Covariance Matrix

https://janakiev.com/blog/covariance-matrix/

Variance, Covariance

• Variance measures the variation of a single random variable (like height of a person in a population)

  $$\sigma_{x}^{2}=\mathbb E \left(\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}\right)$$

• Whereas covariance is a measure of how much two random variables vary together (like the height of a person and the weight of a person in a population)

  $$\sigma(x, y)=\mathbb E\left(\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)\right)$$

所以方差也可以看作一个随机变量自己与自己的协方差:

$$\sigma_{x}^{2} = \frac{1}{n-1} \sum_{i=1}^{n}(x_{i}-\bar{x})(x_{i}-\bar{x}) = \sigma(x, x)$$

协方差矩阵

Wikipedia:

假设 $X$是以 $n$个随机变量组成的列向量，

$\mathbf{X} = \begin{bmatrix} X_1 \\ X_2 \\ \vdots \\ X_n \end{bmatrix}$

并且 $\mu_i$是 $X_i$的期望值，即, $\mu_i = \mathrm{E}(X_i)$。协方差矩阵的第 $(i,j)$項（第 $(i,j)$項是一个协方差）被定义为如下形式：

$$\Sigma_{ij} = \mathrm{cov}(X_i, X_j) = \mathrm{E}\begin{bmatrix} (X_i - \mu_i) (X_j - \mu_j) \end{bmatrix}$$

而协方差矩阵为：

$$\Sigma=\mathrm{E} \left[ \left( \mathbf{X} - \mathrm{E}[\mathbf{X}] \right) \left( \mathbf{X} - \mathrm{E}[\mathbf{X}] \right)^{\rm T} \right]$$

$$= \begin{bmatrix} \mathrm{E}[(X_1 - \mu_1)(X_1 - \mu_1)] & \mathrm{E}[(X_1 - \mu_1)(X_2 - \mu_2)] & \cdots & \mathrm{E}[(X_1 - \mu_1)(X_n - \mu_n)] \\ \\ \mathrm{E}[(X_2 - \mu_2)(X_1 - \mu_1)] & \mathrm{E}[(X_2 - \mu_2)(X_2 - \mu_2)] & \cdots & \mathrm{E}[(X_2 - \mu_2)(X_n - \mu_n)] \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \mathrm{E}[(X_n - \mu_n)(X_1 - \mu_1)] & \mathrm{E}[(X_n - \mu_n)(X_2 - \mu_2)] & \cdots & \mathrm{E}[(X_n - \mu_n)(X_n - \mu_n)] \end{bmatrix}$$

矩阵中的第 $(i,j)$个元素是 $X_i$与 $X_j$的协方差

进一步

矩阵的奇异值分解可以将数据还原为普通的形式, 这在LDA等许多算法中都有应用
进一步可以阅读以下文章:
https://janakiev.com/blog/covariance-matrix/