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@@ -21,8 +21,8 @@ $$\Delta \theta_j = -\eta \cfrac{\partial E_k}{\partial \theta_j}$$
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$$
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\begin{aligned}
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\cfrac{\partial E_k}{\partial \theta_j} &= \cfrac{\partial E_k}{\partial \hat{y}_j^k} \cdot\cfrac{\partial \hat{y}_j^k}{\partial \theta_j} \\
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-&= (\hat{y}_j^k-y_j^k) \cdot f’(\beta_j-\theta_j) \cdot (-1) \\
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-&= -(\hat{y}_j^k-y_j^k)f’(\beta_j-\theta_j) \\
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+&= (\hat{y}_j^k-y_j^k) \cdot f^{\prime}(\beta_j-\theta_j) \cdot (-1) \\
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+&= -(\hat{y}_j^k-y_j^k)f^{\prime}(\beta_j-\theta_j) \\
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&= g_j
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\end{aligned}
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$$
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@@ -37,10 +37,10 @@ $$
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\begin{aligned}
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\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}} \\
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&= \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 \\
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-&= \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 \\
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-&= \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 \\
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-&= \sum_{j=1}^{l} (-g_j) \cdot w_{hj} \cdot f’(\alpha_h-\gamma_h) \cdot x_i \\
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-&= -f’(\alpha_h-\gamma_h) \cdot \sum_{j=1}^{l} g_j \cdot w_{hj} \cdot x_i\\
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+&= \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 x_i \\
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+&= \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) \cdot x_i \\
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+&= \sum_{j=1}^{l} (-g_j) \cdot w_{hj} \cdot f^{\prime}(\alpha_h-\gamma_h) \cdot x_i \\
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+&= -f^{\prime}(\alpha_h-\gamma_h) \cdot \sum_{j=1}^{l} g_j \cdot w_{hj} \cdot x_i\\
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&= -b_h(1-b_h) \cdot \sum_{j=1}^{l} g_j \cdot w_{hj} \cdot x_i \\
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&= -e_h \cdot x_i
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\end{aligned}
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@@ -55,8 +55,8 @@ $$\Delta \gamma_h = -\eta \cfrac{\partial E_k}{\partial \gamma_h}$$
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$$
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\begin{aligned}
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\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} \\
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-&= \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) \\
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-&= -\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)\\
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+&= \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) \\
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+&= -\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)\\
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&=e_h
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\end{aligned}
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$$
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Sm1les