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June 15, 2011 17:10
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Gaussian Mixture Model
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#!/usr/bin/python | |
""" | |
Author: Jeremy M. Stober | |
Program: GMM.PY | |
Date: Thursday, September 4 2008 | |
Description: Fit a Gaussian Mixture Model with EM. | |
""" | |
import os, sys, getopt, pdb, string | |
from numpy import * | |
import pylab | |
import matplotlib | |
faithful = [array([float(elem) for elem in line.split()]) for line in open('faithful.txt').readlines()] | |
xvals = array([elem[0] for elem in faithful]) | |
yvals = array([elem[1] for elem in faithful]) | |
# Standard normalization. | |
xmean = mean(xvals) | |
xstd = std(xvals) | |
ymean = mean(yvals) | |
ystd = std(yvals) | |
xnorm = (xvals - xmean) / xstd | |
ynorm = (yvals - ymean) / ystd | |
nfaithful= array(zip(xnorm, ynorm)) | |
# Does the data look reasonable? | |
# pylab.plot(xnorm,ynorm, '+') | |
# pylab.show() | |
# initialize the means and covariances | |
mus = [array([-1.5,1.0]), array([1.0,-1.0])] | |
sigmas = [eye(2),eye(2)] | |
mixings = [0.5, 0.5] | |
def normal(x,mu,sigma): | |
""" Return the normal density at a point. """ | |
D = len(sigma) | |
det = linalg.det(sigma) | |
div = (2.0 * pi)**(D / 2.0) * (det)**(0.5) | |
return exp(-0.5 * dot(dot(x - mu, linalg.inv(sigma)), x - mu)) / div | |
k = 2 # 2 dimensional data | |
n = len(nfaithful) # number of data points | |
for l in range(100): | |
print l | |
# E step | |
print mus, mixings | |
responses = zeros((k,n)) | |
for j in range(n): | |
for i in range(k): | |
responses[i,j] = mixings[i] * normal(nfaithful[j],mus[i],sigmas[i]) | |
responses = responses / sum(responses,axis=0) # normalize the weights | |
# M step | |
N = sum(responses,axis=1) | |
for i in range(k): | |
mus[i] = dot(responses[i,:],nfaithful) / N[i] | |
sigmas[i] = zeros((2,2)) | |
for j in range(n): | |
sigmas[i] += responses[i,j] * outer(nfaithful[j,:] - mus[i],nfaithful[j,:] - mus[i]) | |
sigmas[i] = sigmas[i] / N[i] | |
mixings[i] = N[i] / sum(N) | |
def shownormal(mus,sigmas): | |
# Plot the normalized faithful data points. | |
fig = pylab.figure(num = 1, figsize=(4,4)) | |
axes = fig.add_subplot(111) | |
axes.plot(xnorm,ynorm, '+') | |
# Plot the ellipses representing the principle components of the normals. | |
k = len(mus) | |
for i in range(k): | |
color = None | |
if i == 0: | |
color = 'red' | |
else: | |
color = 'blue' | |
[u,s,v] = linalg.svd(sigmas[i]) | |
angle = arccos(dot(u[1],array([1,0]))) * 180.0 / pi | |
ellipse = matplotlib.patches.Ellipse(mus[i], sqrt(s[1]), sqrt(s[0]), angle=angle, fill=False, ec=color) | |
axes.add_patch(ellipse) | |
pylab.draw() | |
pylab.show() | |
shownormal(mus,sigmas) |
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