| \(\large{y=ax^2e^{-bx}} \qquad (1)\) |
| \(\large{a=} \) | \(\large{b=} \) | \(\large{σ=} \) | |||||||||
| \(\large{a^\left(0\right)=} \) | \(\large{b^\left(0\right)=} \) | ||||||
| \(\large{\varDelta a=} \) | \(\large{\varDelta b=} \) | ||||||
| \(\large{a^\left(k\right)=} \) | \(\large{b^\left(k\right)=} \) |
| \(\large{(k=} \) | \(\large{)} \) |
| \(\large{r\left(a,b \right)=\frac{1}{2}\displaystyle \sum_{i=1}^{n} \left(y_i - a^\left(k\right)x_i^2e^{-b^\left(k\right)x_i} \right)^2 } \qquad (2)\) | |
| \(\small{y_i}\): データ系列 \(\small{\qquad n}\): データ系列の個数 | |
| \(\small{a^\left(k\right), b^\left(k\right)}\): \(\small{k}\)回目の反復での\(\small{a, b}\) |
| \(\large{ \begin{bmatrix} \frac{\partial^2 r}{\partial a^2} & \frac{\partial^2 r}{\partial a \partial b} \\ \frac{\partial^2 r}{\partial b \partial a} & \frac{\partial^2 r}{\partial b^2} \end{bmatrix} \varDelta X= - \begin{bmatrix} \frac{\partial r}{\partial a} \\ \frac{\partial r}{\partial b} \end{bmatrix} \qquad (3) }\) |
| \(\large{ \varDelta X= \begin{bmatrix} a^\left(k+1\right)-a^\left(k\right) \\ b^\left(k+1\right)-b^\left(k\right) \end{bmatrix} }\) |
| \(\large{g\left(x\right)=x^2 e^{-bx} } \qquad \left(x=x_i\right) \qquad(4)\) | |
| \(\large{\frac{\partial r}{\partial a}=\displaystyle \sum_{i=1}^{n} -g\left(y_i - ag \right) } \qquad (5)\) | |
| \(\large{\frac{\partial r}{\partial b}=\displaystyle \sum_{i=1}^{n} axg\left(y_i - ag \right) } \qquad (6)\) | |
| \(\large{\frac{\partial^2 r}{\partial a^2}=\displaystyle \sum_{i=1}^{n} g^2 } \qquad (7)\) | |
| \(\large{\frac{\partial^2 r}{\partial a \partial b} = \frac{\partial^2 r}{\partial b \partial a}=\displaystyle \sum_{i=1}^{n} xg\left(y_i - 2ag \right) } \qquad (8)\) | |
| \(\large{\frac{\partial^2 r}{\partial b^2}=\displaystyle \sum_{i=1}^{n} -ax^2g \left(y_i - 2ag \right) } \qquad (9)\) |