2019-01-08

2019-01-08  本文已影响0人  一重大礼

\Delta \omega_i=\eta(t-y)x_i \tag{4.1}

E=\frac {1} {2N}(T-Y)^2=\frac {1} {2N}\sum_{i=1}^{N} {(t_i-y_i)^2}\tag{4.1}

f'(x_0)=\lim_{\Delta x\to 0}\frac {\Delta y } {\Delta X}=\lim_{\Delta X\to 0}\frac {f(x_0+\Delta x)-f(x_0)} {\Delta X}\tag{4.2}

\frac {\delta} {\delta x_i}f(x_0,x_1,\ldots,x_n)=\lim_{\Delta x\to 0}\frac {\Delta y } {\Delta x}=\lim_{\Delta x\to 0}\frac {f(x_0,\ldots,x_i+\Delta x, \ldots,x_n)-f(x_0,\ldots,x_i,\ldots,x_n)} {\Delta x}\tag{4.3}

\frac {\delta} {\delta l}f(x_0,x_1,\ldots,x_n)=\lim_{\Delta \rho\to 0}\frac {\Delta y } {\Delta x}=\lim_{\Delta \rho\to 0}\frac {f(x_0+\Delta x_0,\ldots,x_i+\Delta x_i, \ldots,x_n+\Delta x_n)-f(x_0,\ldots,x_i,\ldots,x_n)} { \rho}\tag{4.4}

\rho=\sqrt {(\Delta x_0)^2+\ldots+(\Delta x_i)^2+\ldots+(\Delta x_n)^2}

gradf(x_0,x_1,\ldots,x_n)=(\frac {\delta f} {\delta x_0},\ldots,\frac {\delta f} {\delta x_i},\ldots,\frac {\delta f} {\delta x_n})\tag{4.5}

|gradf(x_0,x_1,\ldots,x_n)|=\sqrt{(\frac {\delta f} {\delta x_0})^2 ,\ldots,(\frac {\delta f} {\delta x_i})^2 ,\ldots,(\frac {\delta f} {\delta x_n})^2}\tag{4.6}

x_0=x_0-\eta\frac {\delta f} {\delta x_0}

x_i=x_i-\eta\frac {\delta f} {\delta x_i}

x_n=x_n-\eta\frac {\delta f} {\delta x_n}

x=x_n-\eta\frac {\delta f} {\delta x}

w=w-\eta\frac {\delta f} {\delta w}\tag{4.7}

f(x)=\frac {1} {1+e^{-x}}\tag{4.8}

f(x)=\frac {e^x-e^{-x}} {e^x+e^{-x}}\tag{4.9}

f(x)=\frac {x} {1+|x|}\tag{4.10}

f(x)=max(0,x)\tag{4.11}

X=(x_1,x_2,\ldots,x_i,\ldots,x_n)

Y^1=( y_1^1,y_2^1,\ldots,y_j^1,\ldots,y_m^1)

Y^2=( y_1^2,y_2^2,\ldots,y_k^2,\ldots,y_l^2)
T=(t_1,t_2,\ldots,t_k,\ldots,t_l)

net_j^1=\sum_{i=0}^{n} {w^1_{ij}x_i}\tag{4.12}
j=1,2,\ldots,m\tag{4.12}

y_j^1=f(net_j^1)\tag{4.13}
j=1,2,\ldots,m\tag{4.13}

net_k^2=\sum_{i=0}^{n} {w^2_{jk}y_j^1}\tag{4.14}
k=1,2,\ldots,l\tag{4.14}

y_k^2=f(net_k^2)\tag{4.15}
k=1,2,\ldots,l\tag{4.15}

f'(x)=f(x)[1-f(x)]\tag{4.16}

E=\frac {1} {2}(T-Y^2)^2=\frac {1} {2}\sum_{k=1}^{l} {t_k-y_k^2)^2}\tag{4.17}

E=\frac {1} {2}\sum_{k=1}^{l} {[t_k-f(net_k^2)]^2}=\frac {1} {2}\sum_{k=1}^{l} { \left [t_k-f(\sum_{j=0}^{m}{w_{jk}^2}y_j^1 )\right]^2}\tag{4.18}

E=\frac {1} {2}\sum_{k=1}^{l} { \left [t_k-f\left(\sum_{j=0}^{m}{w_{jk}^2}f(net_j^1)\right)\right]^2}=\frac {1} {2}\sum_{k=1}^{l} { \left [t_k-f\left(\sum_{j=0}^{m}{w_{jk}^2}f\left(\sum_{j=0}^{m}{w_{ij}^1x_i}\right)\right)\right]^2}\tag{4.19}

\Delta w_{ij}^1=-\eta\frac {\delta E} {\delta w_{ij}^1}
i=0,1,2,\ldots,n;j=1,2,\ldots,m\tag{4.20}

\Delta w_{ij}^2=-\eta\frac {\delta E} {\delta w_{jk}^2 }
j=0,1,2,\ldots,m;k=1,2,\ldots,l\tag{4.21}

\Delta w_{ij}^1=-\eta\frac {\delta E} {\delta w_{ij}^1 }=-\eta\frac {\delta E} {\delta net_j^1 }\frac {\delta net_j^1 } {\delta w_{ij}^1 }\tag{4.22}

\Delta w_{jk}^2=-\eta\frac {\delta E} {\delta w_{jk}^2 }=-\eta\frac {\delta E} {\delta net_k^2 }\frac {\delta net_k^2 } {\delta w_{jk}^2 }\tag{4.23}

\delta_j^1=-\frac {\delta E} {\delta net_j^1 }\tag{4.24}

\delta_k^1=-\frac {\delta E} {\delta net_k^2 }\tag{4.25}

\Delta w_{ij}^1=\eta\delta_j^1x_i\tag{4.26}

\Delta w_{jk}^2=\eta\delta_k^2y_i^1\tag{4.27}

\delta_j^1=-\frac {\delta E} {\delta net_j^1 }=-\frac {\delta E} {\delta y_j^1 }\frac {\delta y_j^1} {\delta net_j^1 }=-\frac {\delta E} {\delta y_j^1 }f'(net_j^1)\tag{4.28}

\delta_k^2=-\frac {\delta E} {\delta net_k^2 }=-\frac {\delta E} {\delta y_k^2 }\frac {\delta y_k^2} {\delta net_k^2 }=-\frac {\delta E} {\delta y_k^2 }f'(net_k^2)\tag{4.29}

\frac {\delta E} {\delta y_k^2 }=-(t_k-y_k^2)\tag{4.30}

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