Covariance

November 10, 2019

In probability theory and statistics, covariance is a measure of the joint variability of two random variables.

协方差（Covariance）在概率论和统计学中用于衡量两个变量的总体误差。而方差是协方差的一种特殊情况，即当两个变量是相同的情况。维基百科定义

Definition

Let $X$ and $Y$ be random variables with means $μ_X$ and $μ_Y$ and standard deviations $σ_X$ and $σ_Y$ . Define the covariance between $X$ and $Y$ by

$\begin{align*} Cov(X,Y)=\mathbb{E}((X-\mu_X)(Y-\mu_Y)) \end{align*}$

and the correlation by

$\rho=\rho_{X,Y}=\rho(X,Y)=\frac{Cov(X,Y)}{\rho_{X}\rho_{Y}}$

Code

def cov(X):
    """
    just like np.cov(X.T)
    """
    sub_mean  = X - np.mean(X, axis=0, keepdims=True)
    cov = np.dot(sub_mean.T, sub_mean) / (X.shape[0] - 1)
    return cov

a = [-2.1, 3, 1]
b = [-1, 1, 2]
c = [4.3, 0.12, 3]
d = [5, 0, 4]
X = np.stack([a, b, c, d]) # 4 samples, 3 variances

print cov(X)

Reference

All of Statistics