修订5.12/5.13/5.14

docs/chapter5/chapter5.md

@@ -21,8 +21,13 @@ $$\Delta \theta_j = -\eta \cfrac{\partial E_k}{\partial \theta_j}$$
 $$
 \begin{aligned}	
 \cfrac{\partial E_k}{\partial \theta_j} &= \cfrac{\partial E_k}{\partial \hat{y}_j^k} \cdot\cfrac{\partial \hat{y}_j^k}{\partial \theta_j} \\
-&= (\hat{y}_j^k-y_j^k) \cdot f^{\prime}(\beta_j-\theta_j) \cdot (-1) \\
-&= -(\hat{y}_j^k-y_j^k)f^{\prime}(\beta_j-\theta_j) \\
+&= \cfrac{\partial E_k}{\partial \hat{y}_j^k} \cdot\cfrac{\partial [f(\beta_j-\theta_j)]}{\partial \theta_j} \\
+&=\cfrac{\partial E_k}{\partial \hat{y}_j^k} \cdot f^{\prime}(\beta_j-\theta_j) \times (-1) \\
+&=\cfrac{\partial E_k}{\partial \hat{y}_j^k} \cdot f\left(\beta_{j}-\theta_{j}\right)\times\left[1-f\left(\beta_{j}-\theta_{j}\right)\right]  \times (-1) \\
+&=\cfrac{\partial E_k}{\partial \hat{y}_j^k} \cdot \hat{y}_j^k\left(1-\hat{y}_j^k\right)  \times (-1) \\
+&=\cfrac{\partial\left[ \cfrac{1}{2} \sum\limits_{j=1}^{l}\left(\hat{y}_{j}^{k}-y_{j}^{k}\right)^{2}\right]}{\partial \hat{y}_{j}^{k}} \cdot \hat{y}_j^k\left(1-\hat{y}_j^k\right) \times (-1)  \\
+&=\cfrac{1}{2}\times 2(\hat{y}_j^k-y_j^k)\times 1 \cdot\hat{y}_j^k\left(1-\hat{y}_j^k\right)  \times (-1) \\
+&=(y_j^k-\hat{y}_j^k)\hat{y}_j^k\left(1-\hat{y}_j^k\right)  \\
 &= g_j
 \end{aligned}
 $$
@@ -46,7 +51,7 @@ $$
 \end{aligned}
 $$
 所以
-$$\Delta v_{ih} = -\eta \cdot -e_h \cdot x_i=\eta e_h x_i$$
+$$\Delta v_{ih} =-\eta \cfrac{\partial E_k}{\partial v_{ih}} =\eta e_h x_i$$
 ## 5.14
 $$\Delta \gamma_h= -\eta e_h$$
 [推导]：因为
@@ -57,8 +62,10 @@ $$
 \cfrac{\partial E_k}{\partial \gamma_h} &= \sum_{j=1}^{l} \cfrac{\partial E_k}{\partial \hat{y}_j^k} \cdot \cfrac{\partial \hat{y}_j^k}{\partial \beta_j} \cdot \cfrac{\partial \beta_j}{\partial b_h} \cdot \cfrac{\partial b_h}{\partial \gamma_h} \\
 &= \sum_{j=1}^{l} \cfrac{\partial E_k}{\partial \hat{y}_j^k} \cdot \cfrac{\partial \hat{y}_j^k}{\partial \beta_j} \cdot \cfrac{\partial \beta_j}{\partial b_h} \cdot f^{\prime}(\alpha_h-\gamma_h) \cdot (-1) \\
 &= -\sum_{j=1}^{l} \cfrac{\partial E_k}{\partial \hat{y}_j^k} \cdot \cfrac{\partial \hat{y}_j^k}{\partial \beta_j} \cdot w_{hj} \cdot f^{\prime}(\alpha_h-\gamma_h)\\
+&= -\sum_{j=1}^{l} \cfrac{\partial E_k}{\partial \hat{y}_j^k} \cdot \cfrac{\partial \hat{y}_j^k}{\partial \beta_j} \cdot w_{hj} \cdot b_h(1-b_h)\\
+&= \sum_{j=1}^{l}g_j\cdot w_{hj} \cdot b_h(1-b_h)\\
 &=e_h
 \end{aligned}
 $$
 所以
-$$\Delta \gamma_h= -\eta e_h$$
+$$\Delta \gamma_h=-\eta\cfrac{\partial E_k}{\partial \gamma_h} = -\eta e_h$$