Jointly Gaussian Random Variables

Implementation of the Jointly Gaussian Random Variable

Step 1: Import all the required libraries such as numpy, matplotlib, etc. 

import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from mpl_toolkits.mplot3d import Axes3D

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Step 2: Create a 2-dimensional distribution over variables X and Y. We take our N as 1000 to visualize over it. 

N = 1000
X = np.linspace(-3, 3, N)
Y = np.linspace(-3, 4, N)
X, Y = np.meshgrid(X, Y)

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Step 3: We create our mean vector and covariance matrix. We use mu for mean vectors and sigma to represent covariance matrix. 

mu = np.array([0, 1])
Sigma = np.array([[1 , 0.6], [0.6,  2]])

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Step 4: We pack X and Y into a single three-dimensional array to visualize our created data.

pos = np.empty(X.shape + (2,))
pos[:, :, 0] = X
pos[:, :, 1] = Y

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Step 5: Return the multivariate Gaussian distribution on array pos.

pos is an array constructed by packing the meshed arrays of variables

x_1, x_2, x_3, ..., x_k into its _last_ dimension.

def multivariate_gaussian(pos, mu, Sigma):

    n = mu.shape[0]
    Sigma_det = np.linalg.det(Sigma)
    Sigma_inv = np.linalg.inv(Sigma)
    N = np.sqrt((2*np.pi)**n * Sigma_det)
    # This einsum call calculates (x-mu)T.Sigma-1.(x-mu) in a vectorized

    # way across all the input variables.
    fac = np.einsum('...k,kl,...l->...', pos-mu, Sigma_inv, pos-mu)

    return np.exp(-fac / 2) / N

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Step 6: We assign the variable Z to mu, sigma, and pos values. 

Z = multivariate_gaussian(pos, mu, Sigma)

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Step 7: We create a surface plot and project a filled contour plot under it. 

ig = plt.figure()
ax = Axes3D(fig)
ax.plot_surface(X, Y, Z, rstride=3, cstride=3, linewidth=1, antialiased=False,
                cmap=cm.viridis)

cset = ax.contourf(X, Y, Z, zdir='z', offset=-0.15, cmap=cm.viridis)

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Step 8: In the last step, we need to adjust the limits, ticks, and view angle. 

ax.set_zlim(-0.15,0.2)
ax.set_zticks(np.linspace(0,0.2,5))
ax.view_init(27, -21)
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('P D F', labelpad=1)
plt.title('Bivariate Normal Distribution (1000 samples)')
plt.savefig('1000bivariatePDF.png', dpi = 300)
plt.show()

iks = np.array([1,2])
fx = (1/(Sigma * np.sqrt(2 * np.pi)) *
            np.exp( - (iks - mu)**2 / (2 * Sigma**2) ))
print(iks, fx)

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Output:

Thus we have successfully visualized the implementation of the jointly Gaussian random variable in python for a better understanding of the concepts. 

Frequently Asked Questions

1. How do we acknowledge that the given two variables are jointly Gaussian random variables?
   To confirm the fact that the two given variables are jointly Gaussian random variables, If aX+bY has a normal distribution for every a,b∈R, then the two random variables X and Y are said to be jointly gaussian random variables.
    
2. How do we simulate the jointly gaussian random variable? 
   The conditional distribution of a random variable X2 given X1. As a result, a bivariate normal distribution may be simulated by selecting one random variable from the marginal normal distribution and another from the conditional normal distribution. 
    
3. What is a jointly Gaussian random variable in python? 
   One or both variables might be discontinuous in the bivariate distribution histogram. Another technique to compare conditional univariate distributions is to plot one discrete and one continuous variable.  

Conclusion

In machine learning and data science, Gaussians are a required distribution. Knowing how to use them in multivariate circumstances in order to become a better data scientist or machine learning professional.  This article has thoroughly discussed the joint Gaussian random variable and all the required prerequisites such as joint Gaussian pdf, covariance matrix, and Gaussian random variable. We have also visualized how the jointly gaussian random variable works in practical-based scenarios.  
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