Modify the EM implementation (see the notebook file EM_algorithm.ipynb shared with you on the google drive: https://drive.google.com/drive/folders/1_rehaUOvRAp_uKK3_A2Hqak1BI8qIfQ L?usp=share_link) to allow for three classes (each following a Gaussian distribution with different mean and standard deviation) in the data. You should modify the code to generate three classes of the data and use the EM algorithm to learn the parameters for the three Gaussian distribution. You should submit your modified implementation in a notebook file through canvas, which should include the implementation as well as a short description of your simulation result.

HERE IS CODE

-----------------------------------------------------------

import numpy as np

import matplotlib.pyplot as plt

# set random seed so that everytime we get the same result

np.random.seed(1)

# prepare simulation data

N1 = 1000

N2 = 1000

N = N1 + N2

real_mu1 = 0.2

real_std1 = 1

real_mu2 = 0.8

real_std2 = 1

y = np.concatenate((np.random.normal(real_mu1, real_std1, N1), np.random.normal(real_mu2, real_std2, N2)))

# EM algorithm

# Initialization

nits = 1000

count = 0 # should be initialized as 0

p1 = 0.5

p2 = 0.5

mu1 = np.random.random()

mu2 = np.random.random()

s1 = np.std(y)

s2 = s1

ll = np.zeros(nits)

gamma1 = np.zeros(N)

gamma2 = np.zeros(N)

while count < nits:

count = count + 1

# E-step

for i in range(N):

num1 = p1 * np.exp(-(y[i]-mu1)**2/(2*s1)) / np.sqrt(s1)

num2 = p2 * np.exp(-(y[i]-mu2)**2/(2*s2)) / np.sqrt(s2)

gamma1[i] = num1 / (num1 + num2)

gamma2[i] = num2 / (num1 + num2)

# M-step

mu1 = np.sum(gamma1*y) / np.sum(gamma1)

mu2 = np.sum(gamma2*y) / np.sum(gamma2)

s1 = np.sum(gamma1*(y-mu1)**2) / np.sum(gamma1)

s2 = np.sum(gamma2*(y-mu2)**2) / np.sum(gamma2)

p1 = np.sum(gamma1) / N

p2 = np.sum(gamma2) / N

ll[count - 1] = np.sum(np.log(p1*np.exp(-(y-mu1)**2/(2*s1)) / np.sqrt(s1) + p2*np.exp(-(y-mu2)**2/(2*s2)) / np.sqrt(s2)))

print('Estimated (mu1, sigma1): (%s, %s), (mu2, sigma2): (%s, %s)' % (mu1, s1, mu2, s2))

plt.plot(range(nits), ll)