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a1_EM-Algorithm.py
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import math
import random
import numpy as np
import matplotlib.pyplot as plt
class EM():
"""
Implementation of the EM algorithm
"""
def __init__(self, mn, mu1_actual, mu2_actual, sig1_actual, sig2_actual, alpha_actual, beta_actual):
"""
Pass in actual parameter values that the parameter estimates should converge to
"""
self.mn = mn
self.mu1_actual = mu1_actual
self.mu2_actual = mu2_actual
self.sig1_actual = sig1_actual
self.sig2_actual = sig2_actual
self.alpha_actual = alpha_actual
self.beta_actual = beta_actual
self.LL = np.empty((0,1))
def initialize(self, mu1_init, mu2_init, sig1_init, sig2_init, alpha_init, beta_init):
"""
Initialize parameters to initial parameter estimates
"""
self.mu1_est = mu1_init
self.mu2_est = mu2_init
self.sig1_est = sig1_init
self.sig2_est = sig2_init
self.alpha_est = alpha_init
self.beta_est = beta_init
def PD_d(self, d, ab, k) -> float:
"""
Args:
d: float, x value?
ab: string; "alpha" or "beta"
k: int; 1 or 2 (which mixture)
Returns:
float; value of pD(d) for the given data point
"""
if ab == "alpha":
if k == 1:
return self.alpha_est*(math.exp((-(d-self.mu1_est)**2)/(2*(self.sig1_est)**2))/(math.sqrt(2*(math.pi)*(self.sig1_est)**2)))
if k == 2:
return (1-self.alpha_est)*(math.exp((-(d-self.mu2_est)**2)/(2*(self.sig2_est)**2))/(math.sqrt(2*(math.pi)*(self.sig2_est)**2)))
elif ab == "beta":
if k == 1:
return self.beta_est*(math.exp((-(d-self.mu1_est)**2)/(2*self.sig1_est**2))/(math.sqrt(2*(math.pi)*(self.sig1_est**2))))
if k == 2:
return (1-self.beta_est)*(math.exp((-(d-self.mu2_est)**2)/(2*self.sig2_est**2))/(math.sqrt(2*(math.pi)*(self.sig2_est**2))))
def generateNormal(self, num, mode):
"""
Args:
num: int; 1 or 2
mode: string; "est" or "actual"
Returns:
np.array; normal distribution based on given inputs
"""
if num == 1:
if mode == "est":
return np.random.normal(self.mu1_est, self.sig1_est, self.mn)
elif mode == "actual":
return np.random.normal(self.mu1_actual, self.sig1_actual, int(self.mn/2))
elif num == 2:
if mode == "est":
return np.random.normal(self.mu2_est, self.sig2_est, self.mn)
elif mode == "actual":
return np.random.normal(self.mu2_actual, self.sig2_actual, int(self.mn/2))
def generateData(self, xy):
"""
Generates initial data based on actual values
Args:
xy: string; "x" or "y"
Returns:
np.array: x or y data based on input
"""
if xy == 'x':
x_dist_1 = self.alpha_actual*self.generateNormal(1, "actual")
x_dist_2 = (1-self.alpha_actual)*self.generateNormal(2, "actual")
self.x_data = np.concatenate((x_dist_1, x_dist_2)).flatten()
elif xy == 'y':
y_dist_1 = self.beta_actual*self.generateNormal(1, "actual")
y_dist_2 = (1-self.beta_actual)*self.generateNormal(2, "actual")
self.y_data = np.concatenate((y_dist_1, y_dist_2)).flatten()
def E_step(self):
# Initialize arrays to hold pseudo posteriors
self.og_gammas = np.empty((len(self.x_data),2))
self.gammas = np.empty((len(self.x_data),2))
for i in range(len(self.x_data)):
for mode in [1, 2]:
self.og_gammas[i, mode-1] = em.PD_d(self.x_data[i], "alpha", mode)
self.gammas[:,0] = self.og_gammas[:,0]/np.sum(self.og_gammas, axis=1)
self.gammas[:,1] = self.og_gammas[:,1]/np.sum(self.og_gammas, axis=1)
def M_step(self):
N_1, N_2 = np.sum(self.gammas, axis=0)
self.mu1_est = (1/N_1)*np.sum(np.multiply(self.gammas[:,0], self.x_data))
self.mu2_est = (1/N_2)*np.sum(np.multiply(self.gammas[:,1], self.x_data))
self.sig1_est = math.sqrt((1/N_1)*np.sum((self.gammas[:,0])*np.square((self.x_data - self.mu1_est))))
self.sig2_est = math.sqrt((1/N_2)*np.sum((self.gammas[:,1])*np.square((self.x_data - self.mu2_est))))
self.alpha_est = N_1/self.mn
self.beta_est = N_2/self.mn
def computeLogLikelihood(self):
self.LL = np.append(self.LL, np.sum(np.log(np.sum(self.og_gammas, axis=1))))
num_tests = 1000
max_steps = 100
results = np.empty((0,6))
results_main = np.empty((0,6))
for i in range(num_tests):
em = EM(1000, 10, 12, 2, 0.5, 0.5, 0.7)
em.initialize(1, 1, 1, 1, 0.1, 0.6)
em.generateData('x')
em.generateData('y')
for step in range(max_steps):
em.E_step()
em.M_step()
results = np.append(results, np.array([[em.mu1_est, em.mu2_est, em.sig1_est, em.sig2_est, em.alpha_est, em.beta_est]]), axis=0)
em.computeLogLikelihood()
results_main = np.append(results_main, np.array([[em.mu1_est, em.mu2_est, em.sig1_est, em.sig2_est, em.alpha_est, em.beta_est]]), axis=0)
print(F"[INFO]: {i+1} / {num_tests} complete.")
# Post process data
averages = np.mean(results_main, axis=0)
stddevs = np.std(results_main, axis=0)
print(averages)
print(stddevs)
# Plots
x_range = range(max_steps)
est_labels = ["mu1 est", "mu2 est", "sigma1 est", "sigma2 est", "alpha est"]
actual = [em.mu1_actual, em.mu2_actual, em.sig1_actual, em.sig2_actual, em.alpha_actual]
actual_labels = ["mu1 actual", "mu2 actual", "sigma1 actual", "sigma2 actual", "alpha actual"]
colors = ["r", "b", "g", "c", "m"]
# for i in range(5):
# plt.plot(x_range, [actual[i]]*len(x_range), colors[i], label = actual_labels[i])
# plt.plot(x_range, results[:,i], colors[i], label = est_labels[i])
plt.plot(x_range, em.LL, label = "Log Likelihood")
plt.legend()
plt.title("Luke Davidson - Q7c")
plt.ylabel("Param Val")
plt.xlabel("Iteration")
plt.show()
# print(em.mu1_est, em.mu2_est, em.sig1_est, em.sig2_est, em.alpha_est)