umontreal.iro.lecuyer.randvarmulti
Class MultinormalPCAGen

java.lang.Object
  umontreal.iro.lecuyer.randvarmulti.RandomMultivariateGen
      umontreal.iro.lecuyer.randvarmulti.MultinormalGen
          umontreal.iro.lecuyer.randvarmulti.MultinormalPCAGen

public class MultinormalPCAGen
extends MultinormalGen

Extends MultinormalGen for a multivariate normal distribution, generated via the method of principal components analysis (PCA) of the covariance matrix. The covariance matrix Σ is decomposed (by the constructor) as Σ = VΛV^t where V is an orthogonal matrix and Λ is the diagonal matrix made up of the eigenvalues of Σ. V^t is the transpose matrix of V. The eigenvalues are ordered from the largest (λ₁) to the smallest (λ_d). The random multinormal vector X is generated via

X = μ + AZ,

where A = V()^1/2, and Z is a d-dimensional vector of independent standard normal random variates. The decomposition method uses the SingularValueDecomposition class in colt.

Constructor Summary

MultinormalPCAGen(NormalGen gen1, double[] mu, double[][] sigma)
Equivalent to MultinormalPCAGen(gen1, mu, new DenseDoubleMatrix2D(sigma)).

MultinormalPCAGen(NormalGen gen1, double[] mu, DoubleMatrix2D sigma)
Constructs a multinormal generator with mean vector mu and covariance matrix sigma.

Method Summary

static DoubleMatrix2D	decompPCA(double[][] sigma) Computes the decomposition sigma = Σ = VΛVt.
static DoubleMatrix2D	decompPCA(DoubleMatrix2D sigma) Computes the decomposition sigma = Σ = VΛVt.
double[]	getLambda() Returns the eigenvalues of Σ in decreasing order.
static double[]	getLambda(DoubleMatrix2D sigma) Computes and returns the eigenvalues of sigma in decreasing order.
DoubleMatrix2D	getPCADecompSigma() Returns the matrix A = V()1/2 of this object.
void	nextPoint(double[] p) Generates a point from this multinormal distribution.
static void	nextPoint(NormalGen gen1, double[] mu, double[][] sigma, double[] p) Equivalent to nextPoint(gen1, mu, new DenseDoubleMatrix2D(sigma), p).
static void	nextPoint(NormalGen gen1, double[] mu, DoubleMatrix2D sigma, double[] p) Generates a d-dimensional vector from the multinormal distribution with mean vector mu and covariance matrix sigma, using the one-dimensional normal generator gen1 to generate the coordinates of Z, and using the PCA decomposition of Σ.
void	setSigma(DoubleMatrix2D sigma) Sets the covariance matrix Σ of this multinormal generator to sigma (and recomputes A).

Methods inherited from class umontreal.iro.lecuyer.randvarmulti.MultinormalGen

getMu, getMu, getSigma, setMu, setMu

Methods inherited from class umontreal.iro.lecuyer.randvarmulti.RandomMultivariateGen

getDimension, getStream, nextArrayOfPoints, setStream

Methods inherited from class java.lang.Object

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

Constructor Detail

MultinormalPCAGen

public MultinormalPCAGen(NormalGen gen1,
                         double[] mu,
                         double[][] sigma)

Equivalent to MultinormalPCAGen(gen1, mu, new DenseDoubleMatrix2D(sigma)).

MultinormalPCAGen

public MultinormalPCAGen(NormalGen gen1,
                         double[] mu,
                         DoubleMatrix2D sigma)

Constructs a multinormal generator with mean vector mu and covariance matrix sigma. The mean vector must have the same length as the dimensions of the covariance matrix, which must be symmetric and positive semi-definite. If any of the above conditions is violated, an exception is thrown. The vector Z is generated by calling d times the generator gen1, which must be a standard normal 1-dimensional generator.

Parameters:

gen1 - the one-dimensional generator

mu - the mean vector.

sigma - the covariance matrix.

Throws:

NullPointerException - if any argument is null.

IllegalArgumentException - if the length of the mean vector is incompatible with the dimensions of the covariance matrix.

Method Detail

decompPCA

public static DoubleMatrix2D decompPCA(double[][] sigma)

Computes the decomposition sigma = Σ = VΛV^t. Returns A = V()^1/2.

decompPCA

public static DoubleMatrix2D decompPCA(DoubleMatrix2D sigma)

Computes the decomposition sigma = Σ = VΛV^t. Returns A = V()^1/2.

getPCADecompSigma

public DoubleMatrix2D getPCADecompSigma()

Returns the matrix A = V()^1/2 of this object.

Returns:

the PCA square root of the covariance matrix

getLambda

public static double[] getLambda(DoubleMatrix2D sigma)

Computes and returns the eigenvalues of sigma in decreasing order.

getLambda

public double[] getLambda()

Returns the eigenvalues of Σ in decreasing order.

setSigma

public void setSigma(DoubleMatrix2D sigma)

Sets the covariance matrix Σ of this multinormal generator to sigma (and recomputes A).

Parameters:

sigma - the new covariance matrix.

Throws:

IllegalArgumentException - if sigma has incorrect dimensions.

nextPoint

public static void nextPoint(NormalGen gen1,
                             double[] mu,
                             DoubleMatrix2D sigma,
                             double[] p)

Generates a d-dimensional vector from the multinormal distribution with mean vector mu and covariance matrix sigma, using the one-dimensional normal generator gen1 to generate the coordinates of Z, and using the PCA decomposition of Σ. The resulting vector is put into p. Note that this static method will be very slow for large dimensions, because it recomputes the singular value decomposition at every call. It is therefore recommended to use a MultinormalPCAGen object instead, if the method is to be called more than once.

Parameters:

p - the array to be filled with the generated point.

Throws:

IllegalArgumentException - if the one-dimensional normal generator uses a normal distribution with μ not equal to 0, or σ not equal to 1.

IllegalArgumentException - if the length of the mean vector is different from the dimensions of the covariance matrix, or if the covariance matrix is not symmetric and positive-definite.

NullPointerException - if any argument is null.

nextPoint

public static void nextPoint(NormalGen gen1,
                             double[] mu,
                             double[][] sigma,
                             double[] p)

Equivalent to nextPoint(gen1, mu, new DenseDoubleMatrix2D(sigma), p).

nextPoint

public void nextPoint(double[] p)

Generates a point from this multinormal distribution. This is much faster than the static method as it computes the singular value decomposition matrix only once in the constructor.

Overrides:

nextPoint in class MultinormalGen

Parameters:

p - the array to be filled with the generated point