|
|
@@ -20,9 +20,9 @@ $$\Delta \theta_j = -\eta \cfrac{\partial E_k}{\partial \theta_j}$$
|
|
|
又
|
|
|
$$
|
|
|
\begin{aligned}
|
|
|
-\cfrac{\partial E_k}{\partial \theta_j} &= \cfrac{\partial E_k}{\partial \hat{y_j^k}} \cdot\cfrac{\partial \hat{y_j^k}}{\partial \theta_j} \\\\
|
|
|
-&= (\hat{y_j^k}-y_j^k) \cdot f’(\beta_j-\theta_j) \cdot (-1) \\\\
|
|
|
-&= -(\hat{y_j^k}-y_j^k)f’(\beta_j-\theta_j) \\\\
|
|
|
+\cfrac{\partial E_k}{\partial \theta_j} &= \cfrac{\partial E_k}{\partial \hat{y}_j^k} \cdot\cfrac{\partial \hat{y}_j^k}{\partial \theta_j} \\\\
|
|
|
+&= (\hat{y}_j^k-y_j^k) \cdot f’(\beta_j-\theta_j) \cdot (-1) \\\\
|
|
|
+&= -(\hat{y}_j^k-y_j^k)f’(\beta_j-\theta_j) \\\\
|
|
|
&= g_j
|
|
|
\end{aligned}
|
|
|
$$
|
|
|
@@ -35,10 +35,10 @@ $$\Delta v_{ih} = -\eta \cfrac{\partial E_k}{\partial v_{ih}}$$
|
|
|
又
|
|
|
$$
|
|
|
\begin{aligned}
|
|
|
-\cfrac{\partial E_k}{\partial v_{ih}} &= \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 \alpha_h} \cdot \cfrac{\partial \alpha_h}{\partial v_{ih}} \\\\
|
|
|
-&= \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 \alpha_h} \cdot x_i \\\\
|
|
|
-&= \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’(\alpha_h-\gamma_h) \cdot x_i \\\\
|
|
|
-&= \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’(\alpha_h-\gamma_h) \cdot x_i \\\\
|
|
|
+\cfrac{\partial E_k}{\partial v_{ih}} &= \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 \alpha_h} \cdot \cfrac{\partial \alpha_h}{\partial v_{ih}} \\\\
|
|
|
+&= \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 \alpha_h} \cdot x_i \\\\
|
|
|
+&= \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’(\alpha_h-\gamma_h) \cdot x_i \\\\
|
|
|
+&= \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’(\alpha_h-\gamma_h) \cdot x_i \\\\
|
|
|
&= \sum_{j=1}^{l} (-g_j) \cdot w_{hj} \cdot f’(\alpha_h-\gamma_h) \cdot x_i \\\\
|
|
|
&= -f’(\alpha_h-\gamma_h) \cdot \sum_{j=1}^{l} g_j \cdot w_{hj} \cdot x_i\\\\
|
|
|
&= -b_h(1-b_h) \cdot \sum_{j=1}^{l} g_j \cdot w_{hj} \cdot x_i \\\\
|
|
|
@@ -54,9 +54,9 @@ $$\Delta \gamma_h = -\eta \cfrac{\partial E_k}{\partial \gamma_h}$$
|
|
|
又
|
|
|
$$
|
|
|
\begin{aligned}
|
|
|
-\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’(\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’(\alpha_h-\gamma_h)\\\\
|
|
|
+\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’(\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’(\alpha_h-\gamma_h)\\\\
|
|
|
&=e_h
|
|
|
\end{aligned}
|
|
|
$$
|
Sm1les