一些统计量
均值 (Mean):
\[\overline{x}=\frac{1}{n}\sum_{i=1}^{n} x_i
\]
方差 (Variance): 衡量单类样本偏离均值的程度
\[D(x)=\frac{1}{n}\sum_{i=1}^{n}(x_i-\overline{x})^2
\]
协方差 (Covariance): 反映两个随机变量的相关程度
\[\begin{aligned} \text{Cov}(x,y) &= E[(X-E(X))(Y-E(Y))] \\ &= \frac{1}{n}\sum_{i=1}^{n} (x_i-\overline{x})(y_i-\overline{y}) \\ &= E(XY) - E(X)\cdot E(Y) \\ &= (\frac{1}{n}\sum_{i=1}^{n} x_i y_i) -\overline{x}\cdot\overline{y} \end{aligned}
\]
皮尔森相关系数 (Pearson Correlation Coefficient, PCC): 标准化协方差，消除量纲影响
数据中心化后两个n维向量的夹角余弦
\[\begin{aligned} \rho(x,y)
&= \frac{\text{Cov}(x,y)}{\sqrt{D(x)}\sqrt{D(y)}} \\
&= \frac{\sum_{i=1}^{n} (x_i-\overline{x})(y_i-\overline{y})}{\sqrt{\sum_{i=1}^{n}(x_i-\overline{x})^2}\sqrt{\sum_{i=1}^{n}(y_i-\overline{y})^2}}
\end{aligned}\]
