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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>Data structures</TITLE><META NAME="description" CONTENT="Data structures"><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="node17.html"><LINK REL="previous" HREF="node15.html"><LINK REL="up" HREF="node14.html"><LINK REL="next" HREF="node17.html"></HEAD><BODY BGCOLOR="ivory"><!--Navigation Panel--><B> Next:</B> <A NAME="tex2html288" HREF="node17.html">Examples</A><B>Up:</B> <A NAME="tex2html284" HREF="node14.html">The H2M/cnt extension: models</A><B> Previous:</B> <A NAME="tex2html278" HREF="node15.html">Nomenclature</A><P><!--End of Navigation Panel--><!--Table of Child-Links--><A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A><UL><LI><A NAME="tex2html289" HREF="#SECTION00042100000000000000">Poisson mixture model</A><LI><A NAME="tex2html290" HREF="#SECTION00042200000000000000">Poisson HMM</A><LI><A NAME="tex2html291" HREF="#SECTION00042300000000000000">Negative binomial HMM</A><LI><A NAME="tex2html292" HREF="#SECTION00042400000000000000">Note on the initial distribution of HMMs in <TT>H2M/cnt</TT></A></UL><!--End of Table of Child-Links--><HR><H2><A NAME="SECTION00042000000000000000">Data structures</A></H2><H3><A NAME="SECTION00042100000000000000">Poisson mixture model</A></H3>A Poisson mixture models is defined by the two (1-D) arrays:<DL><DT><STRONG>wght</STRONG></DT><DD>Mixture weights (positive and sum to 1)</DD><DT><STRONG>rate</STRONG></DT><DD>Corresponding rates (parameters of the Poisson distributions)</DD></DL><P><H3><A NAME="SECTION00042200000000000000">Poisson HMM</A></H3>A Poisson HMM is defined by<DL><DT><STRONG>TRANS</STRONG></DT><DD>The transition matrix of the hidden chain</DD><DT><STRONG>rate</STRONG></DT><DD>Corresponding rates (parameters of the Poisson distributions)</DD></DL><P><H3><A NAME="SECTION00042300000000000000">Negative binomial HMM</A></H3>A negative binomial HMM is defined by<DL><DT><STRONG>TRANS</STRONG></DT><DD>The transition matrix of the hidden chain</DD><DT><STRONG>alpha</STRONG></DT><DD>Corresponding (positive) shape parameters</DD><DT><STRONG>beta</STRONG></DT><DD>Corresponding (positive) inverse scales</DD></DL>where the Negative Binomial distribution is such that<!-- MATH \begin{displaymath}\operatorname{P}(N = n) = \left(\begin{array}{c} n+\alpha-1\\ \alpha-1\end{array}\right) \left(\frac{\beta}{1+\beta}\right)^\alpha\left(\frac{1}{1+\beta}\right)^n \qquad \mathrm{for} \quad n \in \mathbb{N}\end{displaymath} --><P></P><DIV ALIGN="CENTER"><IMG WIDTH="377" HEIGHT="46" ALIGN="MIDDLE" BORDER="0" SRC="img11.png" ALT="$\displaystyle \operatorname{P}(N = n) = \left(\begin{array}{c} n+\alpha-1\\\......pha\left(\frac{1}{1+\beta}\right)^n \qquad \mathrm{for} \quad n \in \mathbb{N}$"></DIV><P></P>which has mean <!-- MATH $\alpha/\beta$ --><IMG WIDTH="28" HEIGHT="27" ALIGN="MIDDLE" BORDER="0" SRC="img12.png" ALT="$ \alpha/\beta$"> and variance <!-- MATH $\alpha(1+\beta)/\beta^2$ --><IMG WIDTH="74" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img13.png" ALT="$ \alpha(1+\beta)/\beta^2$">. Thenegative binomial distribution can be viewed as a Poisson (continuous) mixturefor which the rate follows a <!-- MATH $\operatorname{Gamma}(\alpha, \beta)$ --><IMG WIDTH="82" HEIGHT="27" ALIGN="MIDDLE" BORDER="0" SRC="img14.png" ALT="$ \operatorname{Gamma}(\alpha, \beta)$"> distribution(this is the method used for simulating from the model in <code>nbh_gen</code>). Ifyou don't know what the negative binomial distribution is, you should refer,for instance, to [<A HREF="node23.html#JK69dd">8</A>] or perhaps to [<A HREF="node23.html#Grandell:MixPoisson">9</A>].<P><H3><A NAME="SECTION00042400000000000000">Note on the initial distribution of HMMs in <TT>H2M/cnt</TT></A></H3>Contrary to what was the case for the <TT>H2M</TT> main functions, the <TT>H2M/cnt</TT> HMMfunctions are mostly intended to deal with ergodic models which are estimatedfrom a single long observation sequence (whereas left-right HMMs such as thoseused in speech processing need to be trained using multiple observationsequences). With a single (long) training sequence, the initial distribution isa parameter that has little influence and that cannot be estimatedconsistently. Taking this into account, it is assumed that the initialdistribution (usually called <code>pi0</code> in the <TT>H2M</TT> functions is uniform(equal probabilities for all states of the model).<P><P><HR><!--Navigation Panel--><B> Next:</B> <A NAME="tex2html288" HREF="node17.html">Examples</A><B>Up:</B> <A NAME="tex2html284" HREF="node14.html">The H2M/cnt extension: models</A><B> Previous:</B> <A NAME="tex2html278" HREF="node15.html">Nomenclature</A><P><!--End of Navigation Panel--><ADDRESS>Olivier Cappé, Aug 24 2001</ADDRESS></BODY></HTML>
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