协方差矩阵

Covariance Matrix

多维随机向量的协方差构成的对称矩阵，用来描述一组变量的联合分布中的元素之间的线性关系

二维随机变量 $(X_{1}, X_{2})$ 的 4 个二阶中心矩排成矩阵的形式：

\begin{aligned} (\begin{array}{c} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}) \\ c_{11} = E {[X_{1} - E (X_{1})]^{2}} \\ c_{22} = E {[X_{2} - E (X_{2})]^{2}} \\ c_{12} = E {[X_{1} - E (X_{1})] [X_{2} - E (X_{2})]} \\ c_{21} = E {[X_{2} - E (X_{2})] [X_{1} - E (X_{1})]} \end{aligned}

n 维随机变量 $(X_{1}, X_{2}, \dots X_{n})$ 的协方差矩阵：

\begin{array}{r} C = (\begin{array}{c} c_{11} & c_{12} & \dots & c_{1 n} \\ c_{21} & c_{22} & \dots & c_{2 n} \\ ⋮ & ⋮ & ⋮ \\ c_{n 1} & c_{n 2} & \dots & c_{n n} \end{array}) \end{array}

\begin{aligned} c_{i j} & = C o v (X_{i}, X_{j}) = E {[X_{i} - E (X_{i})] [X_{j} - E (X_{j})]} (i, j = 1, 2, \dots, n) \end{aligned}

重要应用：研究多维正态分布

注意

以三个变量举例
可以扩展为 n 个变量的计算

矩阵计算
过渡矩阵 $a$

\begin{aligned} a & = [\begin{array}{c} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3} \end{array}] - \frac{1}{3} [\begin{array}{c} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}] [\begin{array}{c} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3} \end{array}] \end{aligned}

则协方差矩阵 $p$ 为：

\begin{array}{r} p = \frac{1}{3} a^{T} a \end{array}