4.3. 両方の場合をまとめた式

(Formula used in machine_learning header file package; 4. Derivative of the cross entropy error

E

with respect to model parameters

W_{j, i}^{(m)}

; 4.3. Unified formula for the both cases)

前節までで以下の式が得られた。
The formula below were obtained in the previous sections.

4.1節の(3)式
Eq. (3) of section 4.1:
$\begin{array}{rcl} \frac{\partial E}{\partial W_{j, i}^{(m)}} & = & - \frac{1}{N} \sum_{n = 0}^{N - 1} [(1 - δ_{i J^{(m)}}) x_{i}^{(m, n)} + δ_{i J^{(m)}}] \sum_{i^{'} = 0}^{J^{(M + 1)} - 1} \frac{t_{i^{'}}^{(n)}}{x_{i^{'}}^{(M + 1, n)}} D_{i^{'}, j}^{(m, n)} \\ (1) & (m = 0, \dots, M - 1; i = 0, \dots, J^{(m)}; j = 0, \dots, J^{(m + 1)} - 1) \end{array}$
4.2節の(5)式
Eq. (5) of section 4.2:
$\begin{array}{rcl} \frac{\partial E}{\partial W_{j, i}^{(M)}} & = & - \frac{1}{N} \sum_{n = 0}^{N - 1} [(1 - δ_{i J^{(M)}}) x_{i}^{(M, n)} + δ_{i J^{(M)}}] \\ \sum_{i^{'} = 0}^{J^{(M + 1)} - 1} \frac{t_{i^{'}}^{(n)}}{x_{i^{'}}^{(M + 1, n)}} [D_{i^{'}, j}^{(M, n)} - D_{i^{'}, J^{(M + 1)} - 1}^{(M, n)}] \\ (2) & (i = 0, \dots, J^{(M)}; j = 0, \dots, J^{(M + 1)} - 2) \end{array}$

(

1

)(

2

)式はまとめて

\begin{array}{rcl} \frac{\partial E}{\partial W_{j, i}^{(m)}} & = & - \frac{1}{N} \sum_{n = 0}^{N - 1} [(1 - δ_{i J^{(m)}}) x_{i}^{(m, n)} + δ_{i J^{(m)}}] \\ \sum_{i^{'} = 0}^{J^{(M + 1)} - 1} \frac{t_{i^{'}}^{(n)}}{x_{i^{'}}^{(M + 1, n)}} [D_{i^{'}, j}^{(m, n)} - D_{i^{'}, J^{(M + 1)} - 1}^{(M, n)} δ_{m M}] \\ (m = 0, \dots, M; i = 0, \dots, J^{(m)}; j = 0, \dots, J^{(m + 1)} - 1; \\ (3) & m \neq M or j \neq J^{(m + 1)} - 1) \end{array}

と書ける。
Eqs. (

1

) and (

2

) can be unified as (

3