www.pudn.com > Gaussian.zip > images.tex, change:2004-05-18,size:37831b

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A Tutorial for the Course <em>Computational Intelligence</em>\end{rawhtml}

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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]
$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3320}%$\displaystyle \ensuremath\boldsymbol{\Sigma}_3 = \left[ \begin{array}{cc}
8000 & 8400 \\
8400 & 18500
\end{array} \right]
$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3323}%$ \ensuremath\boldsymbol{\Sigma}_2$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3325}%$ \ensuremath\boldsymbol{\Sigma}_3$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3346}%$ \ensuremath\mathbf{x}_i$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3372}%$ \ensuremath\mathbf{x}^{\mathsf T}\ensuremath\boldsymbol{\Sigma}\, \ensuremath\mathbf{x}\ge 0$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3387}%$ \displaystyle \hat{\ensuremath\boldsymbol{\mu}} = \frac{1}{N}
\sum_{i=1}^{N} \ensuremath\mathbf{x}_i$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3389}%$ \displaystyle \hat{\ensuremath\boldsymbol{\Sigma}} =
\frac{1}{N-1} \; \sum_{i=1}^{N} (\ensuremath\mathbf{x}_i-\ensuremath\boldsymbol{\mu})^{\mathsf T}(\ensuremath\mathbf{x}_i-\ensuremath\boldsymbol{\mu}) $% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3394}%$ {\cal
N}(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma}_3)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3396}%$ \hat{\ensuremath\boldsymbol{\mu}}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3398}%$ \hat{\ensuremath\boldsymbol{\Sigma}}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3400}%$ \hat{\ensuremath\boldsymbol{\mu}}_{(10000)}
=$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3402}%$ \hat{\ensuremath\boldsymbol{\Sigma}}_{(10000)} =$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3404}%$ \hat{\ensuremath\boldsymbol{\mu}}_{(1000)}
=$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3406}%$ \hat{\ensuremath\boldsymbol{\Sigma}}_{(1000)} =$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3408}%$ \hat{\ensuremath\boldsymbol{\mu}}_{(100)}
=$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3410}%$ \hat{\ensuremath\boldsymbol{\Sigma}}_{(100)} =$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3420}%$ \|\mathbf{A}-\mathbf{B}\|_2$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3422}%$ \mathbf{A}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3424}%$ \mathbf{B}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} \stepcounter{subsubsection} \stepcounter{subsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3450}%$ \ensuremath\boldsymbol{\Theta}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3457}%$ p(\ensuremath\mathbf{x}_i|\ensuremath\boldsymbol{\Theta})$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3459}%$ \ensuremath\boldsymbol{\Theta}= (\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3463}%$\displaystyle p(X|\ensuremath\boldsymbol{\Theta}) =      \prod_{i=1}^{N} p(\ensuremath\mathbf{x}_i|\ensuremath\boldsymbol{\Theta}) =      \prod_{i=1}^{N} p(\ensuremath\mathbf{x}_i|\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma}) =      \prod_{i=1}^{N} g_{(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})}(\ensuremath\mathbf{x}_i)$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3466}%$\displaystyle p(X|\ensuremath\boldsymbol{\Theta}) = \prod_{i=1}^{N} p(\ensuremath\mathbf{x}_i|\ensuremath\boldsymbol{\Theta}) \quad \Leftrightarrow \quad
\log p(X|\ensuremath\boldsymbol{\Theta}) = \log \prod_{i=1}^{N} p(\ensuremath\mathbf{x}_i|\ensuremath\boldsymbol{\Theta}) = \sum_{i=1}^{N}
\log p(\ensuremath\mathbf{x}_i|\ensuremath\boldsymbol{\Theta})
$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3469}%$\displaystyle p(\ensuremath\mathbf{x}|\ensuremath\boldsymbol{\Theta})$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3473}%$\displaystyle \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})}$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3475}%$\displaystyle \log p(\ensuremath\mathbf{x}|\ensuremath\boldsymbol{\Theta})$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3479}%$\displaystyle \frac{1}{2} \left[-d \log \left( 2\pi \right)
-  \log \left( \det\left(\ensuremath\boldsymbol{\Sigma}\right) \right)
-  (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})^{\mathsf T}\ensuremath\boldsymbol{\Sigma}^{-1} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})\right]$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3483}%$\displaystyle p(x|\ensuremath\boldsymbol{\Theta}_1) > p(x|\ensuremath\boldsymbol{\Theta}_2) \quad \Leftrightarrow \quad
\log p(x|\ensuremath\boldsymbol{\Theta}_1) > \log p(x|\ensuremath\boldsymbol{\Theta}_2),
$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3501}%$ \log \left( \det\left(\ensuremath\boldsymbol{\Sigma}\right)
\right)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3505}%$ (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})^{\mathsf T}\ensuremath\boldsymbol{\Sigma}^{-1} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3531}%$ \ensuremath\boldsymbol{\Theta}_i = (\ensuremath\boldsymbol{\mu}_i,\ensuremath\boldsymbol{\Sigma}_i)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3533}%$ {\cal N}_1: \; \ensuremath\boldsymbol{\Theta}_1 = \left(
\left[\begin{array}{c}730 \\1090\end{array}\right],
\left[\begin{array}{cc}8000 & 0 \\0 & 8000\end{array}\right]
\right)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3535}%$ {\cal N}_2: \; \ensuremath\boldsymbol{\Theta}_2 = \left(
\left[\begin{array}{c}730 \\1090\end{array}\right],
\left[\begin{array}{cc}8000 & 0 \\0 & 18500\end{array}\right]
\right)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3537}%$ {\cal N}_3: \; \ensuremath\boldsymbol{\Theta}_3 = \left(
\left[\begin{array}{c}730 \\1090\end{array}\right],
\left[\begin{array}{cc}8000 & 8400 \\8400 & 18500\end{array}\right]
\right)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3539}%$ {\cal N}_4: \; \ensuremath\boldsymbol{\Theta}_4 = \left(
\left[\begin{array}{c}270 \\1690\end{array}\right],
\left[\begin{array}{cc}8000 & 8400 \\8400 & 18500\end{array}\right]
\right)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3549}%$\displaystyle \log p(X_3|\ensuremath\boldsymbol{\Theta}_1),\; \log p(X_3|\ensuremath\boldsymbol{\Theta}_2),\; \log p(X_3|\ensuremath\boldsymbol{\Theta}_3),\;$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3550}%$\displaystyle \; \log p(X_3|\ensuremath\boldsymbol{\Theta}_4).
$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} \stepcounter{section} \stepcounter{subsection} \stepcounter{subsubsection} \stepcounter{subsubsection} \stepcounter{subsection} \stepcounter{subsubsection} \stepcounter{subsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3608}%$\displaystyle X \in q_k$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3609}%$\displaystyle \quad P(q_k|X,\ensuremath\boldsymbol{\Theta}) \geq P(q_j|X,\ensuremath\boldsymbol{\Theta}),
$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3619}%$ P(q_k|X,\ensuremath\boldsymbol{\Theta})$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3621}%$ P(q_k|\ensuremath\boldsymbol{\Theta})$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3629}%$\displaystyle P(q_k|X,\ensuremath\boldsymbol{\Theta}) = \frac{p(X|q_k,\ensuremath\boldsymbol{\Theta})\; P(q_k|\ensuremath\boldsymbol{\Theta})}{p(X|\ensuremath\boldsymbol{\Theta})}$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3637}%$ p(X|\ensuremath\boldsymbol{\Theta})=$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3643}%$ p(X|q_k,\ensuremath\boldsymbol{\Theta}) P(q_k|\ensuremath\boldsymbol{\Theta})$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3645}%$\displaystyle P(q_k|X,\ensuremath\boldsymbol{\Theta}) \propto p(X|q_k,\ensuremath\boldsymbol{\Theta})\; P(q_k|\ensuremath\boldsymbol{\Theta}), \quad \forall k
$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3649}%$\displaystyle \log P(q_k|X,\ensuremath\boldsymbol{\Theta}) \propto \log p(X|q_k,\ensuremath\boldsymbol{\Theta}) + \log P(q_k|\ensuremath\boldsymbol{\Theta})$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3653}%$ \ensuremath\boldsymbol{\mu}_k$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3655}%$ \ensuremath\boldsymbol{\Sigma}_k$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3663}%$ p(X|q_k,\ensuremath\boldsymbol{\Theta})$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3665}%$ \log p(X|q_k,\ensuremath\boldsymbol{\Theta})$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3673}%$ (\ensuremath\boldsymbol{\mu}_k,\ensuremath\boldsymbol{\Sigma}_k)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3696}%$ p(X|\ensuremath\boldsymbol{\Theta})$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3704}%$ \ensuremath\mathbf{x}_i=[F_1,F_2]^{\mathsf T}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3706}%$ f_k(\ensuremath\mathbf{x}_i)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3710}%$ f_k(\ensuremath\mathbf{x}_i) = \log
p(\ensuremath\mathbf{x}_i|q_k,\ensuremath\boldsymbol{\Theta}) + \log P(q_k|\ensuremath\boldsymbol{\Theta})$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \renewcommand{\arraystretch}{1.5} \setlength{\tabcolsep}{0.12in}% \setlength{\tabcolsep}{0.12in} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3718}%$ f_{\text{/a/}}(\ensuremath\mathbf{x}_i)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3720}%$ f_{\text{/e/}}(\ensuremath\mathbf{x}_i)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3722}%$ f_{\text{/i/}}(\ensuremath\mathbf{x}_i)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3724}%$ f_{\text{/o/}}(\ensuremath\mathbf{x}_i)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3726}%$ f_{\text{/y/}}(\ensuremath\mathbf{x}_i)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3730}%$ [400,1800]^{\mathsf T}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3732}%$ [400,1000]^{\mathsf T}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3734}%$ [530,1000]^{\mathsf T}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3736}%$ [600,1300]^{\mathsf T}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3738}%$ [670,1300]^{\mathsf T}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3740}%$ [420,2500]^{\mathsf T}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3799}%$ \log p(\ensuremath\mathbf{x}_i|q_k,\ensuremath\boldsymbol{\Theta})$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3810}%$ d-1$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3817}%$ f_k(\ensuremath\mathbf{x})$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3825}%$\displaystyle \ensuremath\mathbf{x}\in q_k \quad \Leftrightarrow \quad f_k(\ensuremath\mathbf{x},\ensuremath\boldsymbol{\Theta}_k) \geq f_l(\ensuremath\mathbf{x},\ensuremath\boldsymbol{\Theta}_l),
$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3831}%$ P(q_k|\ensuremath\mathbf{x}_i)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3838}%$\displaystyle \ensuremath\mathbf{x}\in q_k$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3842}%$\displaystyle P(q_k|\ensuremath\mathbf{x}_i) \geq P(q_l|\ensuremath\mathbf{x}_i),\quad \forall l \neq k$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3846}%$\displaystyle p(\ensuremath\mathbf{x}_i|q_k)\; P(q_k) \geq p(\ensuremath\mathbf{x}_i|q_l)\; P(q_l),\quad
\forall l \neq k$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3850}%$\displaystyle \log p(\ensuremath\mathbf{x}_i|q_k)+\log P(q_k) \geq \log
p(\ensuremath\mathbf{x}_i|q_l)+\log P(q_l),\quad \forall l \neq k$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap3855}% \includegraphics[height=0.95\textheight]{iso}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3884}%$ \ensuremath\boldsymbol{\Sigma}_{\text{/e/}}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} \stepcounter{section} \stepcounter{subsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3904}%$ \ensuremath\mathbf{x}_n$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3906}%$ n=1,\ldots,N$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3911}%$\displaystyle d_k(\ensuremath\mathbf{x}_n)$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3915}%$\displaystyle \, \left\| \ensuremath\mathbf{x}_n - \ensuremath\boldsymbol{\mu}_k \right\|^2$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3919}%$\displaystyle (\ensuremath\mathbf{x}_n-\ensuremath\boldsymbol{\mu}_k)^{\mathsf T}(\ensuremath\mathbf{x}_n-\ensuremath\boldsymbol{\mu}_k)$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3929}%$\displaystyle d_k(\ensuremath\mathbf{x}_n) \, < \, d_l(\ensuremath\mathbf{x}_n), \qquad \forall l \neq k
$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3931}%$\displaystyle J = \sum_{k=1}^{K} \sum_{\ensuremath\mathbf{x}_n \in q_k} d_k(\ensuremath\mathbf{x}_n)
$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3936}%$ \{\ensuremath\boldsymbol{\mu}_{\text{/a/}},
\ensuremath\boldsymbol{\mu}_{\text{/e/}}, \ensuremath\boldsymbol{\mu}_{\text{/i/}}, \ensuremath\boldsymbol{\mu}_{\text{/o/}},
\ensuremath\boldsymbol{\mu}_{\text{/y/}}\}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3946}%$ {\cal N}(\ensuremath\boldsymbol{\mu}_{k},\ensuremath\boldsymbol{\Sigma}_{k}),
\; k=1\ldots K$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline3968}%$ p(X,Q|\ensuremath\boldsymbol{\Theta})$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3978}%$\displaystyle \ensuremath\boldsymbol{\mu}_{k}^{(i+1)} =$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3981}%$\displaystyle \ensuremath\boldsymbol{\Sigma}_{k}^{(i+1)} =$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3984}%$\displaystyle P(q_k^{(i+1)}|\ensuremath\boldsymbol{\Theta}^{(i+1)}) = \frac{\mbox{number of training points
belonging to } q_k^{(i)} }{\mbox{total number of training points}}
$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3987}%$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta})$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3991}%$\displaystyle \sum_{X} P(X|\ensuremath\boldsymbol{\Theta}) \;=\; \sum_{Q} \sum_{X} p(X,Q|\ensuremath\boldsymbol{\Theta})$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay3995}%$\displaystyle \sum_{k=1}^{K} \sum_{\ensuremath\mathbf{x}_n \in q_k} \log p(\ensuremath\mathbf{x}_n|\ensuremath\boldsymbol{\Theta}_k),$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline4000}%$ \{\ensuremath\boldsymbol{\Sigma}_{\text{/a/}}, \ensuremath\boldsymbol{\Sigma}_{\text{/e/}}, \ensuremath\boldsymbol{\Sigma}_{\text{/i/}},
\ensuremath\boldsymbol{\Sigma}_{\text{/o/}}, \ensuremath\boldsymbol{\Sigma}_{\text{/y/}}\}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline4012}%$ P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay4019}%$\displaystyle P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay4023}%$\displaystyle \frac{P(q_k^{(i)}|\ensuremath\boldsymbol{\Theta}^{(i)})
\cdot p(\ensuremath\mathbf{x}_n|q_k^{(i)},\ensuremath\boldsymbol{\Theta}^{(i)})}
{p(\ensuremath\mathbf{x}_n|\ensuremath\boldsymbol{\Theta}^{(i)})}$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay4027}%$\displaystyle \frac{P(q_k^{(i)}|\ensuremath\boldsymbol{\Theta}^{(i)}) \cdot p(\ensuremath\mathbf{x}_n|\ensuremath\boldsymbol{\mu}_k^{(i)},\ensuremath\boldsymbol{\Sigma}_k^{(i)}) }
{\sum_j P(q_j^{(i)}|\ensuremath\boldsymbol{\Theta}^{(i)}) \cdot p(\ensuremath\mathbf{x}_n|\ensuremath\boldsymbol{\mu}_j^{(i)},\ensuremath\boldsymbol{\Sigma}_j^{(i)}) }$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay4037}%$\displaystyle \ensuremath\boldsymbol{\mu}_{k}^{(i+1)} = \frac{\sum_{n=1}^{N} \ensuremath\mathbf{x}_n
P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})}
{\sum_{n=1}^{N} P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay4039}%$\displaystyle \ensuremath\boldsymbol{\Sigma}_{k}^{(i+1)} = \frac{\sum_{n=1}^{N} P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})\;
(\ensuremath\mathbf{x}_n - \ensuremath\boldsymbol{\mu}_k^{(i+1)})(\ensuremath\mathbf{x}_n - \ensuremath\boldsymbol{\mu}_k^{(i+1)})^{\mathsf T}}
{\sum_{n=1}^{N} P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay4041}%$\displaystyle P(q_k^{(i+1)}|\ensuremath\boldsymbol{\Theta}^{(i+1)}) = \frac{1}{N} \sum_{n=1}^{N}
P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)}) $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay4044}%$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta}) = \log p(X|\ensuremath\boldsymbol{\Theta})$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay4045}%$\displaystyle = \log \sum_Q p(X,Q|\ensuremath\boldsymbol{\Theta})$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay4046}%$\displaystyle = \log \sum_Q P(Q|X,\ensuremath\boldsymbol{\Theta})p(X|\ensuremath\boldsymbol{\Theta})$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay4047}%$\displaystyle = \log \sum_{k=1}^{K} P(q_k|X,\ensuremath\boldsymbol{\Theta}) p(X|\ensuremath\boldsymbol{\Theta})$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay4050}%$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta}) \ge J(\ensuremath\boldsymbol{\Theta})$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay4051}%$\displaystyle = \sum_{k=1}^{K} P(q_k|X,\ensuremath\boldsymbol{\Theta}) \log p(X|\ensuremath\boldsymbol{\Theta})$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay4052}%$\displaystyle = \sum_{k=1}^{K} \sum_{n=1}^{N} P(q_k|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}) \log p(\ensuremath\mathbf{x}_n|\ensuremath\boldsymbol{\Theta})$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline4054}%$ J(\ensuremath\boldsymbol{\Theta})$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline4056}%$ {\cal
L}(\ensuremath\boldsymbol{\Theta})\$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\end{document}