correlation
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correlation [2023/10/05 16:09] – [e.g. 1,] hkimscil | correlation [2023/10/05 17:19] (current) – [e.g. 1,] hkimscil | ||
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Line 154: | Line 154: | ||
\end{eqnarray} | \end{eqnarray} | ||
- | ---- | + | <WRAP box> |
- | 그런데 왜 다음과 같은 공식인지는 | + | 그런데 왜 다음과 같은 공식인지는 |
+ | \begin{align} | ||
+ | SS_{\small{X}} = \sum X^2 - \frac{(\sum X)^2}{n} | ||
+ | \end{align} | ||
우선 | 우선 | ||
+ | |||
\begin{align} | \begin{align} | ||
- | Var[X] | + | Var[X] & = \frac {SS_{\small{X}}}{df} \;\;\; \nonumber \\ |
& \text{Let' | & \text{Let' | ||
& \text{is n instead of n-1} \nonumber \\ | & \text{is n instead of n-1} \nonumber \\ | ||
& \text{And we also know that} \nonumber \\ | & \text{And we also know that} \nonumber \\ | ||
- | = & E[X^2] − (E[X])^2 \nonumber \\ | + | Var[X] |
- | = & \frac {\Sigma {X^2}}{n} - \left(\frac{\Sigma{X}}{n} \right)^2 \nonumber \\ | + | & = \frac {\Sigma {X^2}}{n} - \left(\frac{\Sigma{X}}{n} \right)^2 \nonumber \\ |
- | = & \frac {\Sigma {X^2}}{n} - \frac{(\Sigma{X})^2}{n^2} \nonumber \\ | + | & = \frac {\Sigma {X^2}}{n} - \frac{(\Sigma{X})^2}{n^2} \nonumber \\ |
- | & \therefore \\ | + | & \therefore |
- | SS_{\small{X}} | + | SS_{\small{X}} & = \Sigma {X^2} - \frac{(\Sigma{X})^2}{n} |
+ | \end{align} | ||
+ | </ | ||
+ | |||
+ | <WRAP box> | ||
+ | 또한 | ||
+ | \begin{align} | ||
+ | SP & = & \sum XY - \frac{\sum X \sum Y}{n} \label{sp.simplified} \tag{SP simplified} \\ | ||
+ | \end{align} | ||
+ | |||
+ | |||
+ | \begin{align} | ||
+ | Cov[X,Y] & = E[(X-\overline{X})(Y-\overline{Y})] \nonumber \\ | ||
+ | & = E[XY - X \overline{Y} - \overline{X} Y - \overline{X} \overline{Y}] \nonumber \\ | ||
+ | & = E[XY] - E[X] \overline{Y} - \overline{X} E[Y] + \overline{X} \overline{Y} \nonumber \\ | ||
+ | & \because \;\;\; E[c] = c \;\;\; \text{and, } \overline{X} = E[X] \nonumber \\ | ||
+ | & = E[XY] - E[X]E[Y] - E[X]E[Y] + E[X]E[Y] \nonumber \\ | ||
+ | & = E[XY] - E[X]E[Y] \nonumber \\ | ||
+ | & = \frac{\Sigma{XY}}{n} - \frac{\Sigma{X}}{n} \frac{\Sigma{Y}}{n} | ||
+ | & \therefore | ||
+ | SP & = \Sigma{XY} - \frac{\Sigma{X} \Sigma{Y}}{n} | ||
\end{align} | \end{align} | ||
+ | </ | ||
이제 r (correlation coefficient) 값은: | 이제 r (correlation coefficient) 값은: |
correlation.1696489784.txt.gz · Last modified: 2023/10/05 16:09 by hkimscil