images.tex

高斯混合模型算法

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$\displaystyle \ensuremath\boldsymbol{\Sigma}_3 = \left[ \begin{array}{cc}
8000 & 8400 \\
8400 & 18500
\end{array} \right]
$%
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$ \ensuremath\boldsymbol{\Sigma}_2$%
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$ \ensuremath\boldsymbol{\Sigma}_3$%
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$ \ensuremath\mathbf{x}_i$%
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$ \ensuremath\mathbf{x}^{\mathsf T}\ensuremath\boldsymbol{\Sigma}\, \ensuremath\mathbf{x}\ge 0$%
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$ \displaystyle \hat{\ensuremath\boldsymbol{\mu}} = \frac{1}{N}
\sum_{i=1}^{N} \ensuremath\mathbf{x}_i$%
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$ \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}) $%
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$ {\cal
N}(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma}_3)$%
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$ \hat{\ensuremath\boldsymbol{\mu}}$%
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$ \hat{\ensuremath\boldsymbol{\Sigma}}$%
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$ \hat{\ensuremath\boldsymbol{\mu}}_{(10000)}
=$%
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$ \hat{\ensuremath\boldsymbol{\Sigma}}_{(10000)} =$%
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$ \hat{\ensuremath\boldsymbol{\mu}}_{(1000)}
=$%
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$ \hat{\ensuremath\boldsymbol{\Sigma}}_{(1000)} =$%
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$ \hat{\ensuremath\boldsymbol{\mu}}_{(100)}
=$%
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$ \hat{\ensuremath\boldsymbol{\Sigma}}_{(100)} =$%
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$ \|\mathbf{A}-\mathbf{B}\|_2$%
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$ \mathbf{A}$%
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$ \mathbf{B}$%
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$ \ensuremath\boldsymbol{\Theta}$%
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$ p(\ensuremath\mathbf{x}_i|\ensuremath\boldsymbol{\Theta})$%
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$ \ensuremath\boldsymbol{\Theta}= (\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})$%
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$\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)$%
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$\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})
$%
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$\displaystyle p(\ensuremath\mathbf{x}|\ensuremath\boldsymbol{\Theta})$%
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$\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})}$%
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$\displaystyle \log p(\ensuremath\mathbf{x}|\ensuremath\boldsymbol{\Theta})$%
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$\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]$%
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$\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),
$%
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$ \log \left( \det\left(\ensuremath\boldsymbol{\Sigma}\right)
\right)$%
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$ (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})^{\mathsf T}\ensuremath\boldsymbol{\Sigma}^{-1} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})$%
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$ \ensuremath\boldsymbol{\Theta}_i = (\ensuremath\boldsymbol{\mu}_i,\ensuremath\boldsymbol{\Sigma}_i)$%
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$ {\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)$%
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$ {\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)$%
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$ {\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)$%
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$ {\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)$%
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$\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),\;$%
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$\displaystyle \; \log p(X_3|\ensuremath\boldsymbol{\Theta}_4).
$%
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$\displaystyle X \in q_k$%
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$\displaystyle \quad P(q_k|X,\ensuremath\boldsymbol{\Theta}) \geq P(q_j|X,\ensuremath\boldsymbol{\Theta}),
\quad\forall j \neq k
$%
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$ P(q_k|X,\ensuremath\boldsymbol{\Theta})$%
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$ P(q_k|\ensuremath\boldsymbol{\Theta})$%
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$\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})}$%
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$ p(X|q_k,\ensuremath\boldsymbol{\Theta}) P(q_k|\ensuremath\boldsymbol{\Theta})$%
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$\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
$%
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$\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})$%
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$ \ensuremath\boldsymbol{\mu}_k$%
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$ \ensuremath\boldsymbol{\Sigma}_k$%
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{\newpage\clearpage
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$ p(X|q_k,\ensuremath\boldsymbol{\Theta})$%
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{\newpage\clearpage
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$ \log p(X|q_k,\ensuremath\boldsymbol{\Theta})$%
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{\newpage\clearpage
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$ (\ensuremath\boldsymbol{\mu}_k,\ensuremath\boldsymbol{\Sigma}_k)$%
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$ p(X|\ensuremath\boldsymbol{\Theta})$%
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{\newpage\clearpage
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$ \ensuremath\mathbf{x}_i=[F_1,F_2]^{\mathsf T}$%
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{\newpage\clearpage
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