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Exercise - 1D Gaussian Mixture Model and Expectation Maximization

Table of Contents

Introduction

In this notebook you will implement an EM algorithm to estimate the mean and variance of a gaussian mixture of two one-dimensional gaussians.

Remark: In order to detect errors in your own code, execute the notebook cells containing assert or assert_almost_equal. These statements raise exceptions, as long as the calculated result is not yet correct.

Requirements

Knowledge

To complete this exercise notebook, you should possess knowledge about the following topics:

  • Maximum-Likelihood
  • Notes by Christian Herta [HER18a]
  • Expectation Maximization:
  • Notes by Christian Herta [HER18b]
  • Coursera Video [NOV18]

Python Modules

import numpy as np
from matplotlib import pyplot as plt
from scipy.stats import norm

%matplotlib inline
np.random.seed(42)

Exercises

Data Generation

Given is a mixture of two one dimensional Gaussian distributions with mean$ \mu_1 $,$ \mu_2 $ and variance$ \sigma^2_1 $ and$ \sigma^2_2 $:

mu_1 = -2.5
sigma_square_1 = 4.
sigma_1 = np.sqrt(sigma_square_1)

mu_2 = 3.5
sigma_square_2 = 4.
sigma_2 = np.sqrt(sigma_square_2)
prob_1 = 0.4
size=40
def get_data(mu_1 = mu_1, sigma_1 = sigma_1,
             mu_2 = mu_2, sigma_2 = sigma_2, 
             prob_1 = prob_1, size=size):
    
    # latent variale - assignments
    z = np.random.binomial(1, prob_1, size)
    
    x_1 = np.random.normal(loc=mu_1, scale=sigma_1, size=size)
    x_2 = np.random.normal(loc=mu_2, scale=sigma_2, size=size)
    x = z * x_1 + (1-z) * x_2
    
    return z, x
z, x = get_data()
x_1 = x[z==1]
x_2 = x[1-z==1]

plt.plot(x_1, np.zeros_like(x_1), 'rx')
plt.plot(x_2, np.zeros_like(x_2), 'bx')
plt.figure(figsize=(10,5))
plt.plot(x_1, np.zeros_like(x_1), 'rx')
plt.plot(x_2, np.zeros_like(x_2), 'bx')

rv_1 = norm(loc = mu_1, scale = sigma_1)
rv_2 = norm(loc = mu_2, scale = sigma_2)
x_ = np.arange(-6, 14, .1)
#plot the pdfs of the normal distributions 
plt.plot(x_, prob_1 * rv_1.pdf(x_), "r-")
plt.plot(x_, (1-prob_1) * rv_2.pdf(x_), "b-")
plt.plot(x_, prob_1 * rv_1.pdf(x_) + (1-prob_1) * rv_2.pdf(x_), "g-")
plt.figure(figsize=(10,5))
plt.plot(x, np.zeros_like(x), 'gx')
plt.plot(x_, prob_1 * rv_1.pdf(x_) + (1-prob_1) * rv_2.pdf(x_), "g-")

Exercise - Expectation Maximization

Recap

  • E-Step: Compute the responsability of the data points$ q(z_i=j) = p(z_i=j \mid x^{(i)}, \theta^{(t)}) $
  • M-Step: Maximize the Likelihood (MLE) w.r.t. the parameters, i.e. compute new$ \theta^{(t+1)} $

The parameters to estimate here are mean and variance of one of the participating Gaussians.

for E-Step: $ q(z_i=j) = p(z_i=j \mid x^{(i)}, \theta) = \frac{p(x^{(i)} \mid z_i=j, \theta) p (z_i=j \mid \theta)}{p(x^{(i)} \mid \theta)} = \frac{p(x^{(i)} \mid z_i=j, \theta) p(z_i=j \mid \theta)}{\sum_{j'} p(x^{(i)}, z_i=j' \mid \theta)} = \frac{p(x^{(i)} \mid z_i=j, \theta) p(z_i=j \mid \theta)}{\sum_{j'} p(x^{(i)} \mid z_i=j', \theta) p(z_i=j' \mid \theta)} $ $ q(z_i=j) $ reflects the responsibility the$ j $'th class (Gaussian) has for the$ i $'th data point.

for M-Step: $ {\mu_j}^{(t+1)} = \frac{ \sum_{i} x^{(i)} q(z_i=j)}{\sum_{i} q(z_i=j) } $

$ {\sigma_j}^{(t+1)} = \left( \frac{ \sum_{i} ({\mu_k}^{(t+1)}-x^{(i)})^2 q(z_i=j)}{\sum_{i} q(z_i=j) }\right)^{1/2} $

$ p_j^{(t+1)} = \frac{\sum_i q(z_i=j)}{m} $

-$ m $: number of training points

Task:

  • Implement the EM-Algorithm for a mixture of two 1D-Gaussians. Apply the algorithm on the train set. Plot the result.

  • Monitor the log-likelihood during the training. When your implementation is correct, it should increase after each step.

mu1=-0.5
mu2=0.5
sigma1=1
sigma2=1
prob1 = px1 = 0.5
px2 = 1-px1
px1, px2 = e_step_(x, mu1, sigma1, mu2, sigma2, prob1)
print (px1, px2)
mu1, sigma1, mu2, sigma2, prob1 = m_step(x, px1, px2)
print (mu1, sigma1, mu2, sigma2, prob1 )

Literature

Licenses

Notebook License (CC-BY-SA 4.0)

The following license applies to the complete notebook, including code cells. It does however not apply to any referenced external media (e.g., images).

Exercise - 1D Gaussian Mixture Model and Expectation Maximization
by Christian Herta
is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at https://gitlab.com/deep.TEACHING.

Code License (MIT)

The following license only applies to code cells of the notebook.

Copyright 2018 Christian Herta

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.