org.metaqtl.algo
Class EMAlgorithm

java.lang.Object
  org.metaqtl.algo.EMAlgorithm

public final class EMAlgorithm
extends java.lang.Object

This class defines methods to perform the EM-algorithm for univariate mixture of gaussians with known variances.

Field Summary

static boolean	DO_SEM If true the best starting point is used to do Supplemental EM (SEM) in order to compute the DM Matrix..
static int	EM_CONTINUE The EM convergence status : continue
static double	EM_ERR The EM convergence tolerance.
static int	EM_FAILURE The EM convergence status : failure
static int	EM_ITER_MAX The EM maximum number of iterations.
static double	EM_MIN_DISTANCE The minimal mahalanobis distance between components.
static int	EM_OK The EM convergence status : success
static int	EM_START The number of replicates for a run of the EM-algorithm.
static double	TINY_Z_PROBA The minimal value for Z proba in EM-clustering

Constructor Summary

EMAlgorithm()

Method Summary

static void	computeCOV(double[] sd, EMResult theta) Computes the variance-covariance matrix of the mixture component estimates.
static void	computeDM(double[] x, double[] sd, double[] mu, EMResult theta) Computes the matrix of derivatives of the update function.
static EMResult	doEM(double[] x, double[] sd, int k, EMResult spoint) Apply the EM-Algorithm on the given data set where x is an array of observed value and sd an array of same size than x which stores the standard deviations of the observed values.
static void	emRate(EMResult theta, double[] mu, double[] pi) Compute the euclidean distance between the new parameters and the old ones to obtain a simple approximation of the EM convergence rate.
static void	eStep(double[] x, double[] sd, EMResult theta) The Expectation step : it just consists in updating the Z matrix, i.e the cluster membership probabilities.
static void	initEM(double[] x, double[] sd, EMResult theta, java.util.Random rng) Initializes the EM algorithm by randomly assigning observations to the clusters and do one M-Step to compute the first values of the parameters.
static int	iterate(double[] x, double[] sd, EMResult theta) This method performs one iteration of the EM-algorithm.
static int	mStep(double[] x, double[] sd, EMResult theta, double err) The Maximization step : here this step is straighforward and simple analytical formula are applied to obtain the new parameter estimates.
static void	updateLoglikelihood(double[] x, double[] sd, EMResult theta) Updates the loglikelihood value.
static double[]	updateMuVector(double[] x, double[] sd, EMResult theta) Updates the vector of mixture components and returns the new values.
static double[]	updatePiVector(EMResult theta) Updates the vector of mixture mixings and returns the new values.
static void	updateZMatrix(double[] x, double[] sd, EMResult theta) Updates the Z matrix.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

EM_START

public static int EM_START

The number of replicates for a run of the EM-algorithm.

EM_ITER_MAX

public static int EM_ITER_MAX

The EM maximum number of iterations.

EM_ERR

public static double EM_ERR

The EM convergence tolerance.

EM_MIN_DISTANCE

public static double EM_MIN_DISTANCE

The minimal mahalanobis distance between components.

DO_SEM

public static boolean DO_SEM

If true the best starting point is used to do Supplemental EM (SEM) in order to compute the DM Matrix..

TINY_Z_PROBA

public static final double TINY_Z_PROBA

The minimal value for Z proba in EM-clustering

See Also:

Constant Field Values

EM_OK

public static final int EM_OK

The EM convergence status : success

See Also:

Constant Field Values

EM_CONTINUE

public static final int EM_CONTINUE

The EM convergence status : continue

See Also:

Constant Field Values

EM_FAILURE

public static final int EM_FAILURE

The EM convergence status : failure

See Also:

Constant Field Values

Constructor Detail

EMAlgorithm

public EMAlgorithm()

Method Detail

doEM

public static EMResult doEM(double[] x,
                            double[] sd,
                            int k,
                            EMResult spoint)

Apply the EM-Algorithm on the given data set where x is an array of observed value and sd an array of same size than x which stores the standard deviations of the observed values. The underlying mixture model will have k distinct components and its parameter estimates will be returned as an EMResult. If soint is not null then the EM will start at the given point specified by the mixture parameters of soint. To control some behaviours of the EM-algorithm use the global variables of the class.

Parameters:

x - the observed positions.

sd - the standard deviation of the observed positions.

k - the number of mixture components.

spoint - the starting point for the EM-algortihm

Returns:

a EMResult from which the parameter estimates can be obtained.

See Also:

EMResult

initEM

public static void initEM(double[] x,
                          double[] sd,
                          EMResult theta,
                          java.util.Random rng)

Initializes the EM algorithm by randomly assigning observations to the clusters and do one M-Step to compute the first values of the parameters.

Parameters:

x - the observed values.

sd - the standard deviations of the observed values.

theta - the current parameter estimates.

iterate

public static int iterate(double[] x,
                          double[] sd,
                          EMResult theta)

This method performs one iteration of the EM-algorithm. An iteration consists in two steps :

• Expectation (E-step) : where the expectation of the cluster membership missing variable is computed.
• Maximization (M-step) : where the parameters are updated to the values which maximize the expectation.

If the iteration leads to an update of the parameters which is under the suited range of convergence error then the method will return EM_OK, otherwise it will return EM_CONTINUE. In some cases it can happen - mainly due to numerical roundoff errors - that the likelihood value after the iteration is lower than the likelihood before the iteration which in theory cannot happen. Then in this case the method will return EM_FAILURE.

Parameters:

x - the observed values.

sd - the standard deviations of the observed values.

theta - the current parameter estimates.

Returns:

the status of the iteration.

eStep

public static void eStep(double[] x,
                         double[] sd,
                         EMResult theta)

The Expectation step : it just consists in updating the Z matrix, i.e the cluster membership probabilities.

Parameters:

x - the observed values.

sd - the standard deviations of the observed values.

theta - the current parameter estimates.

See Also:

updateZMatrix(double[], double[], EMResult)

mStep

public static int mStep(double[] x,
                        double[] sd,
                        EMResult theta,
                        double err)

The Maximization step : here this step is straighforward and simple analytical formula are applied to obtain the new parameter estimates. The method returns the status of the maximization. If the maximization leads to an new value of the parameters which is under the suited range of convergence error then the method will return EM_OK, otherwise it will return EM_CONTINUE. In some cases it can happen - mainly due to numerical roundoff errors - that the likelihood value after the maximization is lower than the likelihood before the iteration which in theory cannot happen. Then in this case the method will return EM_FAILURE

Parameters:

x - the observed values.

sd - the standard deviations of the observed values.

theta - the current parameter estimates.

See Also:

updateMuVector(double[], double[], EMResult), EMAlgorithm#updatePiVector(double[], double[], EMResult), updateLoglikelihood(double[], double[], EMResult)

emRate

public static void emRate(EMResult theta,
                          double[] mu,
                          double[] pi)

Compute the euclidean distance between the new parameters and the old ones to obtain a simple approximation of the EM convergence rate.

Parameters:

theta - the current parameter estimates.

mu - the last estimates of the means.

pi - the last estimates of the mixing.

updateMuVector

public static double[] updateMuVector(double[] x,
                                      double[] sd,
                                      EMResult theta)

Updates the vector of mixture components and returns the new values.

Parameters:

x - the observed values.

sd - the standard deviations of the observed values.

theta - the current parameter estimates.

Returns:

the new vector of means as an array of double.

updatePiVector

public static double[] updatePiVector(EMResult theta)

Updates the vector of mixture mixings and returns the new values.

Parameters:

theta - the current parameter estimates.

Returns:

the new vector of mixings as an array of double.

updateZMatrix

public static void updateZMatrix(double[] x,
                                 double[] sd,
                                 EMResult theta)

Updates the Z matrix.

Parameters:

x - the observed values.

sd - the standard deviations of the observed values.

theta - the current parameter estimates.

updateLoglikelihood

public static void updateLoglikelihood(double[] x,
                                       double[] sd,
                                       EMResult theta)

Updates the loglikelihood value.

Parameters:

x - the observed values.

sd - the standard deviations of the observed values.

theta - the current parameter estimates.

computeDM

public static void computeDM(double[] x,
                             double[] sd,
                             double[] mu,
                             EMResult theta)

Computes the matrix of derivatives of the update function. This is done by applying one iteration of the EM algorithm to a sligh modified vector of the current parameter estimates.

Parameters:

x - the data points.

sd - the standard deviation of the data points.

mu - the previous estimate of the means

theta - the current parameter estimates.

computeCOV

public static void computeCOV(double[] sd,
                              EMResult theta)

Computes the variance-covariance matrix of the mixture component estimates. If the global variable DO_SEM has been set to false then the observed information won't be computed and the variance-covariance matrix will be obtained by using only the complete information.

Parameters:

sd - the standard deviations.

theta - the maximum likelihood estimate.