Java tutorial
/* * 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.fitting; import java.util.ArrayList; import java.util.Arrays; import java.util.List; import org.apache.commons.math3.distribution.MultivariateNormalDistribution; import org.apache.commons.math3.distribution.MixtureMultivariateNormalDistribution; import org.apache.commons.math3.exception.ConvergenceException; import org.apache.commons.math3.exception.DimensionMismatchException; import org.apache.commons.math3.exception.NotStrictlyPositiveException; import org.apache.commons.math3.exception.NumberIsTooSmallException; import org.apache.commons.math3.exception.NumberIsTooLargeException; import org.apache.commons.math3.exception.util.LocalizedFormats; import org.apache.commons.math3.linear.Array2DRowRealMatrix; import org.apache.commons.math3.linear.RealMatrix; import org.apache.commons.math3.linear.SingularMatrixException; import org.apache.commons.math3.stat.correlation.Covariance; import org.apache.commons.math3.util.FastMath; import org.apache.commons.math3.util.MathArrays; import org.apache.commons.math3.util.Pair; /** * Expectation-Maximization</a> algorithm for fitting the parameters of * multivariate normal mixture model distributions. * * This implementation is pure original code based on <a * href="https://www.ee.washington.edu/techsite/papers/documents/UWEETR-2010-0002.pdf"> * EM Demystified: An Expectation-Maximization Tutorial</a> by Yihua Chen and Maya R. Gupta, * Department of Electrical Engineering, University of Washington, Seattle, WA 98195. * It was verified using external tools like <a * href="http://cran.r-project.org/web/packages/mixtools/index.html">CRAN Mixtools</a> * (see the JUnit test cases) but it is <strong>not</strong> based on Mixtools code at all. * The discussion of the origin of this class can be seen in the comments of the <a * href="https://issues.apache.org/jira/browse/MATH-817">MATH-817</a> JIRA issue. * @since 3.2 */ public class MultivariateNormalMixtureExpectationMaximization { /** * Default maximum number of iterations allowed per fitting process. */ private static final int DEFAULT_MAX_ITERATIONS = 1000; /** * Default convergence threshold for fitting. */ private static final double DEFAULT_THRESHOLD = 1E-5; /** * The data to fit. */ private final double[][] data; /** * The model fit against the data. */ private MixtureMultivariateNormalDistribution fittedModel; /** * The log likelihood of the data given the fitted model. */ private double logLikelihood = 0d; /** * Creates an object to fit a multivariate normal mixture model to data. * * @param data Data to use in fitting procedure * @throws NotStrictlyPositiveException if data has no rows * @throws DimensionMismatchException if rows of data have different numbers * of columns * @throws NumberIsTooSmallException if the number of columns in the data is * less than 2 */ public MultivariateNormalMixtureExpectationMaximization(double[][] data) throws NotStrictlyPositiveException, DimensionMismatchException, NumberIsTooSmallException { if (data.length < 1) { throw new NotStrictlyPositiveException(data.length); } this.data = new double[data.length][data[0].length]; for (int i = 0; i < data.length; i++) { if (data[i].length != data[0].length) { // Jagged arrays not allowed throw new DimensionMismatchException(data[i].length, data[0].length); } if (data[i].length < 2) { throw new NumberIsTooSmallException(LocalizedFormats.NUMBER_TOO_SMALL, data[i].length, 2, true); } this.data[i] = MathArrays.copyOf(data[i], data[i].length); } } /** * Fit a mixture model to the data supplied to the constructor. * * The quality of the fit depends on the concavity of the data provided to * the constructor and the initial mixture provided to this function. If the * data has many local optima, multiple runs of the fitting function with * different initial mixtures may be required to find the optimal solution. * If a SingularMatrixException is encountered, it is possible that another * initialization would work. * * @param initialMixture Model containing initial values of weights and * multivariate normals * @param maxIterations Maximum iterations allowed for fit * @param threshold Convergence threshold computed as difference in * logLikelihoods between successive iterations * @throws SingularMatrixException if any component's covariance matrix is * singular during fitting * @throws NotStrictlyPositiveException if numComponents is less than one * or threshold is less than Double.MIN_VALUE * @throws DimensionMismatchException if initialMixture mean vector and data * number of columns are not equal */ public void fit(final MixtureMultivariateNormalDistribution initialMixture, final int maxIterations, final double threshold) throws SingularMatrixException, NotStrictlyPositiveException, DimensionMismatchException { if (maxIterations < 1) { throw new NotStrictlyPositiveException(maxIterations); } if (threshold < Double.MIN_VALUE) { throw new NotStrictlyPositiveException(threshold); } final int n = data.length; // Number of data columns. Jagged data already rejected in constructor, // so we can assume the lengths of each row are equal. final int numCols = data[0].length; final int k = initialMixture.getComponents().size(); final int numMeanColumns = initialMixture.getComponents().get(0).getSecond().getMeans().length; if (numMeanColumns != numCols) { throw new DimensionMismatchException(numMeanColumns, numCols); } int numIterations = 0; double previousLogLikelihood = 0d; logLikelihood = Double.NEGATIVE_INFINITY; // Initialize model to fit to initial mixture. fittedModel = new MixtureMultivariateNormalDistribution(initialMixture.getComponents()); while (numIterations++ <= maxIterations && FastMath.abs(previousLogLikelihood - logLikelihood) > threshold) { previousLogLikelihood = logLikelihood; double sumLogLikelihood = 0d; // Mixture components final List<Pair<Double, MultivariateNormalDistribution>> components = fittedModel.getComponents(); // Weight and distribution of each component final double[] weights = new double[k]; final MultivariateNormalDistribution[] mvns = new MultivariateNormalDistribution[k]; for (int j = 0; j < k; j++) { weights[j] = components.get(j).getFirst(); mvns[j] = components.get(j).getSecond(); } // E-step: compute the data dependent parameters of the expectation // function. // The percentage of row's total density between a row and a // component final double[][] gamma = new double[n][k]; // Sum of gamma for each component final double[] gammaSums = new double[k]; // Sum of gamma times its row for each each component final double[][] gammaDataProdSums = new double[k][numCols]; for (int i = 0; i < n; i++) { final double rowDensity = fittedModel.density(data[i]); sumLogLikelihood += FastMath.log(rowDensity); for (int j = 0; j < k; j++) { gamma[i][j] = weights[j] * mvns[j].density(data[i]) / rowDensity; gammaSums[j] += gamma[i][j]; for (int col = 0; col < numCols; col++) { gammaDataProdSums[j][col] += gamma[i][j] * data[i][col]; } } } logLikelihood = sumLogLikelihood / n; // M-step: compute the new parameters based on the expectation // function. final double[] newWeights = new double[k]; final double[][] newMeans = new double[k][numCols]; for (int j = 0; j < k; j++) { newWeights[j] = gammaSums[j] / n; for (int col = 0; col < numCols; col++) { newMeans[j][col] = gammaDataProdSums[j][col] / gammaSums[j]; } } // Compute new covariance matrices final RealMatrix[] newCovMats = new RealMatrix[k]; for (int j = 0; j < k; j++) { newCovMats[j] = new Array2DRowRealMatrix(numCols, numCols); } for (int i = 0; i < n; i++) { for (int j = 0; j < k; j++) { final RealMatrix vec = new Array2DRowRealMatrix(MathArrays.ebeSubtract(data[i], newMeans[j])); final RealMatrix dataCov = vec.multiply(vec.transpose()).scalarMultiply(gamma[i][j]); newCovMats[j] = newCovMats[j].add(dataCov); } } // Converting to arrays for use by fitted model final double[][][] newCovMatArrays = new double[k][numCols][numCols]; for (int j = 0; j < k; j++) { newCovMats[j] = newCovMats[j].scalarMultiply(1d / gammaSums[j]); newCovMatArrays[j] = newCovMats[j].getData(); } // Update current model fittedModel = new MixtureMultivariateNormalDistribution(newWeights, newMeans, newCovMatArrays); } if (FastMath.abs(previousLogLikelihood - logLikelihood) > threshold) { // Did not converge before the maximum number of iterations throw new ConvergenceException(); } } /** * Fit a mixture model to the data supplied to the constructor. * * The quality of the fit depends on the concavity of the data provided to * the constructor and the initial mixture provided to this function. If the * data has many local optima, multiple runs of the fitting function with * different initial mixtures may be required to find the optimal solution. * If a SingularMatrixException is encountered, it is possible that another * initialization would work. * * @param initialMixture Model containing initial values of weights and * multivariate normals * @throws SingularMatrixException if any component's covariance matrix is * singular during fitting * @throws NotStrictlyPositiveException if numComponents is less than one or * threshold is less than Double.MIN_VALUE */ public void fit(MixtureMultivariateNormalDistribution initialMixture) throws SingularMatrixException, NotStrictlyPositiveException { fit(initialMixture, DEFAULT_MAX_ITERATIONS, DEFAULT_THRESHOLD); } /** * Helper method to create a multivariate normal mixture model which can be * used to initialize {@link #fit(MixtureMultivariateNormalDistribution)}. * * This method uses the data supplied to the constructor to try to determine * a good mixture model at which to start the fit, but it is not guaranteed * to supply a model which will find the optimal solution or even converge. * * @param data Data to estimate distribution * @param numComponents Number of components for estimated mixture * @return Multivariate normal mixture model estimated from the data * @throws NumberIsTooLargeException if {@code numComponents} is greater * than the number of data rows. * @throws NumberIsTooSmallException if {@code numComponents < 2}. * @throws NotStrictlyPositiveException if data has less than 2 rows * @throws DimensionMismatchException if rows of data have different numbers * of columns */ public static MixtureMultivariateNormalDistribution estimate(final double[][] data, final int numComponents) throws NotStrictlyPositiveException, DimensionMismatchException { if (data.length < 2) { throw new NotStrictlyPositiveException(data.length); } if (numComponents < 2) { throw new NumberIsTooSmallException(numComponents, 2, true); } if (numComponents > data.length) { throw new NumberIsTooLargeException(numComponents, data.length, true); } final int numRows = data.length; final int numCols = data[0].length; // sort the data final DataRow[] sortedData = new DataRow[numRows]; for (int i = 0; i < numRows; i++) { sortedData[i] = new DataRow(data[i]); } Arrays.sort(sortedData); // uniform weight for each bin final double weight = 1d / numComponents; // components of mixture model to be created final List<Pair<Double, MultivariateNormalDistribution>> components = new ArrayList<Pair<Double, MultivariateNormalDistribution>>( numComponents); // create a component based on data in each bin for (int binIndex = 0; binIndex < numComponents; binIndex++) { // minimum index (inclusive) from sorted data for this bin final int minIndex = (binIndex * numRows) / numComponents; // maximum index (exclusive) from sorted data for this bin final int maxIndex = ((binIndex + 1) * numRows) / numComponents; // number of data records that will be in this bin final int numBinRows = maxIndex - minIndex; // data for this bin final double[][] binData = new double[numBinRows][numCols]; // mean of each column for the data in the this bin final double[] columnMeans = new double[numCols]; // populate bin and create component for (int i = minIndex, iBin = 0; i < maxIndex; i++, iBin++) { for (int j = 0; j < numCols; j++) { final double val = sortedData[i].getRow()[j]; columnMeans[j] += val; binData[iBin][j] = val; } } MathArrays.scaleInPlace(1d / numBinRows, columnMeans); // covariance matrix for this bin final double[][] covMat = new Covariance(binData).getCovarianceMatrix().getData(); final MultivariateNormalDistribution mvn = new MultivariateNormalDistribution(columnMeans, covMat); components.add(new Pair<Double, MultivariateNormalDistribution>(weight, mvn)); } return new MixtureMultivariateNormalDistribution(components); } /** * Gets the log likelihood of the data under the fitted model. * * @return Log likelihood of data or zero of no data has been fit */ public double getLogLikelihood() { return logLikelihood; } /** * Gets the fitted model. * * @return fitted model or {@code null} if no fit has been performed yet. */ public MixtureMultivariateNormalDistribution getFittedModel() { return new MixtureMultivariateNormalDistribution(fittedModel.getComponents()); } /** * Class used for sorting user-supplied data. */ private static class DataRow implements Comparable<DataRow> { /** One data row. */ private final double[] row; /** Mean of the data row. */ private Double mean; /** * Create a data row. * @param data Data to use for the row */ DataRow(final double[] data) { // Store reference. row = data; // Compute mean. mean = 0d; for (int i = 0; i < data.length; i++) { mean += data[i]; } mean /= data.length; } /** * Compare two data rows. * @param other The other row * @return int for sorting */ public int compareTo(final DataRow other) { return mean.compareTo(other.mean); } /** {@inheritDoc} */ @Override public boolean equals(Object other) { if (this == other) { return true; } if (other instanceof DataRow) { return MathArrays.equals(row, ((DataRow) other).row); } return false; } /** {@inheritDoc} */ @Override public int hashCode() { return Arrays.hashCode(row); } /** * Get a data row. * @return data row array */ public double[] getRow() { return row; } } }