协方差与相关性Covariance & Correlation

先中心化Center first

对每个特征分别减去自己的样本均值，得到相对于中心的偏差。每个点的两个偏差同号时乘积为正，异号时为负。Subtract each feature’s sample mean to get deviations from the center. The product of a point’s deviations is positive if their signs agree, negative if they differ.

平均偏差乘积Average deviation products

协方差是这些乘积的平均。正值表示在这组数据中常同向变化，负值表示常反向变化；协方差矩阵对称，对角线是各特征的描述性方差。Covariance is the average of these products. A positive value indicates that deviations often move together in this dataset, while a negative value indicates opposite movement. The covariance matrix is symmetric and its diagonal contains descriptive variances.

去掉尺度Remove scale

相关系数用两个标准差归一化协方差。任一特征方差为零时分母为零，相关系数未定义。这里用 1/n 描述手中的样本；估计总体时另有 1/(n−1) 的条件与用途。Correlation normalizes covariance by the two standard deviations. If either variance is zero, the denominator is zero and correlation is undefined. We use 1/n to describe the available sample; 1/(n−1) has a separate estimation purpose and conditions.

公式与符号Formula and symbols

n 是样本数，xᵢⱼ 是样本 i 的特征 j，x̄ⱼ 是该特征均值；C 是采用 1/n 的经验协方差矩阵。相关系数 r 要求两个方差 C₁₁、C₂₂ 均大于零。n is sample count, xᵢⱼ feature j of sample i, and x̄ⱼ that feature’s mean. C is the empirical covariance matrix using 1/n. Correlation r requires both variances C₁₁ and C₂₂ to be positive.

继续探索Explore further

• 后续阅读：协方差主轴与 PCAFurther reading: covariance axes and PCA