ITK  5.0.0 Insight Segmentation and Registration Toolkit
Examples/Statistics/ExpectationMaximizationMixtureModelEstimator.cxx
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// Software Guide : BeginLatex
//
// \index{Statistics!Mixture model estimation}
// \index{Statistics!Expectation maximization}
//
// \index{itk::Statistics::Gaussian\-Mixture\-Model\-Component}
// \index{itk::Statistics::Expectation\-Maximization\-Mixture\-Model\-Estimator}
//
// In this example, we present an implementation of the expectation
// maximization (EM) process to generates parameter estimates for a two
// Gaussian component mixture model.
//
// The Bayesian plug-in classifier example (see Section
// \ref{sec:BayesianPluginClassifier}) used two Gaussian probability
// density functions (PDF) to model two Gaussian distribution classes (two
// models for two class). However, in some cases, we want to model a
// distribution as a mixture of several different
// distributions. Therefore, the probability density function ($p(x)$)
// of a mixture model can be stated as follows :
//
//
// p(x) = \sum^{c}_{i=0}\alpha_{i}f_{i}(x)
//
// where $i$ is the index of the component,
// $c$ is the number of components,
// $\alpha_{i}$ is the proportion of the component,
// and $f_{i}$ is the probability density function of the component.
//
// Now the task is to find the parameters(the component PDF's
// parameters and the proportion values) to maximize the likelihood of
// the parameters. If we know which component a measurement vector
// belongs to, the solutions to this problem is easy to solve.
// However, we don't know the membership of each measurement
// vector. Therefore, we use the expectation of membership instead of
// the exact membership. The EM process splits into two steps:
// \begin{enumerate}
// \item{ E step: calculate the expected membership values for each
// measurement vector to each classes.}
// \item{ M step: find the next parameter sets that maximize the
// likelihood with the expected membership values and the current set of
// parameters.}
// \end{enumerate}
//
// The E step is basically a step that calculates the \emph{a posteriori}
// probability for each measurement vector.
//
// The M step is dependent on the type of each PDF. Most of
// distributions belonging to exponential family such as Poisson,
// Binomial, Exponential, and Normal distributions have analytical
// solutions for updating the parameter set. The
// \subdoxygen{Statistics}{ExpectationMaximizationMixtureModelEstimator}
// class assumes that such type of components.
//
// In the following example we use the \subdoxygen{Statistics}{ListSample} as
// the sample (test and training). The \subdoxygen{Vector} is our measurement
// vector class. To store measurement vectors into two separate sample
// container, we use the \subdoxygen{Statistics}{Subsample} objects.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
#include "itkVector.h"
#include "itkListSample.h"
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// The following two files provides us the parameter estimation algorithms.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// We will fill the sample with random variables from two normal
// distribution using the \subdoxygen{Statistics}{NormalVariateGenerator}.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
// Software Guide : EndCodeSnippet
int main()
{
// Software Guide : BeginLatex
//
// Since the NormalVariateGenerator class only supports 1-D, we
// define our measurement vector type as a one component vector. We
// then, create a ListSample object for data inputs.
//
// We also create two Subsample objects that will store the
// measurement vectors in the \code{sample} into two separate sample
// containers. Each Subsample object stores only the
// measurement vectors belonging to a single class. This
// \textit{class sample} will be used by the parameter estimation
// algorithms.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
unsigned int numberOfClasses = 2;
using MeasurementVectorType = itk::Vector< double, 1 >;
SampleType::Pointer sample = SampleType::New();
sample->SetMeasurementVectorSize( 1 ); // length of measurement vectors
// in the sample.
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// The following code snippet creates a NormalVariateGenerator object.
// Since the random variable generator returns values according to the
// standard normal distribution (the mean is zero, and the standard
// deviation is one) before pushing random values into the \code{sample},
// we change the mean and standard deviation. We want two normal (Gaussian)
// distribution data. We have two for loops. Each for loop uses different
// mean and standard deviation. Before we fill the \code{sample} with the
// second distribution data, we call \code{Initialize()} method to recreate
// the pool of random variables in the \code{normalGenerator}. In the
// second for loop, we fill the two class samples with measurement vectors
// using the \code{AddInstance()} method.
//
// To see the probability density plots from the two distribution,
// refer to Figure~\ref{fig:TwoNormalDensityFunctionPlot}.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
using NormalGeneratorType = itk::Statistics::NormalVariateGenerator;
NormalGeneratorType::Pointer normalGenerator = NormalGeneratorType::New();
normalGenerator->Initialize( 101 );
MeasurementVectorType mv;
double mean = 100;
double standardDeviation = 30;
for ( unsigned int i = 0; i < 100; ++i )
{
mv[0] = ( normalGenerator->GetVariate() * standardDeviation ) + mean;
sample->PushBack( mv );
}
normalGenerator->Initialize( 3024 );
mean = 200;
standardDeviation = 30;
for ( unsigned int i = 0; i < 100; ++i )
{
mv[0] = ( normalGenerator->GetVariate() * standardDeviation ) + mean;
sample->PushBack( mv );
}
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// In the following code snippet notice that the template argument
// for the MeanCalculator and CovarianceCalculator is
// \code{ClassSampleType} (i.e., type of Subsample) instead of
// \code{SampleType} (i.e., type of ListSample). This is because the
// parameter estimation algorithms are applied to the class sample.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
using ParametersType = itk::Array< double >;
ParametersType params( 2 );
std::vector< ParametersType > initialParameters( numberOfClasses );
params[0] = 110.0;
params[1] = 800.0;
initialParameters[0] = params;
params[0] = 210.0;
params[1] = 850.0;
initialParameters[1] = params;
using ComponentType =
std::vector< ComponentType::Pointer > components;
for ( unsigned int i = 0; i < numberOfClasses; i++ )
{
components.push_back( ComponentType::New() );
(components[i])->SetSample( sample );
(components[i])->SetParameters( initialParameters[i] );
}
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// We run the estimator.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
using EstimatorType =
EstimatorType::Pointer estimator = EstimatorType::New();
estimator->SetSample( sample );
estimator->SetMaximumIteration( 200 );
itk::Array< double > initialProportions(numberOfClasses);
initialProportions[0] = 0.5;
initialProportions[1] = 0.5;
estimator->SetInitialProportions( initialProportions );
for (unsigned int i = 0; i < numberOfClasses; ++i)
{
(components[i]).GetPointer() );
}
estimator->Update();
// Software Guide : EndCodeSnippet
// Software Guide : BeginLatex
//
// We then print out the estimated parameters.
//
// Software Guide : EndLatex
// Software Guide : BeginCodeSnippet
for (unsigned int i = 0; i < numberOfClasses; ++i)
{
std::cout << "Cluster[" << i << "]" << std::endl;
std::cout << " Parameters:" << std::endl;
std::cout << " " << (components[i])->GetFullParameters()
<< std::endl;
std::cout << " Proportion: ";
std::cout << " " << estimator->GetProportions()[i] << std::endl;
}
// Software Guide : EndCodeSnippet
return EXIT_SUCCESS;
}