images.tex

高斯混合模型算法

$ f_k(\ensuremath\mathbf{x}_i)$%
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$ f_k(\ensuremath\mathbf{x}_i) = \log
p(\ensuremath\mathbf{x}_i|q_k,\ensuremath\boldsymbol{\Theta}) + \log P(q_k|\ensuremath\boldsymbol{\Theta})$%
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$ f_{\text{/a/}}(\ensuremath\mathbf{x}_i)$%
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$ f_{\text{/e/}}(\ensuremath\mathbf{x}_i)$%
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$ f_{\text{/i/}}(\ensuremath\mathbf{x}_i)$%
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$ f_{\text{/o/}}(\ensuremath\mathbf{x}_i)$%
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$ f_{\text{/y/}}(\ensuremath\mathbf{x}_i)$%
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$ [400,1800]^{\mathsf T}$%
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$ [400,1000]^{\mathsf T}$%
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$ [530,1000]^{\mathsf T}$%
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$ [600,1300]^{\mathsf T}$%
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$ [670,1300]^{\mathsf T}$%
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$ [420,2500]^{\mathsf T}$%
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$ \log p(\ensuremath\mathbf{x}_i|q_k,\ensuremath\boldsymbol{\Theta})$%
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$ d-1$%
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$ f_k(\ensuremath\mathbf{x})$%
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$\displaystyle \ensuremath\mathbf{x}\in q_k \quad \Leftrightarrow \quad f_k(\ensuremath\mathbf{x},\ensuremath\boldsymbol{\Theta}_k) \geq f_l(\ensuremath\mathbf{x},\ensuremath\boldsymbol{\Theta}_l),
\quad \forall l \neq k
$%
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$ P(q_k|\ensuremath\mathbf{x}_i)$%
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$\displaystyle \ensuremath\mathbf{x}\in q_k$%
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$\displaystyle P(q_k|\ensuremath\mathbf{x}_i) \geq P(q_l|\ensuremath\mathbf{x}_i),\quad \forall l \neq k$%
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$\displaystyle p(\ensuremath\mathbf{x}_i|q_k)\; P(q_k) \geq p(\ensuremath\mathbf{x}_i|q_l)\; P(q_l),\quad
\forall l \neq k$%
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$\displaystyle \log p(\ensuremath\mathbf{x}_i|q_k)+\log P(q_k) \geq \log
p(\ensuremath\mathbf{x}_i|q_l)+\log P(q_l),\quad \forall l \neq k$%
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$ \ensuremath\boldsymbol{\Sigma}_{\text{/e/}}$%
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$ \ensuremath\mathbf{x}_n$%
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$ n=1,\ldots,N$%
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$\displaystyle d_k(\ensuremath\mathbf{x}_n)$%
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$\displaystyle \, \left\| \ensuremath\mathbf{x}_n - \ensuremath\boldsymbol{\mu}_k \right\|^2$%
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$\displaystyle (\ensuremath\mathbf{x}_n-\ensuremath\boldsymbol{\mu}_k)^{\mathsf T}(\ensuremath\mathbf{x}_n-\ensuremath\boldsymbol{\mu}_k)$%
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$\displaystyle d_k(\ensuremath\mathbf{x}_n) \, < \, d_l(\ensuremath\mathbf{x}_n), \qquad \forall l \neq k
$%
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$\displaystyle J = \sum_{k=1}^{K} \sum_{\ensuremath\mathbf{x}_n \in q_k} d_k(\ensuremath\mathbf{x}_n)
$%
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$ \{\ensuremath\boldsymbol{\mu}_{\text{/a/}},
\ensuremath\boldsymbol{\mu}_{\text{/e/}}, \ensuremath\boldsymbol{\mu}_{\text{/i/}}, \ensuremath\boldsymbol{\mu}_{\text{/o/}},
\ensuremath\boldsymbol{\mu}_{\text{/y/}}\}$%
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$ {\cal N}(\ensuremath\boldsymbol{\mu}_{k},\ensuremath\boldsymbol{\Sigma}_{k}),
\; k=1\ldots K$%
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$ p(X,Q|\ensuremath\boldsymbol{\Theta})$%
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$\displaystyle \ensuremath\boldsymbol{\mu}_{k}^{(i+1)} =$%
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$\displaystyle \ensuremath\boldsymbol{\Sigma}_{k}^{(i+1)} =$%
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$\displaystyle P(q_k^{(i+1)}|\ensuremath\boldsymbol{\Theta}^{(i+1)}) = \frac{\mbox{number of training points
belonging to } q_k^{(i)} }{\mbox{total number of training points}}
$%
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$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta})$%
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$\displaystyle \sum_{X} P(X|\ensuremath\boldsymbol{\Theta}) \;=\; \sum_{Q} \sum_{X} p(X,Q|\ensuremath\boldsymbol{\Theta})$%
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$\displaystyle \sum_{k=1}^{K} \sum_{\ensuremath\mathbf{x}_n \in q_k} \log p(\ensuremath\mathbf{x}_n|\ensuremath\boldsymbol{\Theta}_k),$%
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$ \{\ensuremath\boldsymbol{\Sigma}_{\text{/a/}}, \ensuremath\boldsymbol{\Sigma}_{\text{/e/}}, \ensuremath\boldsymbol{\Sigma}_{\text{/i/}},
\ensuremath\boldsymbol{\Sigma}_{\text{/o/}}, \ensuremath\boldsymbol{\Sigma}_{\text{/y/}}\}$%
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$ P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})$%
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$\displaystyle P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})$%
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$\displaystyle \frac{P(q_k^{(i)}|\ensuremath\boldsymbol{\Theta}^{(i)})
\cdot p(\ensuremath\mathbf{x}_n|q_k^{(i)},\ensuremath\boldsymbol{\Theta}^{(i)})}
{p(\ensuremath\mathbf{x}_n|\ensuremath\boldsymbol{\Theta}^{(i)})}$%
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$\displaystyle \frac{P(q_k^{(i)}|\ensuremath\boldsymbol{\Theta}^{(i)}) \cdot p(\ensuremath\mathbf{x}_n|\ensuremath\boldsymbol{\mu}_k^{(i)},\ensuremath\boldsymbol{\Sigma}_k^{(i)}) }
{\sum_j P(q_j^{(i)}|\ensuremath\boldsymbol{\Theta}^{(i)}) \cdot p(\ensuremath\mathbf{x}_n|\ensuremath\boldsymbol{\mu}_j^{(i)},\ensuremath\boldsymbol{\Sigma}_j^{(i)}) }$%
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$\displaystyle \ensuremath\boldsymbol{\mu}_{k}^{(i+1)} = \frac{\sum_{n=1}^{N} \ensuremath\mathbf{x}_n
P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})}
{\sum_{n=1}^{N} P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})} $%
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$\displaystyle \ensuremath\boldsymbol{\Sigma}_{k}^{(i+1)} = \frac{\sum_{n=1}^{N} P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})\;
(\ensuremath\mathbf{x}_n - \ensuremath\boldsymbol{\mu}_k^{(i+1)})(\ensuremath\mathbf{x}_n - \ensuremath\boldsymbol{\mu}_k^{(i+1)})^{\mathsf T}}
{\sum_{n=1}^{N} P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})} $%
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$\displaystyle P(q_k^{(i+1)}|\ensuremath\boldsymbol{\Theta}^{(i+1)}) = \frac{1}{N} \sum_{n=1}^{N}
P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)}) $%
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$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta}) = \log p(X|\ensuremath\boldsymbol{\Theta})$%
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$\displaystyle = \log \sum_Q p(X,Q|\ensuremath\boldsymbol{\Theta})$%
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$\displaystyle = \log \sum_Q P(Q|X,\ensuremath\boldsymbol{\Theta})p(X|\ensuremath\boldsymbol{\Theta})$%
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$\displaystyle = \log \sum_{k=1}^{K} P(q_k|X,\ensuremath\boldsymbol{\Theta}) p(X|\ensuremath\boldsymbol{\Theta})$%
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$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta}) \ge J(\ensuremath\boldsymbol{\Theta})$%
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$\displaystyle = \sum_{k=1}^{K} P(q_k|X,\ensuremath\boldsymbol{\Theta}) \log p(X|\ensuremath\boldsymbol{\Theta})$%
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$\displaystyle = \sum_{k=1}^{K} \sum_{n=1}^{N} P(q_k|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}) \log p(\ensuremath\mathbf{x}_n|\ensuremath\boldsymbol{\Theta})$%
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$ J(\ensuremath\boldsymbol{\Theta})$%
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$ {\cal
L}(\ensuremath\boldsymbol{\Theta})$%
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