images.pl

高斯混合模型算法

$key = q/displaystyleSigma=left({array}{*{4}{c}}c_{11}&c_{12}&cdots&c_{1n}c_{21}&c_{22}&cc_{2n}vdots&vdots&ddots&vdotsc_{n1}&c_{n2}&cdots&c_{nn}{array}right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="90" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.gif"
 ALT="$\displaystyle \Sigma =
 \left(
 \begin{array}{*{4}{c}}
 c_{11} &amp; c_{12} &amp; \cdot...
...ddots &amp; \vdots  
 c_{n1} &amp; c_{n2} &amp; \cdots &amp; c_{nn}  
 \end{array}
 \right)$">|; 

$key = q/displaystyle=logsum_QP(Q|X,Theta)p(X|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img159.gif"
 ALT="$\displaystyle = \log \sum_Q P(Q\vert X,\Theta)p(X\vert\Theta)$">|; 

$key = q/frac{1}{2}left[ldotsright];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img73.gif"
 ALT="$ \frac{1}{2} \left[\ldots\right]$">|; 

$key = q/displaystyleXinq_k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img95.gif"
 ALT="$\displaystyle X \in q_k$">|; 

$key = q/displaystyleEleft((x_{d1}-mu_{d1})^{mathsfT}(x_{d2}-mu_{d2})right)=0;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.gif"
 ALT="$\displaystyle E\left((x_{d1}-\mu_{d1})^{\mathsf T}(x_{d2}-\mu_{d2})\right) = 0 $">|; 

$key = q/N;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img30.gif"
 ALT="$ N$">|; 

$key = q/displaystylelogp(X_3|ensuremathboldsymbol{Theta}_1),;logp(X_3|ensuremathboldsymbol{Theta}_2),;logp(X_3|ensuremathboldsymbol{Theta}_3),;;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="268" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img82.gif"
 ALT="$\displaystyle \log p(X_3\vert\ensuremath\boldsymbol{\Theta}_1),\; \log p(X_3\ve...
...th\boldsymbol{\Theta}_2),\; \log p(X_3\vert\ensuremath\boldsymbol{\Theta}_3),\;$">|; 

$key = q/{ensuremathboldsymbol{Sigma}_{text{slashaslash}},ensuremathboldsymbol{Sigma}_{te{text{slashoslash}},ensuremathboldsymbol{Sigma}_{text{slashyslash}}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img164.gif"
 ALT="$ \{\ensuremath\boldsymbol{\Sigma}_{\text{/a/}}, \ensuremath\boldsymbol{\Sigma}_...
...\boldsymbol{\Sigma}_{\text{/o/}}, \ensuremath\boldsymbol{\Sigma}_{\text{/y/}}\}$">|; 

$key = q/{calN}(ensuremathboldsymbol{mu}_{k},ensuremathboldsymbol{Sigma}_{k}),;k=1ldotsK;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img151.gif"
 ALT="$ {\cal N}(\ensuremath\boldsymbol{\mu}_{k},\ensuremath\boldsymbol{\Sigma}_{k}),
\; k=1\ldots K$">|; 

$key = q/xcirclearrowleft{calN}(0,I);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.gif"
 ALT="$ x \circlearrowleft {\cal N}(0,I)$">|; 

$key = q/X;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img29.gif"
 ALT="$ X$">|; 

$key = q/displaystylemu_{k}^{(i+1)}=frac{sum_{n=1}^{N}x_nP(q_k^{(i)}|x_n,Theta^{(i)})}{sum_{n=1}^{N}P(q_k^{(i)}|x_n,Theta^{(i)})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img154.gif"
 ALT="$\displaystyle \mu_{k}^{(i+1)} = \frac{\sum_{n=1}^{N} x_n P(q_k^{(i)}\vert x_n,\Theta^{(i)})}
{\sum_{n=1}^{N} P(q_k^{(i)}\vert x_n,\Theta^{(i)})} $">|; 

$key = q/displaystylehat{ensuremathboldsymbol{mu}}=frac{1}{N}sum_{i=1}^{N}ensuremathmathbf{x}_i;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.gif"
 ALT="$ \displaystyle \hat{\ensuremath\boldsymbol{\mu}} = \frac{1}{N}
\sum_{i=1}^{N} \ensuremath\mathbf{x}_i$">|; 

$key = q/[400,1000]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img121.gif"
 ALT="$ [400,1000]^{\mathsf T}$">|; 

$key = q/displaystyleP(q_k^{(i+1)}|ensuremathboldsymbol{Theta}^{(i+1)})=frac{1}{N}sum_{n=P(q_k^{(i)}|ensuremathmathbf{x}_n,ensuremathboldsymbol{Theta}^{(i)});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img175.gif"
 ALT="$\displaystyle P(q_k^{(i+1)}\vert\ensuremath\boldsymbol{\Theta}^{(i+1)}) = \frac...
...
P(q_k^{(i)}\vert\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)}) $">|; 

$key = q/hat{mu};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.gif"
 ALT="$ \hat{\mu}$">|; 

$key = q/P(q_k|x);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img112.gif"
 ALT="$ P(q_k\vert x)$">|; 

$key = q/displaystylehat{ensuremathboldsymbol{Sigma}}=frac{1}{N-1};sum_{i=1}^{N}(ensuremaymbol{mu})^{mathsfT}(ensuremathmathbf{x}_i-ensuremathboldsymbol{mu});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.gif"
 ALT="$ \displaystyle \hat{\ensuremath\boldsymbol{\Sigma}} =
\frac{1}{N-1} \; \sum_{i...
...dsymbol{\mu})^{\mathsf T}(\ensuremath\mathbf{x}_i-\ensuremath\boldsymbol{\mu}) $">|; 

$key = q/displaystylelogp(x|q_k)+logP(q_k)geqlogp(x|q_l)+logP(q_l);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img117.gif"
 ALT="$\displaystyle \log p(x\vert q_k)+\log P(q_k) \geq \log p(x\vert q_l)+\log P(q_l)$">|; 

$key = q/displaystyleP(q_k|X,Theta)=frac{p(X|q_k,Theta)P(q_k|Theta)}{p(X|Theta)};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.gif"
 ALT="$\displaystyle P(q_k\vert X,\Theta) = \frac{p(X\vert q_k,\Theta) P(q_k\vert\Theta)}{p(X\vert\Theta)}
$">|; 

$key = q/displaystyled_k(x_n),leq,d_l(x_n),;;foralllneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img128.gif"
 ALT="$\displaystyle d_k(x_n)   \leq   d_l(x_n),
\;\; \forall l \neq k
$">|; 

$key = q/{calN}(ensuremathboldsymbol{mu}_{text{slashislash}},ensuremathboldsymbol{Sigma}_{text{slashislash}});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.gif"
 ALT="$ {\cal N}(\ensuremath \boldsymbol {\mu }_{\text {/i/}},\ensuremath \boldsymbol {\Sigma }_{\text {/i/}})$">|; 

$key = q/mathbf{A};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img59.gif"
 ALT="$ \mathbf{A}$">|; 

$key = q/displaystyle;;P(q_k|X,Theta)geqP(q_j|X,Theta),;foralljneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img85.gif"
 ALT="$\displaystyle \;\; P(q_k\vert X,\Theta) \geq P(q_j\vert X,\Theta), \; \forall j \neq k
$">|; 

$key = q/displaystyleJ=sum_{k=1}^{K}sum_{ensuremathmathbf{x}_ninq_k}d_k(ensuremathmathbf{x}_n);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img148.gif"
 ALT="$\displaystyle J = \sum_{k=1}^{K} \sum_{\ensuremath\mathbf{x}_n \in q_k} d_k(\ensuremath\mathbf{x}_n)
$">|; 

$key = q/Sigma;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img9.gif"
 ALT="$ \Sigma$">|; 

$key = q/mu;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.gif"
 ALT="$ \mu$">|; 

$key = q/displaystylefrac{1}{2}left[-dlogleft(2piright)-logleft(detleft(ensuremathboldsymmbol{Sigma}^{-1}(ensuremathmathbf{x}-ensuremathboldsymbol{mu})right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.gif"
 ALT="$\displaystyle \frac{1}{2} \left[-d \log \left( 2\pi \right)
- \log \left( \det\...
...dsymbol{\Sigma}^{-1} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})\right]$">|; 

$key = q/displaystylesum_{X}P(X|ensuremathboldsymbol{Theta});=;sum_{Q}sum_{X}p(X,Q|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img162.gif"
 ALT="$\displaystyle \sum_{X} P(X\vert\ensuremath\boldsymbol{\Theta}) \;=\; \sum_{Q} \sum_{X} p(X,Q\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/displaystyleP(q_k^{(i+1)}|ensuremathboldsymbol{Theta}^{(i+1)})=frac{mbox{numberoiningpointsbelongingto}q_k^{(i)}}{mbox{totalnumberoftrainingpoints}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img160.gif"
 ALT="$\displaystyle P(q_k^{(i+1)}\vert\ensuremath\boldsymbol{\Theta}^{(i+1)}) = \frac...
...ng points
belonging to } q_k^{(i)} }{\mbox{total number of training points}}
$">|; 

$key = q/f_k(ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img113.gif"
 ALT="$ f_k(\ensuremath\mathbf{x}_i)$">|; 

$key = q/k=1ldotsK;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img152.gif"
 ALT="$ k=1\ldots K$">|; 

$key = q/{calN}(mu_{k},Sigma_{k}),;k=1ldotsK;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img132.gif"
 ALT="$ {\cal N}(\mu_{k},\Sigma_{k}),
\; k=1\ldots K$">|; 

$key = q/displaystyled_k(ensuremathmathbf{x}_n);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img144.gif"
 ALT="$\displaystyle d_k(\ensuremath\mathbf{x}_n)$">|; 

$key = q/p(X,Q|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img136.gif"
 ALT="$ p(X,Q\vert\Theta)$">|; 

$key = q/,slashuslash};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img107.gif"
 ALT="$ ,/u/\}$">|; 

$key = q/[670,1300]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img124.gif"
 ALT="$ [670,1300]^{\mathsf T}$">|; 

$key = q/displaystyle,left|ensuremathmathbf{x}_n-ensuremathboldsymbol{mu}_kright|^2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img145.gif"
 ALT="$\displaystyle   \left\Vert \ensuremath\mathbf{x}_n - \ensuremath\boldsymbol{\mu}_k \right\Vert^2$">|; 

$key = q/ensuremathmathbf{x}circlearrowleft{calN}(ensuremathboldsymbol{mu},ensuremathboldsymbol{Sigma});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.gif"
 ALT="$ \ensuremath\mathbf{x}\circlearrowleft {\cal
N}(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})$">|; 

$key = q/f_{text{slashyslash}}(ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img119.gif"
 ALT="$ f_{\text{/y/}}(\ensuremath\mathbf{x}_i)$">|; 

$key = q/1slashK;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img153.gif"
 ALT="$ 1/K$">|; 

$key = q/displaystyle=logsum_{k=1}^{K}P(q_k|X,Theta)p(X|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img160.gif"
 ALT="$\displaystyle = \log \sum_{k=1}^{K} P(q_k\vert X,\Theta) p(X\vert\Theta)$">|; 

$key = q/displaystyleSigma_1=left[{array}{cc}8000&00&8000{array}right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.gif"
 ALT="$\displaystyle \Sigma_1 = \left[ \begin{array}{cc}
8000 &amp; 0 \\\\
0 &amp; 8000
\end{array} \right]
$">|; 

$key = q/x_i=(F_1,F_2)^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img100.gif"
 ALT="$ x_i=(F_1,F_2)^{\mathsf T}$">|; 

$key = q/displaystyleP(q_k|ensuremathmathbf{x}_i)geqP(q_l|ensuremathmathbf{x}_i),quadforalllneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img135.gif"
 ALT="$\displaystyle P(q_k\vert\ensuremath\mathbf{x}_i) \geq P(q_l\vert\ensuremath\mathbf{x}_i),\quad \forall l \neq k$">|; 

$key = q/k=1,ldots,K;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img140.gif"
 ALT="$ k=1,\ldots,K$">|; 

$key = q/ensuremathboldsymbol{Theta}=(ensuremathboldsymbol{mu},ensuremathboldsymbol{Sigma});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.gif"
 ALT="$ \ensuremath\boldsymbol{\Theta}= (\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})$">|; 

$key = q/P(q_k)=1slashK;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img166.gif"
 ALT="$ P(q_k) = 1/K$">|; 

$key = q/k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img130.gif"
 ALT="$ k$">|; 

$key = q/hat{ensuremathboldsymbol{mu}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.gif"
 ALT="$ \hat{\ensuremath\boldsymbol{\mu}}$">|; 

$key = q/X={x_1,x_2,cdots,x_N};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG