差分
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| 次のリビジョン | 前のリビジョン | ||
| research:memos:chi2 [2019/12/19 12:39] – 作成 kobayash | research:memos:chi2 [2019/12/19 17:30] (現在) – kobayash | ||
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| 行 1: | 行 1: | ||
| - | ===== Memo of chi 2 ===== | + | ===== Memo of chi square ===== |
| + | |||
| + | ==== Exponential function + constant ==== | ||
| + | \begin{align} | ||
| + | \chi^2 | ||
| + | &= \sum_i \left[ \ln(y_i-c) - \ln a +bx_i \right]^2\\ | ||
| + | &= \sum_i \left[ \left[ \ln(y_i-c) \right]^2 + \left(\ln a\right)^2 + b^2 x_i^2 - 2 \ln a\ln(y_i-c) + 2 bx_i\ln(y_i-c) - 2b x_i \ln a\right]\\ | ||
| + | \end{align} | ||
| + | |||
| + | \begin{align} | ||
| + | \frac{\partial \chi^2 }{\partial a} = \sum_i \left[ 2 \ln a - 2\ln(y_i-c) - 2b x_i \right] | ||
| + | \frac{\partial \chi^2 }{\partial b} = \sum_i \left[2 b x_i^2 + 2x_i \ln\left( y_i-c\right) - 2x_i \ln a \right] = 0\\ | ||
| + | \frac{\partial \chi^2 }{\partial c} = \sum_i \left[ 2\frac{\ln(y_i-c)}{y_i-c} - 2\frac{\ln a}{y_i-c} + 2\frac{b x_i}{y_i-c}\right] | ||
| + | \end{align} | ||
| + | |||
| + | \begin{align} | ||
| + | n\ln a - \sum_i \ln(y_i-c) - b \sum_i x_i = 0\\ | ||
| + | b \sum_i x_i^2 + \sum_i x_i \ln\left( y_i-c\right) - \ln a \sum_i x_i = 0\\ | ||
| + | \sum_i \frac{\ln(y_i-c)}{y_i-c} | ||
| + | \end{align} | ||
| + | |||
| + | \begin{align} | ||
| + | n \ln a \sum_i x_i - \sum_i x_i \sum_i \ln(y_i-c) - b \left(\sum_i x_i\right)^2 = 0\\ | ||
| + | n b \sum_i x_i^2 + n \sum_i x_i \ln\left( y_i-c\right) - n\ln a \sum_i x_i = 0\\ | ||
| + | \end{align} | ||
| + | |||
| + | \begin{align} | ||
| + | \left[n \sum_i x_i^2 + \left(\sum_i x_i\right)^2 \right] b = \sum_i x_i \sum_i \ln(y_i-c) - n \sum_i x_i \ln\left( y_i-c\right) \\ | ||
| + | \end{align} | ||
| + | |||
| + | \begin{align} | ||
| + | b = \frac{\sum_i x_i \sum_i \ln(y_i-c) - n \sum_i x_i \ln\left( y_i-c\right)}{n \sum_i x_i^2 + \left(\sum_i x_i\right)^2} \\ | ||
| + | \end{align} | ||
| + | |||
| + | 書きかけ | ||
| + | |||
| + | |||
| + | |||
