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2 years ago

MA2102

Probability and Statistics

Lecture-23

Bivariate Normal Distributions:

A continuous bivariate random variable (X,Y)(X,Y) is said to have Bivariate Normal distribution with parameters μ1,μ2,σ12,σ22\mu_1,\mu_2,\sigma_1^2,\sigma_2^2, and ρ\rho

if it's JPDFJPDF is given by, fX,Y(x,y)=12πσ1σ21−ρ2e−12(1−ρ2)[(x−μ1σ1)2+(y−μ2σ2)2−2ρ(x−μ1σ1)(y−μ2σ2)]f_{X,Y}(x,y)=\frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}e^{{-\frac{1}{2(1-\rho^2)}}\left[\left(\frac{x-\mu_1}{\sigma_1}\right)^2+\left(\frac{y-\mu_2}{\sigma_2}\right)^2-2\rho\left(\frac{x-\mu_1}{\sigma_1}\right)\left(\frac{y-\mu_2}{\sigma_2}\right)\right]}


x,y,μ1,μ2∈Rx,y,\mu_1,\mu_2\in\mathbb{R}, σ1,σ2>0\sigma_1,\sigma_2>0,−1<ρ<1-1<\rho<1

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import multivariate_normal
from mpl_toolkits.mplot3d import Axes3D
import ipyvolume as ipv
from matplotlib import cm
from ipywidgets import interact ,interactive, fixed, interact_manual
import ipywidgets as widgets
%matplotlib inline

def BVN1(mu1,mu2,s1,s2,r): # Interactive 3D surface plot using ipyvolume
    
    #Create grid and Bivariate normal
    
    x = np.linspace(-10,10,100)
    y = np.linspace(-10,10,100)
    X, Y = np.meshgrid(x,y)
    # Z=f(X,Y)
    Z=(1/(2*np.pi*s1*s2*np.sqrt(1-r**2)))*(np.exp((-1/(2*(1-r**2)))*( ((X-mu1)/s1)**2 + ((Y-mu2)/s2)**2  -2*r* ((X-mu1)/s1)*((Y-mu2)/s2))))
    colormap = cm.coolwarm
    znorm = Z - Z.min()
    znorm /= znorm.ptp()
    znorm.min(), znorm.max()
    color = colormap(znorm)
    #Make a 3D plot
    ipv.figure()
    ipv.pylab.plot_surface(X,Y,Z,color=color[...,:3])
    #ipv.plot_wireframe(X, Z, Y, color="red")
    ipv.pylab.zlim(0,Z.max())
    #ipv.pylab.style.box_off()
    ipv.show()



def BVN2(mu1,mu2,s1,s2,r):  #  3D surface plot using matplotlib
    #Create grid and multivariate normal
    x = np.linspace(-10,10,500)
    y = np.linspace(-10,10,500)
    X, Y = np.meshgrid(x,y)
    pos = np.empty(X.shape + (2,))
    pos[:, :, 0] = X; pos[:, :, 1] = Y
    brv = multivariate_normal([mu1, mu2], [[s1**2, r*s1*s2], [r*s1*s2, s2**2]])

    #Make a 3D plot
    fig= plt.figure(figsize=(10,10))
   # ax = fig.add_subplot(1,1,1,projection='3d')
    ax = fig.gca(projection='3d')
    ax.plot_surface(X, Y, brv.pdf(pos),cmap='viridis',linewidth=0)
    ax.set_xlabel('X axis')
    ax.set_ylabel('Y axis')
    ax.set_zlabel('Z axis')
    ax.set_xlim(-10,10)
    ax.set_ylim(-10,10)
    plt.show()

mu_1=widgets.FloatSlider(min=-3, max=3, step=0.5, value=0,description=r'$\mu_1$ :')
mu_2=widgets.FloatSlider(min=-3, max=3, step=0.5, value=0,description=r'$\mu_2$ :')
s_1=widgets.FloatSlider(min=0, max=3, step=0.5, value=1,description=r'$\sigma_1$ :')
s_2=widgets.FloatSlider(min=0, max=3, step=0.5, value=1,description=r'$\sigma_2$ :')
r=widgets.FloatSlider(min=-1, max=1, step=0.1, value=0,description=r'$\rho$ :')