let y 1, y 2, …, y n denote a sample from the density: where and

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Suppose that m = 2 and let x 1, x 2, …, x 1 denote independent random variables taking on the value 1 with probability and 0 with probability 1- . Suppose that y i comes from the density We will also assume that g(y| i ) is normal with mean i and standard deviation i.

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Thus the joint distribution of x 1, x 2, …, x n and let y 1, y 2, …, y n is:

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In the case of an HMM the log-Likelihood is given by:

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Recall and Expected no. of transitions from state i.

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Let Expected no. of transitions from state i to state j.

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The E-M Re-estimation Formulae Case 1: The observations {Y 1, Y 2,..., Y T } are discrete with K possible values and

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Case 2: The observations {Y 1, Y 2,..., Y T } are continuous Multivariate Normal with mean vector and covariance matrix when, i.e.

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Measuring distance between two HMM’s Let and denote the parameters of two different HMM models. We now consider defining a distance between these two models.

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The Kullback-Leibler distance Consider the two discrete distributions and ( and in the continuous case) then define

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and in the continuous case:

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These measures of distance between the two distributions are not symmetric but can be made symmetric by the following:

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In the case of a Hidden Markov model. where The computation of in this case is formidable

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Juang and Rabiner distance Let denote a sequence of observations generated from the HMM with parameters: Let denote the optimal (Viterbi) sequence of states assuming HMM model.

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Then define: and

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