Differences

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--- correlation [2020/11/17 20:23] – hkimscil
+++ correlation [2023/10/05 17:19] (current) – [e.g. 1,] hkimscil
@@ Line 60: / Line 60: @@
 ===== 공분산 =====
-$$ \text{cov(x, y)} = \frac{\Sigma_{i-1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{n-1}$$
+\begin{eqnarray*}
+\text{cov(x, y)} & = & \frac{\Sigma_{i-1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{n-1} \\
+& = & \frac{SP}{(n-1)}
+\end{eqnarray*}
 공분산 값은 x와 y의 단위에 의한 영향을 받는다. 따라서 이 값을 x와 y의 표준편차 값으로 나누어 준것을 피어슨의 상관계수 (Pearson's correlation)라고 한다.
 \begin{eqnarray*}
 \text{corr(x, y)} & = & \frac{\text{cov(x, y)}}{\text{sd(x)} \text{sd(y)}} \\
-& = & \frac{\text{cov(x, y)}}{\sqrt{\text{V(x)} \text{V(y)} } }
+& = & \frac{\text{cov(x, y)}}{\sqrt{\text{var(x)} \text{var(y)} } }
 \end{eqnarray*}
@@ Line 150: / Line 153: @@
  & = & 10 \nonumber
 \end{eqnarray}
+<WRAP box>
+그런데 왜 다음과 같은 공식인지는
+\begin{align}
+SS_{\small{X}} = \sum X^2 - \frac{(\sum X)^2}{n} \label{ss.simplified} \tag{SS simplified} \\
+\end{align}
+우선
+\begin{align}
+Var[X] & = \frac {SS_{\small{X}}}{df} \;\;\; \nonumber \\
+& \text{Let's assume that  } df \nonumber \\
+& \text{is n instead of n-1} \nonumber \\
+& \text{And we also know that} \nonumber \\
+Var[X] & = E[X^2] − (E[X])^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 \\
+& \therefore \nonumber \\
+SS_{\small{X}} & = \Sigma {X^2} - \frac{(\Sigma{X})^2}{n}  \;\;\;\;\; \text{That is,  } \; \ref{ss.simplified} \nonumber \\
+\end{align}
+</WRAP>
+<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}  \nonumber \\
+ & \therefore  \nonumber \\
+SP & = \Sigma{XY} - \frac{\Sigma{X} \Sigma{Y}}{n}  \;\;\;\;\; \text{That is,  } \; \ref{sp.simplified} \nonumber \\
+\end{align}
+</WRAP>
 이제 r (correlation coefficient) 값은: