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📁 高斯混合模型算法
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\batchmode\documentclass[a4paper]{article}\RequirePackage{ifthen}\usepackage{array,amsmath,amssymb,rotating,graphicx}\PassOptionsToPackage{colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue}{hyperref}\usepackage{html}\usepackage[dvips]{color}\pagecolor[gray]{1.0}%\providecommand{\HREF}[2]{\htmladdnormallink{#2}{#1}}%\providecommand{\href}[2]{\htmladdnormallink{#2}{#1}} %\providecommand{\MakeTutorialTitle}[1]{\title{Tutorial: #1}\begin{rawhtml}<h1>\end{rawhtml}#1\begin{rawhtml}</h1>A Tutorial for the Course <em>Computational Intelligence</em>\end{rawhtml}} %\providecommand{\mat}[1]{{\tt >> #1} \\}%\providecommand{\com}[1]{{\tt #1}} %\providecommand{\tab}{\hspace{1em}} %\providecommand{\trn}{^{\mathsf T}}  % transposition%\providecommand{\xv}{\ensuremath\mathbf{x}}   % vector x%\providecommand{\muv}{\ensuremath\boldsymbol{\mu}}   % vector mu%\providecommand{\Sm}{\ensuremath\boldsymbol{\Sigma}}   % matrix Sigma%\providecommand{\Tm}{\ensuremath\boldsymbol{\Theta}}   % matrix Sigma%\providecommand{\Rf}{\ensuremath\mathbb{R}} \setlength{\hoffset}{-1in} 
\setlength{\voffset}{-1in} \setlength{\topskip}{0cm} 
\setlength{\headheight}{0cm} 
\setlength{\headsep}{0cm} \setlength{\textwidth}{16cm} 
\setlength{\evensidemargin}{2.5cm} 
\setlength{\oddsidemargin}{2.5cm} \setlength{\textheight}{24cm} 
\setlength{\topmargin}{2.5cm} 
\setlength{\headheight}{0.5cm} 
\setlength{\headsep}{0.5cm} \usepackage[latin1]{inputenc}\makeatletter\makeatletter\count@=\the\catcode`\_ \catcode`\_=8 \newenvironment{tex2html_wrap}{}{}%\catcode`\<=12\catcode`\_=\count@\newcommand{\providedcommand}[1]{\expandafter\providecommand\csname #1\endcsname}%\newcommand{\renewedcommand}[1]{\expandafter\providecommand\csname #1\endcsname{}%  \expandafter\renewcommand\csname #1\endcsname}%\newcommand{\newedenvironment}[1]{\newenvironment{#1}{}{}\renewenvironment{#1}}%\let\newedcommand\renewedcommand\let\renewedenvironment\newedenvironment\makeatother\let\mathon=$\let\mathoff=$\ifx\AtBeginDocument\undefined \newcommand{\AtBeginDocument}[1]{}\fi\newbox\sizebox\setlength{\hoffset}{0pt}\setlength{\voffset}{0pt}\addtolength{\textheight}{\footskip}\setlength{\footskip}{0pt}\addtolength{\textheight}{\topmargin}\setlength{\topmargin}{0pt}\addtolength{\textheight}{\headheight}\setlength{\headheight}{0pt}\addtolength{\textheight}{\headsep}\setlength{\headsep}{0pt}\setlength{\textwidth}{451pt}\setlength{\textheight}{554pt}\newwrite\lthtmlwrite\makeatletter\let\realnormalsize=\normalsize\global\topskip=2sp\def\preveqno{}\let\real@float=\@float \let\realend@float=\end@float\def\@float{\let\@savefreelist\@freelist\real@float}\def\liih@math{\ifmmode$\else\bad@math\fi}\def\end@float{\realend@float\global\let\@freelist\@savefreelist}\let\real@dbflt=\@dbflt \let\end@dblfloat=\end@float\let\@largefloatcheck=\relax\let\if@boxedmulticols=\iftrue\def\@dbflt{\let\@savefreelist\@freelist\real@dbflt}\def\adjustnormalsize{\def\normalsize{\mathsurround=0pt \realnormalsize \parindent=0pt\abovedisplayskip=0pt\belowdisplayskip=0pt}% \def\phantompar{\csname par\endcsname}\normalsize}%\def\lthtmltypeout#1{{\let\protect\string \immediate\write\lthtmlwrite{#1}}}%\newcommand\lthtmlhboxmathA{\adjustnormalsize\setbox\sizebox=\hbox\bgroup\kern.05em }%\newcommand\lthtmlhboxmathB{\adjustnormalsize\setbox\sizebox=\hbox to\hsize\bgroup\hfill }%\newcommand\lthtmlvboxmathA{\adjustnormalsize\setbox\sizebox=\vbox\bgroup % \let\ifinner=\iffalse \let\)\liih@math }%\newcommand\lthtmlboxmathZ{\@next\next\@currlist{}{\def\next{\voidb@x}}% \expandafter\box\next\egroup}%\newcommand\lthtmlmathtype[1]{\gdef\lthtmlmathenv{#1}}%\newcommand\lthtmllogmath{\lthtmltypeout{l2hSize %:\lthtmlmathenv:\the\ht\sizebox::\the\dp\sizebox::\the\wd\sizebox.\preveqno}}%\newcommand\lthtmlfigureA[1]{\let\@savefreelist\@freelist       \lthtmlmathtype{#1}\lthtmlvboxmathA}%\newcommand\lthtmlpictureA{\bgroup\catcode`\_=8 \lthtmlpictureB}%\newcommand\lthtmlpictureB[1]{\lthtmlmathtype{#1}\egroup       \let\@savefreelist\@freelist \lthtmlhboxmathB}%\newcommand\lthtmlpictureZ[1]{\hfill\lthtmlfigureZ}%\newcommand\lthtmlfigureZ{\lthtmlboxmathZ\lthtmllogmath\copy\sizebox       \global\let\@freelist\@savefreelist}%\newcommand\lthtmldisplayA{\bgroup\catcode`\_=8 \lthtmldisplayAi}%\newcommand\lthtmldisplayAi[1]{\lthtmlmathtype{#1}\egroup\lthtmlvboxmathA}%\newcommand\lthtmldisplayB[1]{\edef\preveqno{(\theequation)}%  \lthtmldisplayA{#1}\let\@eqnnum\relax}%\newcommand\lthtmldisplayZ{\lthtmlboxmathZ\lthtmllogmath\lthtmlsetmath}%\newcommand\lthtmlinlinemathA{\bgroup\catcode`\_=8 \lthtmlinlinemathB}\newcommand\lthtmlinlinemathB[1]{\lthtmlmathtype{#1}\egroup\lthtmlhboxmathA  \vrule height1.5ex width0pt }%\newcommand\lthtmlinlineA{\bgroup\catcode`\_=8 \lthtmlinlineB}%\newcommand\lthtmlinlineB[1]{\lthtmlmathtype{#1}\egroup\lthtmlhboxmathA}%\newcommand\lthtmlinlineZ{\egroup\expandafter\ifdim\dp\sizebox>0pt %  \expandafter\centerinlinemath\fi\lthtmllogmath\lthtmlsetinline}\newcommand\lthtmlinlinemathZ{\egroup\expandafter\ifdim\dp\sizebox>0pt %  \expandafter\centerinlinemath\fi\lthtmllogmath\lthtmlsetmath}\newcommand\lthtmlindisplaymathZ{\egroup %  \centerinlinemath\lthtmllogmath\lthtmlsetmath}\def\lthtmlsetinline{\hbox{\vrule width.1em \vtop{\vbox{%  \kern.1em\copy\sizebox}\ifdim\dp\sizebox>0pt\kern.1em\else\kern.3pt\fi  \ifdim\hsize>\wd\sizebox \hrule depth1pt\fi}}}\def\lthtmlsetmath{\hbox{\vrule width.1em\kern-.05em\vtop{\vbox{%  \kern.1em\kern0.7 pt\hbox{\hglue.17em\copy\sizebox\hglue0.7 pt}}\kern.3pt%  \ifdim\dp\sizebox>0pt\kern.1em\fi \kern0.7 pt%  \ifdim\hsize>\wd\sizebox \hrule depth1pt\fi}}}\def\centerinlinemath{%  \dimen1=\ifdim\ht\sizebox<\dp\sizebox \dp\sizebox\else\ht\sizebox\fi  \advance\dimen1by.5pt \vrule width0pt height\dimen1 depth\dimen1  \dp\sizebox=\dimen1\ht\sizebox=\dimen1\relax}\def\lthtmlcheckvsize{\ifdim\ht\sizebox<\vsize   \ifdim\wd\sizebox<\hsize\expandafter\hfill\fi \expandafter\vfill  \else\expandafter\vss\fi}%\providecommand{\selectlanguage}[1]{}%\makeatletter \tracingstats = 1 \providecommand{\Eta}{\textrm{H}}\providecommand{\Mu}{\textrm{M}}\providecommand{\Alpha}{\textrm{A}}\providecommand{\Iota}{\textrm{J}}\providecommand{\Nu}{\textrm{N}}\providecommand{\Omicron}{\textrm{O}}\providecommand{\omicron}{\textrm{o}}\providecommand{\Chi}{\textrm{X}}\providecommand{\Beta}{\textrm{B}}\providecommand{\Kappa}{\textrm{K}}\providecommand{\Tau}{\textrm{T}}\providecommand{\Epsilon}{\textrm{E}}\providecommand{\Zeta}{\textrm{Z}}\providecommand{\Rho}{\textrm{R}}\begin{document}\pagestyle{empty}\thispagestyle{empty}\lthtmltypeout{}%\lthtmltypeout{latex2htmlLength hsize=\the\hsize}\lthtmltypeout{}%\lthtmltypeout{latex2htmlLength vsize=\the\vsize}\lthtmltypeout{}%\lthtmltypeout{latex2htmlLength hoffset=\the\hoffset}\lthtmltypeout{}%\lthtmltypeout{latex2htmlLength voffset=\the\voffset}\lthtmltypeout{}%\lthtmltypeout{latex2htmlLength topmargin=\the\topmargin}\lthtmltypeout{}%\lthtmltypeout{latex2htmlLength topskip=\the\topskip}\lthtmltypeout{}%\lthtmltypeout{latex2htmlLength headheight=\the\headheight}\lthtmltypeout{}%\lthtmltypeout{latex2htmlLength headsep=\the\headsep}\lthtmltypeout{}%\lthtmltypeout{latex2htmlLength parskip=\the\parskip}\lthtmltypeout{}%\lthtmltypeout{latex2htmlLength oddsidemargin=\the\oddsidemargin}\lthtmltypeout{}%\makeatletter\if@twoside\lthtmltypeout{latex2htmlLength evensidemargin=\the\evensidemargin}%\else\lthtmltypeout{latex2htmlLength evensidemargin=\the\oddsidemargin}\fi%\lthtmltypeout{}%\makeatother\setcounter{page}{1}\onecolumn% !!! IMAGES START HERE !!!{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3212}%$ {\cal  N}(\ensuremath  \boldsymbol  {\mu }_{\text  {/i/}},\ensuremath  \boldsymbol  {\Sigma }_{\text  {/i/}})$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3214}%$ {\cal  N}(\ensuremath  \boldsymbol  {\mu }_{\text  {/e/}},\ensuremath  \boldsymbol  {\Sigma }_{\text  {/e/}})$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3216}%$ {\cal  N}(\ensuremath  \boldsymbol  {\mu }_{\text  {/i/}},\ensuremath  \boldsymbol  {\Sigma }_{\text  {/e/}})$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}\stepcounter{section}\stepcounter{subsection}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3233}%$ \ensuremath\mathbf{x}\circlearrowleft {\cal
N}(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3235}%$ \ensuremath\mathbf{x}\in \ensuremath\mathbb{R}^d$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_indisplay3237}%$\displaystyle g_{(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})}(\ensuremath\mathbf{x}) = \frac{1}{\sqrt{2\pi}^d
       \sqrt{\det\left(\ensuremath\boldsymbol{\Sigma}\right)}} \, e^{-\frac{1}{2} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})^{\mathsf T}
       \ensuremath\boldsymbol{\Sigma}^{-1} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})}$%\lthtmlindisplaymathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3239}%$ \ensuremath\boldsymbol{\mu}$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3241}%$ \ensuremath\boldsymbol{\Sigma}$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3249}%$ \mu_i = E(x_i)$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_indisplay3263}%$\displaystyle \ensuremath\boldsymbol{\Sigma}=
   \left[
     \begin{array}{*{4}{c}}
       c_{11} & c_{12} & \cdots & c_{1n} \\
       c_{21} & c_{22} & \cdots & c_{2n} \\
       \vdots & \vdots & \ddots & \vdots \\
       c_{n1} & c_{n2} & \cdots & c_{nn} \\
     \end{array}
   \right]$%\lthtmlindisplaymathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3267}%$ x_i$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3269}%$ x_j$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3271}%$ \ensuremath\mathbf{x}$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_indisplay3273}%$\displaystyle c_{ij} = E\left((x_i-\mu_i)^{\mathsf T}\,(x_j-\mu_j)\right).$%\lthtmlindisplaymathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3279}%$ i\ne j$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3281}%$ c_{ij} = 0$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3285}%$ \sqrt{\ensuremath\boldsymbol{\Sigma}}$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3289}%$ \ensuremath\mathbf{x}\circlearrowleft {\cal N}(\mathbf{0},\mathbf{I})$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3293}%$ \mathbf{I}$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3295}%$ \mathbf{y} = \ensuremath\boldsymbol{\mu}+
\sqrt{\ensuremath\boldsymbol{\Sigma}}\,\ensuremath\mathbf{x}$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3297}%$ \mathbf{y} \circlearrowleft {\cal
N}(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}\stepcounter{subsubsection}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_inline3304}%$ X=\{\ensuremath\mathbf{x}_1,
\ensuremath\mathbf{x}_2,\ldots,\ensuremath\mathbf{x}_N\}$%\lthtmlinlinemathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_indisplay3308}%$\displaystyle \ensuremath\boldsymbol{\mu}= \left[ \begin{array}{c} 730 \\1090 \end{array} \right]
$%\lthtmlindisplaymathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_indisplay3312}%$\displaystyle \ensuremath\boldsymbol{\Sigma}_1 = \left[ \begin{array}{cc}
8000 & 0 \\
0    & 8000
\end{array} \right]
$%\lthtmlindisplaymathZ\lthtmlcheckvsize\clearpage}{\newpage\clearpage\lthtmlinlinemathA{tex2html_wrap_indisplay3316}%$\displaystyle \ensuremath\boldsymbol{\Sigma}_2 = \left[ \begin{array}{cc}
8000 & 0 \\
0    & 18500
\end{array} \right]
$%

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