PRML 4章演習問題 4.4～4.7

4.3

わからん

4.4

クラス分離基準(4.22) $w^{T}(m_{2}-m_{1})$ を $w$ に関して最大化すれば $w\propto(m_{2}-m_{1})$ となること示せ。

$L=w^{T}(m_{2}-m_{1})+\lambda(w^{T}w-1)\\ \frac{\partial}{\partial w}L=m_{2}-m_{1}+\lambda 2w=0\\ w = -\frac{1}{2\lambda}(m_{2}-m_{1})\propto(m_{2}-m_{1})\\$

4.5

(4.20),(4.23),(4.24)を使って(4.25)が(4.26)の形で書けることを示せ。

$(m_{2}-m_{1})^{2}=(w^{T}m_{n}-w^{T}m_{2})(w^{T}m_{1}-w^{T}m_{2})^{T}=w^{T}(m_{1}-m_{2})(m_{2}-m_{1})^{T}w=w^{T}S_{B}w \\ (y_{n}-m_{k})^{2}=(w^{T}x_{n}-w^{T}m_{k})(w^{T}x_{n}-w^{T}m_{k})^{T}=w^{T}(x_{n}-m_{k})(x_{n}-m_{k})^{T}w \\ s_{k}^{2}=\sum_{n\in C_{k}}(y_{n}-m_{k})^{2}=w^{T}\sum_{n\in C_{k}}(x_{n}-m_{k})(x_{n}-m_{k})^{T}w \\ s_{1}^{2}+s_{2}^{2}=w^{T}(\sum_{n\in C_{1}}(x_{n}-m_{1})(x_{n}-m_{1})^{T}+\sum_{n\in C_{2}}(x_{n}-m_{2})(x_{n}-m_{2})^{T})w=w^{T}S_{W}w$ $J(w)=\frac{(m_{2}-m_{1})^{2}}{s_{1}^{2}+s_{2}^{2}}=\frac{w^{T}S_{B}w}{w^{T}S_{W}w}$

4.6

(4.27),(4.28),(4.34),(4.36)および4.1.5節で述べた目的値を使って(4.33)が(4.37)の形で書けることを示せ。

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4.7

(4.59)が $\sigma(-a)=1-\sigma(a)$ を満たすことを示せ。

$\sigma(-a)=\frac{1}{1+\exp(a)}=\frac{1}{1+\exp(a)}\frac{\exp(-a)}{\exp(-a)}=\frac{\exp(-a)}{1+\exp(-a)}\\ \hspace{25pt}=\frac{1+\exp(-a)-1}{1+\exp(-a)}=1-\frac{1}{1+\exp(-a)}=1-\sigma(a)$