I am graphing a contour plot of MVN density using Python. The code is copied below but my question can be answered without reading the code.
import numpy as np
from scipy.stats import multivariate_normal as mvn
import matplotlib.pyplot as plt
D = 2
x = np.random.rand(D)
mu = np.random.rand(D)
A = np.random.rand(D,D)
# random symmetric matrix
cov = A.T.dot(A)
# Generate grid points
x, y = np.meshgrid(np.linspace(-1,2,100),np.linspace(-1,2,100))
xy = np.column_stack([x.flat, y.flat])
# density values at the grid points
Z = mvn.pdf(xy, mu, cov).reshape(x.shape)
# arbitrary contour levels
contour_level = [0.1, 0.2, 0.3]
fig = plt.contour(X, Y, Z, levels = contour_level)
I am trying to pick a meaningful levels of the contour plot (i.e., the curve with the same density value). In the univariate case, $\pm \sigma$ and 2$\sigma$ represent 68% and 95% of the data, respectively.
Is there an analogous concept for the bivariate normal distribution such that with in the first contour line, some percentage of data is contained and so on?
