numpy.random usage documentation

正态分布

二维正态分布

官方使用文档
https://numpy.org/doc/stable/reference/random/generated/numpy.random.RandomState.multivariate_normal.html

random.RandomState.multivariate_normal(mean, cov, size=None, check_valid='warn', tol=1e-8)

mean = (1, 2)
cov = [[1, 0], [0, 1]]
x = np.random.multivariate_normal(mean, cov, (3, 3))

The first parameter is the mean vector,The second parameter is the covariance matrix.The relationship between the covariance matrix parameters and the correlation coefficient is as follows.
$${\rho }_{xy}=\frac{Cov(X,Y)}{{\sigma }_X{\sigma }_Y}$$
The following shows how different correlation coefficients affect the resulting distribution,

import numpy as np
from numpy.random import multivariate_normal
import matplotlib.pyplot as plt

def generate_data(mu1, mu2, stv1, stv2, rho, num):
	cov = np.array([[stv1**2, rho*stv1*stv2],[rho*stv1*stv2, stv2**2]])
	mu = np.array([mu1, mu2])
	X = multivariate_normal(mu, cov, size=num)
	plt.scatter(X[:,0], X[:,1], alpha=0.5, label=r'$\rho$={:.2f}'.format(rho))

for rho in [0, 0.23, 0.45, 0.68, 0.90]:
	generate_data(0,0,1,1,rho,200)
plt.legend()
plt.show()

生成结果如下图：相关系数越接近1,The more obvious the linear relationship between the variables,That is, the ellipsoid is flatter.

另外,According to the properties of the multivariate Gaussian,If a multivariate Gaussian distribution is formed,Any subset of variables constitutes a Gaussian distribution：If independently projected toX轴和Y轴,It will be found that both obey a Gaussian distribution.

import numpy as np
from numpy.random import multivariate_normal
import matplotlib.pyplot as plt

def show_1d_gauss(mu1, mu2, stv1, stv2, rho, num, ax):
	cov = np.array([[stv1**2, rho*stv1*stv2],[rho*stv1*stv2, stv2**2]])
	mu = np.array([mu1, mu2])
	X = multivariate_normal(mu, cov, size=num)
	X1 = X[:, 0]
	X2 = X[:, 1]
	ax.scatter(X1, X2, alpha=0.5)
	ax.scatter(X1, np.zeros_like(X1), alpha=0.2)
	ax.scatter(np.zeros_like(X2), X2, alpha=0.2)
	ax.set_title(r'$\rho$={:.2f}'.format(rho))

fig, axes = plt.subplots(2, 3)
rholist = [0, 0.23, 0.45, 0.68, 0.90, 1]

for i, rho in enumerate(rholist):
	show_1d_gauss(0,0,1,1,rho,200, axes.flatten()[i])

plt.show()

结果如图所示