images.pl

高斯混合模型算法

# LaTeX2HTML 2002-2-1 (1.70)
# Associate images original text with physical files.


$key = q/ensuremathmathbf{x}circlearrowleft{calN}(mathbf{0},mathbf{I});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.gif"
 ALT="$ \ensuremath\mathbf{x}\circlearrowleft {\cal N}(\mathbf{0},\mathbf{I})$">|; 

$key = q/ensuremathmathbf{x}inensuremathmathbb{R}^d;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7.gif"
 ALT="$ \ensuremath\mathbf{x}\in \ensuremath\mathbb{R}^d$">|; 

$key = q/displaystylep(x|ensuremathboldsymbol{Theta}_1)>p(x|ensuremathboldsymbol{Theta}_2nsuremathboldsymbol{Theta}_1)>logp(x|ensuremathboldsymbol{Theta}_2),;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.gif"
 ALT="$\displaystyle p(x\vert\ensuremath\boldsymbol{\Theta}_1) &gt; p(x\vert\ensuremath\b...
...emath\boldsymbol{\Theta}_1) &gt; \log p(x\vert\ensuremath\boldsymbol{\Theta}_2),
$">|; 

$key = q/displaystylefrac{P(q_k^{(i)}|ensuremathboldsymbol{Theta}^{(i)})cdotp(ensuremathm^{(i)})}{p(ensuremathmathbf{x}_n|ensuremathboldsymbol{Theta}^{(i)})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img170.gif"
 ALT="$\displaystyle \frac{P(q_k^{(i)}\vert\ensuremath\boldsymbol{\Theta}^{(i)})
\cdot...
...}^{(i)})}
{p(\ensuremath\mathbf{x}_n\vert\ensuremath\boldsymbol{\Theta}^{(i)})}$">|; 

$key = q/n=1,cdots,N;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img123.gif"
 ALT="$ n=1,\cdots,N$">|; 

$key = q/displaystyleg_{(mu,Sigma)}(x)=frac{1}{sqrt{2pi}^dsqrt{detleft(Sigmaright)}},e^{-frac{1}{2}(x-mu)^{mathsfT}Sigma^{-1}(x-mu)};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7.gif"
 ALT="$\displaystyle g_{(\mu,\Sigma)}(x) = \frac{1}{\sqrt{2\pi}^d
\sqrt{\det\left(\Sigma\right)}}   e^{-\frac{1}{2} (x-\mu)^{\mathsf T}
\Sigma^{-1} (x-\mu)}
$">|; 

$key = q/p(ensuremathmathbf{x}_i|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.gif"
 ALT="$ p(\ensuremath\mathbf{x}_i\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/Sigma_2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.gif"
 ALT="$ \Sigma_2$">|; 

$key = q/displaystyleensuremathboldsymbol{Sigma}_2=left[{array}{cc}8000&00&18500{array}right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img37.gif"
 ALT="$\displaystyle \ensuremath\boldsymbol{\Sigma}_2 = \left[ \begin{array}{cc}
8000 &amp; 0 \\\\
0 &amp; 18500
\end{array} \right]
$">|; 

$key = q/Theta;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img86.gif"
 ALT="$ \Theta$">|; 

$key = q/f_{text{slashoslash}}(ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img118.gif"
 ALT="$ f_{\text{/o/}}(\ensuremath\mathbf{x}_i)$">|; 

$key = q/{calN}_1:;Theta_1=left(left[{array}{c}7301090{array}right],left[{array}{cc}8000&00&8000{array}right]right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.gif"
 ALT="$ {\cal N}_1: \; \Theta_1 = \left(
\left[\begin{array}{c}730  1090\end{array}\right],
\left[\begin{array}{cc}8000 &amp; 0  0 &amp; 8000\end{array}\right]
\right)$">|; 

$key = q/displaystyled_k(x_n);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img125.gif"
 ALT="$\displaystyle d_k(x_n)$">|; 

$key = q/p(X|q_k,Theta)P(q_k|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img90.gif"
 ALT="$ p(X\vert q_k,\Theta)
P(q_k\vert\Theta)$">|; 

$key = q/displaystyleP(q_k^{(i)}|ensuremathmathbf{x}_n,ensuremathboldsymbol{Theta}^{(i)});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img169.gif"
 ALT="$\displaystyle P(q_k^{(i)}\vert\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})$">|; 

$key = q/displaystylelogP(q_k|X,ensuremathboldsymbol{Theta})proptologp(X|q_k,ensuremathboldsymbol{Theta})+logP(q_k|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img104.gif"
 ALT="$\displaystyle \log P(q_k\vert X,\ensuremath\boldsymbol{\Theta}) \propto \log p(...
...ensuremath\boldsymbol{\Theta}) + \log P(q_k\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/p(X,Q|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img155.gif"
 ALT="$ p(X,Q\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/P(q_k|X,ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img97.gif"
 ALT="$ P(q_k\vert X,\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/displaystyle{calL}(Theta)=logp(X|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img157.gif"
 ALT="$\displaystyle {\cal L}(\Theta) = \log p(X\vert\Theta)$">|; 

$key = q/P(q_k^{(i)}|x_n,Theta^{(i)});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img149.gif"
 ALT="$ P(q_k^{(i)}\vert x_n,\Theta^{(i)})$">|; 

$key = q/displaystyleP(q_k|x)geqP(q_l|x),foralllneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img115.gif"
 ALT="$\displaystyle P(q_k\vert x) \geq P(q_l\vert x), \forall l \neq k$">|; 

$key = q/{calN}_2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img85.gif"
 ALT="$ {\cal N}_2$">|; 

$key = q/[530,1000]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.gif"
 ALT="$ [530,1000]^{\mathsf T}$">|; 

$key = q/ensuremathboldsymbol{Sigma}_{text{slasheslash}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img139.gif"
 ALT="$ \ensuremath\boldsymbol{\Sigma}_{\text{/e/}}$">|; 

$key = q/X_1;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.gif"
 ALT="$ X_1$">|; 

$key = q/|mathbf{A}-mathbf{B}|_2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.gif"
 ALT="$ \Vert\mathbf{A}-\mathbf{B}\Vert _2$">|; 

$key = q/displaystyleensuremathboldsymbol{Sigma}_{k}^{(i+1)}=frac{sum_{n=1}^{N}P(q_k^{(i)(q_k^{(i)}|ensuremathmathbf{x}_n,ensuremathboldsymbol{Theta}^{(i)})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="372" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img174.gif"
 ALT="$\displaystyle \ensuremath\boldsymbol{\Sigma}_{k}^{(i+1)} = \frac{\sum_{n=1}^{N}...
...P(q_k^{(i)}\vert\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})} $">|; 

$key = q/p(X|q_k,Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img97.gif"
 ALT="$ p(X\vert q_k,\Theta)$">|; 

$key = q/{ensuremathboldsymbol{mu}_{text{slashaslash}},ensuremathboldsymbol{mu}_{text{slau}_{text{slashoslash}},ensuremathboldsymbol{mu}_{text{slashyslash}}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img149.gif"
 ALT="$ \{\ensuremath\boldsymbol{\mu}_{\text{/a/}},
\ensuremath\boldsymbol{\mu}_{\tex...
...emath\boldsymbol{\mu}_{\text{/o/}},
\ensuremath\boldsymbol{\mu}_{\text{/y/}}\}$">|; 

$key = q/y=sqrt{Sigma},x+mu;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.gif"
 ALT="$ y = \sqrt{\Sigma}   x + \mu$">|; 

$key = q/displaystylequadP(q_k|X,ensuremathboldsymbol{Theta})geqP(q_j|X,ensuremathboldsymbol{Theta}),quadforalljneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img96.gif"
 ALT="$\displaystyle \quad P(q_k\vert X,\ensuremath\boldsymbol{\Theta}) \geq P(q_j\vert X,\ensuremath\boldsymbol{\Theta}),
\quad\forall j \neq k
$">|; 

$key = q/displaystyleforallk,;;P(q_k|X,Theta)proptop(X|q_k,Theta)P(q_k|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img91.gif"
 ALT="$\displaystyle \forall k, \;\; P(q_k\vert X,\Theta) \propto p(X\vert q_k,\Theta) P(q_k\vert\Theta)
$">|; 

$key = q/q_k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img91.gif"
 ALT="$ q_k$">|; 

$key = q/displaystyle=sum_{k=1}^{K}sum_{n=1}^{N}P(q_k|ensuremathmathbf{x}_n,ensuremathbolymbol{Theta})logp(ensuremathmathbf{x}_n|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img183.gif"
 ALT="$\displaystyle = \sum_{k=1}^{K} \sum_{n=1}^{N} P(q_k\vert\ensuremath\mathbf{x}_n...
...bol{\Theta}) \log p(\ensuremath\mathbf{x}_n\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/ensuremathmathbf{x}_i;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.gif"
 ALT="$ \ensuremath\mathbf{x}_i$">|; 

$key = q/displaystyleP(q_k|X,ensuremathboldsymbol{Theta})=frac{p(X|q_k,ensuremathboldsymb(q_k|ensuremathboldsymbol{Theta})}{p(X|ensuremathboldsymbol{Theta})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img99.gif"
 ALT="$\displaystyle P(q_k\vert X,\ensuremath\boldsymbol{\Theta}) = \frac{p(X\vert q_k...
...k\vert\ensuremath\boldsymbol{\Theta})}{p(X\vert\ensuremath\boldsymbol{\Theta})}$">|; 

$key = q/[400,1800]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img120.gif"
 ALT="$ [400,1800]^{\mathsf T}$">|; 

$key = q/displaystyle{calL}(ensuremathboldsymbol{Theta})=logp(X|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img176.gif"
 ALT="$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta}) = \log p(X\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/p(X|ensuremathboldsymbol{Theta})=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img100.gif"
 ALT="$ p(X\vert\ensuremath\boldsymbol{\Theta})=$">|; 

$key = q/displaystyleensuremathboldsymbol{mu}=left[{array}{c}7301090{array}right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.gif"
 ALT="$\displaystyle \ensuremath\boldsymbol{\mu}= \left[ \begin{array}{c} 730  1090 \end{array} \right]
$">|; 

$key = q/displaystyle=logsum_{k=1}^{K}P(q_k|X,ensuremathboldsymbol{Theta})p(X|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img179.gif"
 ALT="$\displaystyle = \log \sum_{k=1}^{K} P(q_k\vert X,\ensuremath\boldsymbol{\Theta}) p(X\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/displaystyleensuremathboldsymbol{mu}_{k}^{(i+1)}=frac{sum_{n=1}^{N}ensuremathmat(q_k^{(i)}|ensuremathmathbf{x}_n,ensuremathboldsymbol{Theta}^{(i)})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img173.gif"
 ALT="$\displaystyle \ensuremath\boldsymbol{\mu}_{k}^{(i+1)} = \frac{\sum_{n=1}^{N} \e...
...P(q_k^{(i)}\vert\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})} $">|; 

$key = q/q_k^{(i)};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img168.gif"
 ALT="$ q_k^{(i)}$">|; 

$key = q/mu_i=E(x_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.gif"
 ALT="$ \mu_i = E(x_i)$">|; 

$key = q/logleft(detleft(ensuremathboldsymbol{Sigma}right)right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.gif"
 ALT="$ \log \left( \det\left(\ensuremath\boldsymbol{\Sigma}\right)
\right)$">|; 

$key = q/ensuremathmathbf{x};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img20.gif"
 ALT="$ \ensuremath\mathbf{x}$">|; 

$key = q/Theta_kinTheta;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img99.gif"
 ALT="$ \Theta_k \in \Theta$">|; 

$key = q/logp(X|q_k,Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img98.gif"
 ALT="$ \log p(X\vert q_k,\Theta)$">|; 

$key = q/{calN}_1;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.gif"
 ALT="$ {\cal
N}_1$">|;