images.pl

高斯混合模型算法

$key = q/displaystyle=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.gif"
 ALT="$\displaystyle =$">|; 

$key = q/P(q_k|ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img132.gif"
 ALT="$ P(q_k\vert\ensuremath\mathbf{x}_i)$">|; 

$key = q/{calN}_2:;Theta_2=left(left[{array}{c}7301090{array}right],left[{array}{cc}8000&00&18500{array}right]right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.gif"
 ALT="$ {\cal N}_2: \; \Theta_2 = \left(
\left[\begin{array}{c}730  1090\end{array}\right],
\left[\begin{array}{cc}8000 &amp; 0  0 &amp; 18500\end{array}\right]
\right)$">|; 

$key = q/displaystyleensuremathboldsymbol{mu}_{k}^{(i+1)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img157.gif"
 ALT="$\displaystyle \ensuremath\boldsymbol{\mu}_{k}^{(i+1)} =$">|; 

$key = q/I;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img19.gif"
 ALT="$ I$">|; 

$key = q/dtimesd;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.gif"
 ALT="$ d\times d$">|; 

$key = q/ensuremathboldsymbol{mu};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.gif"
 ALT="$ \ensuremath\boldsymbol{\mu}$">|; 

$key = q/p(X|q_k,ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img108.gif"
 ALT="$ p(X\vert q_k,\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/displaystyle{calL}(ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img161.gif"
 ALT="$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/hat{Sigma}_{(100)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.gif"
 ALT="$ \hat{\Sigma}_{(100)} =$">|; 

$key = q/{Sigma_{text{slashaslash}},Sigma_{text{slasheslash}},Sigma_{text{slashislash}},Sigma_{text{slashoslash}},Sigma_{text{slashyslash}}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img146.gif"
 ALT="$ \{\Sigma_{\text{/a/}}, \Sigma_{\text{/e/}}, \Sigma_{\text{/i/}},
\Sigma_{\text{/o/}}, \Sigma_{\text{/y/}}\}$">|; 

$key = q/sqrt{Sigma};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.gif"
 ALT="$ \sqrt{\Sigma}$">|; 

$key = q/displaystyle{calL}(Theta)geJ(Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img162.gif"
 ALT="$\displaystyle {\cal L}(\Theta) \ge J(\Theta)$">|; 

$key = q/hat{ensuremathboldsymbol{Sigma}}_{(100)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.gif"
 ALT="$ \hat{\ensuremath\boldsymbol{\Sigma}}_{(100)} =$">|; 

$key = q/p(X|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img111.gif"
 ALT="$ p(X\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/{calN}(mu,Sigma_3);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.gif"
 ALT="$ {\cal
N}(\mu,\Sigma_3)$">|; 

$key = q/left(logsum_jlambda_jy_jgeqsum_jlambda_jlogy_jmbox{if}sum_jlambda_j=1right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img180.gif"
 ALT="$ \left( \log \sum_j \lambda_j y_j \geq
\sum_j \lambda_j \log y_j \mbox{ if } \sum_j \lambda_j = 1 \right)$">|; 

$key = q/Ntimes2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.gif"
 ALT="$ N\times2$">|; 

$key = q/displaystylemu=left({array}{c}7301090{array}right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.gif"
 ALT="$\displaystyle \mu = \left( \begin{array}{c} 730  1090 \end{array} \right)
$">|; 

$key = q/displaystylelogp(X_3|Theta_1),;logp(X_3|Theta_2),;logp(X_3|Theta_3),;;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.gif"
 ALT="$\displaystyle \log p(X_3\vert\Theta_1),\; \log p(X_3\vert\Theta_2),\; \log p(X_3\vert\Theta_3),\;$">|; 

$key = q/displaystyleensuremathmathbf{x}inq_kquadLeftrightarrowquadf_k(ensuremathmathbf{xl(ensuremathmathbf{x},ensuremathboldsymbol{Theta}_l),quadforalllneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="281" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img131.gif"
 ALT="$\displaystyle \ensuremath\mathbf{x}\in q_k \quad \Leftrightarrow \quad f_k(\ens...
...suremath\mathbf{x},\ensuremath\boldsymbol{\Theta}_l),
\quad \forall l \neq k
$">|; 

$key = q/displaystyle=sum_{k=1}^{K}P(q_k|X,Theta)logp(X|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img163.gif"
 ALT="$\displaystyle = \sum_{k=1}^{K} P(q_k\vert X,\Theta) \log p(X\vert\Theta)$">|; 

$key = q/2times;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img150.gif"
 ALT="$ 2\times$">|; 

$key = q/f_{text{slashaslash}}(ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img115.gif"
 ALT="$ f_{\text{/a/}}(\ensuremath\mathbf{x}_i)$">|; 

$key = q/{mu_{text{slashaslash}},mu_{text{slasheslash}},mu_{text{slashislash}},mu_{text{slashoslash}},mu_{text{slashyslash}}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img130.gif"
 ALT="$ \{\mu_{\text{/a/}},
\mu_{\text{/e/}}, \mu_{\text{/i/}}, \mu_{\text{/o/}},
\mu_{\text{/y/}}\}$">|; 

$key = q/displaystyle=logsum_Qp(X,Q|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img158.gif"
 ALT="$\displaystyle = \log \sum_Q p(X,Q\vert\Theta)$">|; 

$key = q/displaystyleensuremathmathbf{x}inq_k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img133.gif"
 ALT="$\displaystyle \ensuremath\mathbf{x}\in q_k$">|; 

$key = q/displaystylep(ensuremathmathbf{x}|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.gif"
 ALT="$\displaystyle p(\ensuremath\mathbf{x}\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/displaystyleLeftrightarrow;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img134.gif"
 ALT="$\displaystyle \Leftrightarrow$">|; 

$key = q/displaystyle=sum_{k=1}^{K}P(q_k|X,ensuremathboldsymbol{Theta})logp(X|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img182.gif"
 ALT="$\displaystyle = \sum_{k=1}^{K} P(q_k\vert X,\ensuremath\boldsymbol{\Theta}) \log p(X\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/P(q_k);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img90.gif"
 ALT="$ P(q_k)$">|; 

$key = q/Theta=(mu,Sigma);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.gif"
 ALT="$ \Theta = (\mu,\Sigma)$">|; 

$key = q/mu_{di}=E(x_{di});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.gif"
 ALT="$ \mu_{di} = E(x_{di})$">|; 

$key = q/c_{ij}=c_{ji};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.gif"
 ALT="$ c_{ij} =
c_{ji}$">|; 

$key = q/[0,1];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img172.gif"
 ALT="$ [0,1]$">|; 

$key = q/displaystyleq_k^{(i)};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img158.gif"
 ALT="$\displaystyle q_k^{(i)} $">|; 

$key = q/displaystylefrac{1}{2}left[-dlogleft(2piright)-logleft(detleft(Sigmaright)right)-(x-mu)^{mathsfT}Sigma^{-1}(x-mu)right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="325" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.gif"
 ALT="$\displaystyle \frac{1}{2} \left[-d \log \left( 2\pi \right)
- \log \left( \det\left(\Sigma\right) \right)
- (x-\mu)^{\mathsf T}\Sigma^{-1} (x-\mu)\right]$">|; 

$key = q/log(x);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.gif"
 ALT="$ \log(x)$">|; 

$key = q/{calL}(Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img166.gif"
 ALT="$ {\cal
L}(\Theta)$">|; 

$key = q/f_k(x);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img109.gif"
 ALT="$ f_k(x)$">|; 

$key = q/Sigma_{text{slasheslash}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img119.gif"
 ALT="$ \Sigma_{\text{/e/}}$">|; 

$key = q/displaystylexinq_k;;Leftrightarrow;;f_k(x,Theta_k)geqf_l(x,Theta_l),foralllneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img111.gif"
 ALT="$\displaystyle x \in q_k \;\; \Leftrightarrow \;\; f_k(x,\Theta_k) \geq f_l(x,\Theta_l),
 \forall l \neq k
$">|; 

$key = q/displaystyle;logp(X_3|ensuremathboldsymbol{Theta}_4).;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img83.gif"
 ALT="$\displaystyle \; \log p(X_3\vert\ensuremath\boldsymbol{\Theta}_4).
$">|; 

$key = q/displaystylep(X|Theta)=prod_{i=1}^{N}p(x_i|Theta)=prod_{i=1}^{N}p(x_i|mu,Sigma)=prod_{i=1}^{N}g_{(mu,Sigma)}(x_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.gif"
 ALT="$\displaystyle p(X\vert\Theta) =
\prod_{i=1}^{N} p(x_i\vert\Theta) =
\prod_{i=1}^{N} p(x_i\vert\mu,\Sigma) =
\prod_{i=1}^{N} g_{(\mu,\Sigma)}(x_i)
$">|; 

$key = q/displaystyleg_{(ensuremathboldsymbol{mu},ensuremathboldsymbol{Sigma})}(ensurematoldsymbol{Sigma}^{-1}(ensuremathmathbf{x}-ensuremathboldsymbol{mu})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.gif"
 ALT="$\displaystyle g_{(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})}(...
...th\boldsymbol{\Sigma}^{-1} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})}$">|; 

$key = q/displaystyle(x_n-mu_k)^{mathsfT}(x_n-mu_k);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img127.gif"
 ALT="$\displaystyle (x_n-\mu_k)^{\mathsf T}(x_n-\mu_k)$">|; 

$key = q/logP(q_{text{slashislash}}|x);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img105.gif"
 ALT="$ \log P(q_{\text{/i/}}\vert x)$">|; 

$key = q/J(Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img165.gif"
 ALT="$ J(\Theta)$">|; 

$key = q/displaystylelogp(ensuremathmathbf{x}|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.gif"
 ALT="$\displaystyle \log p(\ensuremath\mathbf{x}\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/includegraphics[height=0.95textheight]{iso};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="723" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img138.gif"
 ALT="\includegraphics[height=0.95\textheight]{iso}">|; 

$key = q/hat{Sigma}_{(1