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* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.distribution;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.linear.Array2DRowRealMatrix;
import org.apache.commons.math3.linear.EigenDecomposition;
import org.apache.commons.math3.linear.NonPositiveDefiniteMatrixException;
import org.apache.commons.math3.linear.RealMatrix;
import org.apache.commons.math3.linear.SingularMatrixException;
import org.apache.commons.math3.random.RandomGenerator;
import org.apache.commons.math3.random.Well19937c;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathArrays;
/**
* Implementation of the multivariate normal (Gaussian) distribution.
*
* @see <a href="http://en.wikipedia.org/wiki/Multivariate_normal_distribution">
* Multivariate normal distribution (Wikipedia)</a>
* @see <a href="http://mathworld.wolfram.com/MultivariateNormalDistribution.html">
* Multivariate normal distribution (MathWorld)</a>
*
* @since 3.1
*/
public class MultivariateNormalDistribution
extends AbstractMultivariateRealDistribution {
/** Vector of means. */
private final double[] means;
/** Covariance matrix. */
private final RealMatrix covarianceMatrix;
/** The matrix inverse of the covariance matrix. */
private final RealMatrix covarianceMatrixInverse;
/** The determinant of the covariance matrix. */
private final double covarianceMatrixDeterminant;
/** Matrix used in computation of samples. */
private final RealMatrix samplingMatrix;
/**
* Creates a multivariate normal distribution with the given mean vector and
* covariance matrix.
* <br/>
* The number of dimensions is equal to the length of the mean vector
* and to the number of rows and columns of the covariance matrix.
* It is frequently written as "p" in formulae.
*
* @param means Vector of means.
* @param covariances Covariance matrix.
* @throws DimensionMismatchException if the arrays length are
* inconsistent.
* @throws SingularMatrixException if the eigenvalue decomposition cannot
* be performed on the provided covariance matrix.
* @throws NonPositiveDefiniteMatrixException if any of the eigenvalues is
* negative.
*/
public MultivariateNormalDistribution(final double[] means,
final double[][] covariances)
throws SingularMatrixException,
DimensionMismatchException,
NonPositiveDefiniteMatrixException {
this(new Well19937c(), means, covariances);
}
/**
* Creates a multivariate normal distribution with the given mean vector and
* covariance matrix.
* <br/>
* The number of dimensions is equal to the length of the mean vector
* and to the number of rows and columns of the covariance matrix.
* It is frequently written as "p" in formulae.
*
* @param rng Random Number Generator.
* @param means Vector of means.
* @param covariances Covariance matrix.
* @throws DimensionMismatchException if the arrays length are
* inconsistent.
* @throws SingularMatrixException if the eigenvalue decomposition cannot
* be performed on the provided covariance matrix.
* @throws NonPositiveDefiniteMatrixException if any of the eigenvalues is
* negative.
*/
public MultivariateNormalDistribution(RandomGenerator rng,
final double[] means,
final double[][] covariances)
throws SingularMatrixException,
DimensionMismatchException,
NonPositiveDefiniteMatrixException {
super(rng, means.length);
final int dim = means.length;
if (covariances.length != dim) {
throw new DimensionMismatchException(covariances.length, dim);
}
for (int i = 0; i < dim; i++) {
if (dim != covariances[i].length) {
throw new DimensionMismatchException(covariances[i].length, dim);
}
}
this.means = MathArrays.copyOf(means);
covarianceMatrix = new Array2DRowRealMatrix(covariances);
// Covariance matrix eigen decomposition.
final EigenDecomposition covMatDec = new EigenDecomposition(covarianceMatrix);
// Compute and store the inverse.
covarianceMatrixInverse = covMatDec.getSolver().getInverse();
// Compute and store the determinant.
covarianceMatrixDeterminant = covMatDec.getDeterminant();
// Eigenvalues of the covariance matrix.
final double[] covMatEigenvalues = covMatDec.getRealEigenvalues();
for (int i = 0; i < covMatEigenvalues.length; i++) {
if (covMatEigenvalues[i] < 0) {
throw new NonPositiveDefiniteMatrixException(covMatEigenvalues[i], i, 0);
}
}
// Matrix where each column is an eigenvector of the covariance matrix.
final Array2DRowRealMatrix covMatEigenvectors = new Array2DRowRealMatrix(dim, dim);
for (int v = 0; v < dim; v++) {
final double[] evec = covMatDec.getEigenvector(v).toArray();
covMatEigenvectors.setColumn(v, evec);
}
final RealMatrix tmpMatrix = covMatEigenvectors.transpose();
// Scale each eigenvector by the square root of its eigenvalue.
for (int row = 0; row < dim; row++) {
final double factor = FastMath.sqrt(covMatEigenvalues[row]);
for (int col = 0; col < dim; col++) {
tmpMatrix.multiplyEntry(row, col, factor);
}
}
samplingMatrix = covMatEigenvectors.multiply(tmpMatrix);
}
/**
* Gets the mean vector.
*
* @return the mean vector.
*/
public double[] getMeans() {
return MathArrays.copyOf(means);
}
/**
* Gets the covariance matrix.
*
* @return the covariance matrix.
*/
public RealMatrix getCovariances() {
return covarianceMatrix.copy();
}
/** {@inheritDoc} */
public double density(final double[] vals) throws DimensionMismatchException {
final int dim = getDimension();
if (vals.length != dim) {
throw new DimensionMismatchException(vals.length, dim);
}
return FastMath.pow(2 * FastMath.PI, -0.5 * dim) *
FastMath.pow(covarianceMatrixDeterminant, -0.5) *
getExponentTerm(vals);
}
/**
* Gets the square root of each element on the diagonal of the covariance
* matrix.
*
* @return the standard deviations.
*/
public double[] getStandardDeviations() {
final int dim = getDimension();
final double[] std = new double[dim];
final double[][] s = covarianceMatrix.getData();
for (int i = 0; i < dim; i++) {
std[i] = FastMath.sqrt(s[i][i]);
}
return std;
}
/** {@inheritDoc} */
@Override
public double[] sample() {
final int dim = getDimension();
final double[] normalVals = new double[dim];
for (int i = 0; i < dim; i++) {
normalVals[i] = random.nextGaussian();
}
final double[] vals = samplingMatrix.operate(normalVals);
for (int i = 0; i < dim; i++) {
vals[i] += means[i];
}
return vals;
}
/**
* Computes the term used in the exponent (see definition of the distribution).
*
* @param values Values at which to compute density.
* @return the multiplication factor of density calculations.
*/
private double getExponentTerm(final double[] values) {
final double[] centered = new double[values.length];
for (int i = 0; i < centered.length; i++) {
centered[i] = values[i] - getMeans()[i];
}
final double[] preMultiplied = covarianceMatrixInverse.preMultiply(centered);
double sum = 0;
for (int i = 0; i < preMultiplied.length; i++) {
sum += preMultiplied[i] * centered[i];
}
return FastMath.exp(-0.5 * sum);
}
}