images.pl

高斯混合模型算法

 ALT="$ K$">|; 

$key = q/mathbf{y}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="|."$dir".q|img28.gif"
 ALT="$ \mathbf{y} \circlearrowleft {\cal
N}(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})$">|; 

$key = q/inej;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.gif"
 ALT="$ i\ne j$">|; 

$key = q/{calN}_3:;ensuremathboldsymbol{Theta}_3=left(left[{array}{c}7301090{array}right],left[{array}{cc}8000&84008400&18500{array}right]right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img80.gif"
 ALT="$ {\cal N}_3: \; \ensuremath\boldsymbol{\Theta}_3 = \left(
\left[\begin{array}{...
...
\left[\begin{array}{cc}8000 &amp; 8400  8400 &amp; 18500\end{array}\right]
\right)$">|; 

$key = q/f_k(ensuremathmathbf{x});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img129.gif"
 ALT="$ f_k(\ensuremath\mathbf{x})$">|; 

$key = q/i;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="8" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img156.gif"
 ALT="$ i$">|; 

$key = q/displaystylep(x|Theta_1)>p(x|Theta_2);;Leftrightarrow;;logp(x|Theta_1)>logp(x|Theta_2);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.gif"
 ALT="$\displaystyle p(x\vert\Theta_1) &gt; p(x\vert\Theta_2) \;\; \Leftrightarrow \;\;
\log p(x\vert\Theta_1) &gt; \log p(x\vert\Theta_2)
$">|; 

$key = q/[420,2500]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img125.gif"
 ALT="$ [420,2500]^{\mathsf T}$">|; 

$key = q/};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img94.gif"
 ALT="$ \}$">|; 

$key = q/(x-mu)^{mathsfT}Sigma^{-1}(x-mu);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.gif"
 ALT="$ (x-\mu)^{\mathsf T}\Sigma^{-1} (x-\mu) $">|; 

$key = q/N=10000;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img32.gif"
 ALT="$ N=10000$">|; 

$key = q/{calN}left(mu_{text{slasheslash}},Sigma_{text{slasheslash}}right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.gif"
 ALT="$ {\cal N}\left (\mu _{\text {/e/}},\Sigma _{\text {/e/}}\right )$">|; 

$key = q/displaystylep(X|ensuremathboldsymbol{Theta})=prod_{i=1}^{N}p(ensuremathmathbf{x}sum_{i=1}^{N}logp(ensuremathmathbf{x}_i|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="470" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.gif"
 ALT="$\displaystyle p(X\vert\ensuremath\boldsymbol{\Theta}) = \prod_{i=1}^{N} p(\ensu...
...{i=1}^{N}
\log p(\ensuremath\mathbf{x}_i\vert\ensuremath\boldsymbol{\Theta})
$">|; 

$key = q/{calN}(ensuremathboldsymbol{mu}_{text{slasheslash}},ensuremathboldsymbol{Sigma}_{text{slasheslash}});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.gif"
 ALT="$ {\cal N}(\ensuremath \boldsymbol {\mu }_{\text {/e/}},\ensuremath \boldsymbol {\Sigma }_{\text {/e/}})$">|; 

$key = q/displaystyleensuremathboldsymbol{Sigma}_{k}^{(i+1)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img159.gif"
 ALT="$\displaystyle \ensuremath\boldsymbol{\Sigma}_{k}^{(i+1)} =$">|; 

$key = q/X={ensuremathmathbf{x}_1,ensuremathmathbf{x}_2,ldots,ensuremathmathbf{x}_N};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.gif"
 ALT="$ X=\{\ensuremath\mathbf{x}_1,
\ensuremath\mathbf{x}_2,\ldots,\ensuremath\mathbf{x}_N\}$">|; 

$key = q/[600,1300]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img123.gif"
 ALT="$ [600,1300]^{\mathsf T}$">|; 

$key = q/displaystylesum_{k=1}^{K}sum_{x_ninq_k}logp(x_n|Theta_k),;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img145.gif"
 ALT="$\displaystyle \sum_{k=1}^{K} \sum_{x_n \in q_k} \log p(x_n\vert\Theta_k),$">|; 

$key = q/displaystyleensuremathboldsymbol{Sigma}_1=left[{array}{cc}8000&00&8000{array}right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img35.gif"
 ALT="$\displaystyle \ensuremath\boldsymbol{\Sigma}_1 = \left[ \begin{array}{cc}
8000 &amp; 0 \\\\
0 &amp; 8000
\end{array} \right]
$">|; 

$key = q/displaystylec_{ij}=Eleft((x_i-mu_i)^{mathsfT},(x_j-mu_j)right).;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.gif"
 ALT="$\displaystyle c_{ij} = E\left((x_i-\mu_i)^{\mathsf T} (x_j-\mu_j)\right).$">|; 

$key = q/[F_1,F_2];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img88.gif"
 ALT="$ [F_1,F_2]$">|; 

$key = q/logP(q_{text{slashyslash}}|x);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img107.gif"
 ALT="$ \log P(q_{\text{/y/}}\vert x)$">|; 

$key = q/displaystylesum_{X}P(X|Theta);=;sum_{Q}sum_{X}p(X,Q|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img144.gif"
 ALT="$\displaystyle \sum_{X} P(X\vert\Theta) \; = \; \sum_{Q} \sum_{X} p(X,Q\vert\Theta)$">|; 

$key = q/ensuremathmathbf{x}_i=[F_1,F_2]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img112.gif"
 ALT="$ \ensuremath\mathbf{x}_i=[F_1,F_2]^{\mathsf T}$">|; 

$key = q/displaystyleSigma_{k}^{(i+1)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img141.gif"
 ALT="$\displaystyle \Sigma_{k}^{(i+1)} =$">|; 

$key = q/displaystylehat{Sigma}=frac{1}{N-1};sum_{i=1}^{N}(x_i-mu)^{mathsfT}(x_i-mu);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.gif"
 ALT="$ \displaystyle \hat{\Sigma} =
\frac{1}{N-1} \; \sum_{i=1}^{N} (x_i-\mu)^{\mathsf T}(x_i-\mu) $">|; 

$key = q/{calN}(ensuremathboldsymbol{mu}_{text{slashislash}},ensuremathboldsymbol{Sigma}_{text{slasheslash}});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.gif"
 ALT="$ {\cal N}(\ensuremath \boldsymbol {\mu }_{\text {/i/}},\ensuremath \boldsymbol {\Sigma }_{\text {/e/}})$">|; 

$key = q/n=1,ldots,N;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img142.gif"
 ALT="$ n=1,\ldots,N$">|; 

$key = q/displaystyled_k(ensuremathmathbf{x}_n),<,d_l(ensuremathmathbf{x}_n),qquadforalllneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img147.gif"
 ALT="$\displaystyle d_k(\ensuremath\mathbf{x}_n)   &lt;   d_l(\ensuremath\mathbf{x}_n), \qquad \forall l \neq k
$">|; 

$key = q/,;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="7" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img93.gif"
 ALT="$ ,$">|; 

$key = q/displaystyle(ensuremathmathbf{x}_n-ensuremathboldsymbol{mu}_k)^{mathsfT}(ensuremathmathbf{x}_n-ensuremathboldsymbol{mu}_k);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img146.gif"
 ALT="$\displaystyle (\ensuremath\mathbf{x}_n-\ensuremath\boldsymbol{\mu}_k)^{\mathsf T}(\ensuremath\mathbf{x}_n-\ensuremath\boldsymbol{\mu}_k)$">|; 

$key = q/P(q_k|X,Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img87.gif"
 ALT="$ P(q_k\vert X,\Theta)$">|; 

$key = q/displaystyle,left|x_n-mu_kright|^2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img126.gif"
 ALT="$\displaystyle   \left\Vert x_n - \mu_k \right\Vert^2$">|; 

$key = q/ensuremathboldsymbol{Theta};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img61.gif"
 ALT="$ \ensuremath\boldsymbol{\Theta}$">|; 

$key = q/dlogleft(2piright);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img74.gif"
 ALT="$ d\log \left( 2\pi \right)$">|; 

$key = q/displaystyleSigma_{k}^{(i+1)}=frac{sum_{n=1}^{N}P(q_k^{(i)}|x_n,Theta^{(i)})(x_n-mu_k^{(i+1)})^{mathsfT}}{sum_{n=1}^{N}P(q_k^{(i)}|x_n,Theta^{(i)})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img155.gif"
 ALT="$\displaystyle \Sigma_{k}^{(i+1)} = \frac{\sum_{n=1}^{N} P(q_k^{(i)}\vert x_n,\T...
...u_k^{(i+1)})^{\mathsf T}}
{\sum_{n=1}^{N} P(q_k^{(i)}\vert x_n,\Theta^{(i)})} $">|; 

$key = q/ycirclearrowleft{calN}(mu,Sigma);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.gif"
 ALT="$ y \circlearrowleft {\cal
N}(\mu,\Sigma)$">|; 

$key = q/{calN}_4:;ensuremathboldsymbol{Theta}_4=left(left[{array}{c}2701690{array}right],left[{array}{cc}8000&84008400&18500{array}right]right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img81.gif"
 ALT="$ {\cal N}_4: \; \ensuremath\boldsymbol{\Theta}_4 = \left(
\left[\begin{array}{...
...
\left[\begin{array}{cc}8000 &amp; 8400  8400 &amp; 18500\end{array}\right]
\right)$">|; 

$key = q/displaystylefrac{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="211" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.gif"
 ALT="$\displaystyle \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/displaystylep(x|q_k)P(q_k)geqp(x|q_l)P(q_l);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img116.gif"
 ALT="$\displaystyle p(x\vert q_k) P(q_k) \geq p(x\vert q_l) P(q_l)$">|; 

$key = q/displaystyle=logsum_Qp(X,Q|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img177.gif"
 ALT="$\displaystyle = \log \sum_Q p(X,Q\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/ensuremathmathbf{x}_n;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img141.gif"
 ALT="$ \ensuremath\mathbf{x}_n$">|; 

$key = q/c_{ij};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.gif"
 ALT="$ c_{ij}$">|; 

$key = q/{calN}_3:;Theta_3=left(left[{array}{c}7301090{array}right],left[{array}{cc}8000&84008400&18500{array}right]right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.gif"
 ALT="$ {\cal N}_3: \; \Theta_3 = \left(
\left[\begin{array}{c}730  1090\end{array}...
...
\left[\begin{array}{cc}8000 &amp; 8400  8400 &amp; 18500\end{array}\right]
\right)$">|; 

$key = q/displaystyleXinq_k;;;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.gif"
 ALT="$\displaystyle X \in q_k \;\;$">|; 

$key = q/displaystyleP(q_k|X,ensuremathboldsymbol{Theta})proptop(X|q_k,ensuremathboldsymbol{Theta});P(q_k|ensuremathboldsymbol{Theta}),quadforallk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img102.gif"
 ALT="$\displaystyle P(q_k\vert X,\ensuremath\boldsymbol{\Theta}) \propto p(X\vert q_k...
...dsymbol{\Theta})\; P(q_k\vert\ensuremath\boldsymbol{\Theta}), \quad \forall k
$">|; 

$key = q/displaystylemu_{k}^{(i+1)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img139.gif"
 ALT="$\displaystyle \mu_{k}^{(i+1)} =$">|; 

$key = q/x^{mathsfT}Sigmaxge0;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.gif"
 ALT="$ x^{\mathsf T}
\Sigma x \ge 0$">|; 

$key = q/ensuremathmathbf{x}^{mathsfT}ensuremathboldsymbol{Sigma},ensuremathmathbf{x}ge0;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.gif"
 ALT="$ \ensuremath\mathbf{x}^{\mathsf T}\ensuremath\boldsymbol{\Sigma}  \ensuremath\mathbf{x}\ge 0$">|; 

$key = q/hat{ensuremathboldsymbol{mu}}_{(100)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.gif"
 ALT="$ \hat{\ensuremath\boldsymbol{\mu}}_{(100)}
=$">|;