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# mblondel/hmm.tex

Created July 12, 2010 14:42
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Good-looking HMM and Lattice diagrams using TikZ
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 % (C) Mathieu Blondel, July 2010 \documentclass[a4paper,10pt]{article} \usepackage[english]{babel} \usepackage[T1]{fontenc} \usepackage[ansinew]{inputenc} \usepackage{lmodern} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{tikz} \begin{document} \tikzstyle{state}=[shape=circle,draw=blue!50,fill=blue!20] \tikzstyle{observation}=[shape=rectangle,draw=orange!50,fill=orange!20] \tikzstyle{lightedge}=[<-,dotted] \tikzstyle{mainstate}=[state,thick] \tikzstyle{mainedge}=[<-,thick] \begin{figure}[htbp] \begin{center} \begin{tikzpicture}[] % states \node[state] (s1) at (0,2) {$s_1$} edge [loop above] (); \node[state] (s2) at (2,2) {$s_2$} edge [<-,bend right=45] node[auto,swap] {$a_{12}$} (s1) edge [->,bend left=45] (s1) edge [loop above] (); \node[state] (s3) at (4,2) {$s_3$} edge [<-,bend right=45] (s2) edge [->,bend left=45] (s2) edge [loop above] (); \node[state] (s4) at (6,2) {$s_4$} edge [<-,bend right=45] (s3) edge [->,bend left=45] (s3) edge [loop above] (); % observations \node[observation] (y1) at (2,0) {$y_1$} edge [lightedge] (s1) edge [lightedge] (s2) edge [lightedge] (s3) edge [lightedge] (s4); \node[observation] (y2) at (4,0) {$y_2$} edge [lightedge] (s1) edge [lightedge] (s2) edge [lightedge] (s3) edge [lightedge] node[auto,swap] {$b_4(y_2)$} (s4); \end{tikzpicture} \end{center} \caption{An HMM with 4 states which can emit 2 discrete symbols $y_1$ or $y_2$. $a_{ij}$ is the probability to transition from state $s_i$ to state $s_j$. $b_j(y_k)$ is the probability to emit symbol $y_k$ in state $s_j$. In this particular HMM, states can only reach themselves or the adjacent state.} \end{figure} \begin{figure}[htbp] \begin{center} \begin{tikzpicture}[] % 1st column \node at (0,6) {$t=1$}; \node[state] (s1_1) at (0,5) {$s_1$}; \node[mainstate] (s2_1) at (0,4) {$s_2$}; \node[state] (s3_1) at (0,3) {$s_3$}; \node[state] (s4_1) at (0,2) {$s_4$}; \node at (0,1) {$y_1$}; % 2nd column \node at (2,6) {$t=2$}; \node[mainstate] (s1_2) at (2,5) {$s_1$} edge[lightedge] (s1_1) edge[mainedge] (s2_1) edge[lightedge] (s3_1) edge[lightedge] (s4_1); \node[state] (s2_2) at (2,4) {$s_2$} edge[lightedge] (s1_1) edge[lightedge] (s2_1) edge[lightedge] (s3_1) edge[lightedge] (s4_1); \node[state] (s3_2) at (2,3) {$s_3$} edge[lightedge] (s1_1) edge[lightedge] (s2_1) edge[lightedge] (s3_1) edge[lightedge] (s4_1); \node[state] (s4_2) at (2,2) {$s_4$} edge[lightedge] (s1_1) edge[lightedge] (s2_1) edge[lightedge] (s3_1) edge[lightedge] (s4_1); \node at (2,1) {$y_2$}; % 3rd column \node at (4,6) {$t=3$}; \node[mainstate] (s1_3) at (4,5) {$s_1$} edge[mainedge] (s1_2) edge[lightedge] (s2_2) edge[lightedge] (s3_2) edge[lightedge] (s4_2); \node[state] (s2_3) at (4,4) {$s_2$} edge[lightedge] (s1_2) edge[lightedge] (s2_2) edge[lightedge] (s3_2) edge[lightedge] (s4_2); \node[state] (s3_3) at (4,3) {$s_3$} edge[lightedge] (s1_2) edge[lightedge] (s2_2) edge[lightedge] (s3_2) edge[lightedge] (s4_2); \node[state] (s4_3) at (4,2) {$s_4$} edge[lightedge] (s1_2) edge[lightedge] (s2_2) edge[lightedge] (s3_2) edge[lightedge] (s4_2); \node at (4,1) {$y_2$}; \end{tikzpicture} \end{center} \caption{Trellis of the observation sequence $y_1$,$y_2$,$y_2$ for the above HMM. The thick arrows indicate the most probable transitions. As an example, the transition between state $s_1$ at time t=2 and state $s_4$ at time t=3 has probability $\alpha_2(1)a_{14}b_4(y_2)$, where $\alpha_t(i)$ is the probability to be in state $s_i$ at time t.} \end{figure} \end{document}

### Jim-Holmstroem commented May 6, 2012

Nice tikz:ing :D thanks :D

Awesome. Thanks

Thanks!

thank s man

### Apdulai commented Feb 18, 2023

Great !
Thanks for sharing