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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}
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\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 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\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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