Exercise 2D Gaussian Mixture
Table of Contents
Introduction
In this notebook you will implement an EM algorithm to find meanvectors and covariance matricies for bivariate gaussians to solve a 2D classfication problem.
Requirements
Knowledge
To complete this exercise notebook, you should possess knowledge about the following topics:
- Mixture Models
- EM Algorihtm
The following material can help you to acquire this knowledge:
- Bishop [BIS06], Chapter 9.2
Python Modules
import numpy as np
from matplotlib import pyplot as plt
from colorutils import Color
%matplotlib inline
np.random.seed(42)
Exercise
Consider the following "Mickey Mouse" like data set.
#Mickey Mouse example
variance1 = [[0.05,0.0 ],[0.0,0.07]]
loc1 = [-1.0,2.0]
x1,y1 = np.random.multivariate_normal(loc1,variance1,size=300).T
variance2 = [[0.05,0.0 ],[0.0,0.07]]
loc2 = [1.0,2.0]
x2,y2 = np.random.multivariate_normal(loc2,variance2,size=300).T
variance3 = [[0.2,0.0 ],[0.0,0.4]]
loc3 = [0.0,0.0]
x3,y3 = np.random.multivariate_normal(loc3,variance3,size=800).T
xData = np.concatenate((x1,x2,x3))
yData = np.concatenate((y1,y2,y3))
xyData = np.ndarray((xData.shape[0], 2))
xyData[:,0] = xData
xyData[:,1] = yData
plt.plot(xyData[:,0],xyData[:,1],'ro',label='Data')
plt.legend()
plt.grid()
plt.show()Task
Read Chapter 9.2 in Bishop's book and implement a Gaussian mixture for multivariate Gaussians using the EM algorithm. As you can see above you may assume three multivariate Gaussians, so you will need to set three initial means, three covariance matrices and an initial mixing as your start values.
def multi_gaussian(data,mean,var):
expo = -1.0/2.0 * np.dot((data-mean).T,np.dot(np.linalg.inv(var),(data-mean)))
return 1.0/np.sqrt((2*np.pi)**np.size(data)*np.linalg.det(var))*np.exp(expo)
def EM_Mixture_Gaussian(initial_mean,initial_cov,initial_mix,Data):
K = np.size(initial_mean,axis=0) #Number of assumed Gaussians
N = np.size(Data,axis=0) #Number of data points
mean = initial_mean
cov = initial_cov
mix = initial_mix
gamma_z_nk = np.zeros((N,K))
#E-Step
# Implement your E-Step here
#M-Step
# Implement your M-Step here
return mean,cov,mix, gamma_z_nk #updated versions
Solution
The following two functions will help to visualize your EM-Steps. "asssigne_class_euclidean" plots a "hard-cut" classification using the minimal euclidean distance to the means and "asssigne_class_color_responsibility" plots a color mixing according to the responsibilties each data point has.
def asssigne_class_euclidean(Data,meanlist):
K = np.size(meanlist,axis=0) #Number of assumed Gaussians
N = np.size(Data,axis=0) #Number of data points
mean = np.asarray(meanlist)
distance = np.zeros(K)
plt.subplot(121)
for n in np.arange(N):
for k in np.arange(K):
distance[k] = np.linalg.norm(Data[n]-mean[k])
maxindex = np.argmin(distance)
col = 'C'+str(maxindex)
plt.plot(Data[n,0],Data[n,1],'o',color=col)
plt.plot(mean[0,0],mean[0,1],'o',markersize=10,color='black',label='mean class 1:('+str(np.round(mean[0,0],3))+','+str(np.round(mean[0,1],3))+')')
plt.plot(mean[1,0],mean[1,1],'s',markersize=10,color='black',label='mean class 2:('+str(np.round(mean[1,0],3))+','+str(np.round(mean[1,1],3))+')')
plt.plot(mean[2,0],mean[2,1],'*',markersize=10,color='black',label='mean class 3:('+str(np.round(mean[2,0],3))+','+str(np.round(mean[2,1],3))+')')
plt.legend(loc='upper center',bbox_to_anchor=(0.5, 1.25))
plt.grid()
return True
def asssigne_class_color_responsibility(Data,gamma_z_nk,meanlist):
N = np.size(Data,axis=0) #Number of data points
mean = np.asarray(meanlist)
plt.subplot(122)
for n in np.arange(N):
colob = Color((gamma_z_nk[n][0],gamma_z_nk[n][1],gamma_z_nk[n][2]))
col = colob.rgb
plt.plot(Data[n,0],Data[n,1],'o',color=col)
plt.plot(mean[0,0],mean[0,1],'o',markersize=10,color='black')
plt.plot(mean[1,0],mean[1,1],'s',markersize=10,color='black')
plt.plot(mean[2,0],mean[2,1],'*',markersize=10,color='black')
plt.grid()
return True
Choose your initial values and number of EM-Steps you wanna do.
initial_mean = [[0.0,0.0],[0.0,0.0],[0.0,0.0]]
initial_mix = [0.0,0.0,0.0]
initial_covariance = [[0.0,0.0],[0.0,0.0]]
L = 0 #number of steps
mean = initial_mean
mix = initial_mix
cov = [initial_covariance,initial_covariance,initial_covariance]Run the following cell to visualize your classification.
print('------------------- Classes with EM -----------------------')
for l in np.arange(L):
mean,cov,mix,gamma_z_nk = EM_Mixture_Gaussian(mean,cov,mix,xyData)
if l%4 == 0:
print('-------- Step='+str(l)+'---------')
plt.figure(figsize=(15,5))
asssigne_class_euclidean(xyData,mean)
asssigne_class_color_responsibility(xyData,gamma_z_nk,mean)
plt.show()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-2D-gaussian-mixture-em
by Oliver Fischer
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 2019 Oliver Fischer
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.
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