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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"><!--Converted with LaTeX2HTML 2K.1beta (1.48)original version by: Nikos Drakos, CBLU, University of Leeds* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan* with significant contributions from: Jens Lippmann, Marek Rouchal, Martin Wilck and others --><HTML><HEAD><TITLE>Simple types: ex_basic</TITLE><META NAME="description" CONTENT="Simple types: ex_basic"><META NAME="keywords" CONTENT="H2M, H2M/cnt, Hidden Markov Model, HMM, Mixture model, Vector Quantization, Expectation Maximization, EM, Multivariate Gaussian, Count data, Poisson, Negative binomial, MATLAB, OCTAVE, GPL"><META NAME="resource-type" CONTENT="document"><META NAME="distribution" CONTENT="global"><META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1"><META NAME="Generator" CONTENT="LaTeX2HTML v2K.1beta"><META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css"><LINK REL="STYLESHEET" HREF="h2m.css"><LINK REL="next" HREF="node11.html"><LINK REL="previous" HREF="node9.html"><LINK REL="up" HREF="node8.html"><LINK REL="next" HREF="node11.html"></HEAD><BODY BGCOLOR="ivory"><!--Navigation Panel--><B> Next:</B> <A NAME="tex2html205" HREF="node11.html">Another mixture example: ex_bic</A><B>Up:</B> <A NAME="tex2html201" HREF="node8.html">Models with multivariate Gaussian</A><B> Previous:</B> <A NAME="tex2html195" HREF="node9.html">Data structures</A><P><!--End of Navigation Panel--><!--Table of Child-Links--><A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A><UL><LI><A NAME="tex2html206" HREF="#SECTION00032100000000000000">Ergodic model with full covariance matrices</A><LI><A NAME="tex2html207" HREF="#SECTION00032200000000000000">Left-right HMM</A><LI><A NAME="tex2html208" HREF="#SECTION00032300000000000000">Mixture model</A></UL><!--End of Table of Child-Links--><HR><H2><A NAME="SECTION00032000000000000000"></A><A NAME="sec:ex"></A><BR>Simple types: ex_basic</H2>The script <code>ex_basic.m</code> contains some code corresponding to lowdimensional examples of the three basic HMM types that are handled by <TT>H2M</TT>.<P><H3><A NAME="SECTION00032100000000000000">Ergodic model with full covariance matrices</A></H3>Let <code>X</code> denote a matrix containing <code>T</code> observed vectors, the EMestimation of the parameters take the following form:<PRE>for i = 1:n_iter [alpha, beta, logscale, dens] = hmm_fb(X, A, pi0, mu, Sigma); logl(i) = log(sum(alpha(T,:))) + logscale; [A, pi0] = hmm_tran(alpha, beta, dens, A, pi0); [mu, Sigma] = hmm_dens(X, alpha, beta, COV_TYPE);end;</PRE><code>COV_TYPE</code> is just a flag that should be set to 0 when using fullcovariance matrices and to 1 for diagonal covariance matrices.<P>Notice that at each step, the log-likelihood is computed from the forwardvariables using a correction term <code>logscale</code> returned by <code>hmm_fb</code>(for forward-backward) which contains the sum of the logarithmic scalingfactors used during the computation of <code>alpha</code> and <code>beta</code>. In thepresent version scaling of the forward variable is performed at each time index<code>t = 2:T</code> (which means that each row of the <code>alpha</code> matrix sums toone, except the first one). This systematic scaling appears to be much saferwhen using input data with largely varying range. The backward variable isscaled using the same normalization factors as indicated in [<A HREF="node23.html#Rabiner:HMM">4</A>](using exactly the same normalization factors is important for there-estimation of the coefficients of the transition matrix). Note that the meresuppression of the scaling procedure would lead to numerical problems in almostevery cases of interest (when the length of the observation sequences ifgreater than 50 for instance) despite the double precision representation usedin MATLAB/OCTAVE.<P>If you don't want to see what's going on you can simply use<PRE>[A, pi0, mu, Sigma, logl] = hmm(X, A, pi0, mu, Sigma, n_iter);</PRE>which calls the very same piece of code except for the fact that the messagesconcerning the execution time are suppressed: all the computational functionsprint those messages by default but this can be suppressed by supplying andoptional argument (named <code>QUIET</code>) different from zero.<P><H3><A NAME="SECTION00032200000000000000">Left-right HMM</A></H3>This case is not really different from the last one except that the case ofmultiple observation sequences has been considered:<PRE>for i = 1:n_iter [A, logl(i), gamma] = hmm_mest(X, st, A, mu, Sigma); [mu, Sigma] = mix_par(X, gamma, COV_TYPE);end;</PRE>In this case, the matrix <code>X</code> contains all the observation sequences andthe vector <code>st</code> yields the index corresponding to the beginning of eachsequence so that <code>X(1:st(2)-1,:)</code> contains the vectors that correspond tothe first observation sequence, and so on until<code>X(st(length(st)),length(X(1,:),:)</code> which corresponds to the lastobservation sequence.<P>The transition parameters are re-estimated inside <code>hmm_mest</code> and the aposteriori distribution of the states are returned in <code>gamma</code>. Once again,if you don't want to see what's happening you could more simply use<PRE>[A, mu, Sigma, logl] = hmm(X, st, A, mu, Sigma, n_iter);</PRE><P>In fact, if you need to estimate the parameter of an ergodic model usingmultiple (independent) observation sequences, you may use <code>hmm_mest</code> aswell, but <code>hmm_mest</code> assumes that <code>pi0 = [1 0 ... 0]</code> (ie. that theMarkov chain starts from the first state).<P><H3><A NAME="SECTION00032300000000000000">Mixture model</A></H3>The EM estimation in the case of mixture model is achieved through<PRE>for i = 1:n_iter [gamma, logl(i)] = mix_post(X, w, mu, Sigma); [mu, Sigma, w] = mix_par(X, gamma, COV_TYPE);end;</PRE>or, more simply<PRE>[w, mu, Sigma, logl] = mix(X, w, mu, Sigma, n_iter);</PRE><P><P><HR><!--Navigation Panel--><B> Next:</B> <A NAME="tex2html205" HREF="node11.html">Another mixture example: ex_bic</A><B>Up:</B> <A NAME="tex2html201" HREF="node8.html">Models with multivariate Gaussian</A><B> Previous:</B> <A NAME="tex2html195" HREF="node9.html">Data structures</A><P><!--End of Navigation Panel--><ADDRESS>Olivier Cappé, Aug 24 2001</ADDRESS></BODY></HTML>
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