001 /* 002 * Licensed to the Apache Software Foundation (ASF) under one or more 003 * contributor license agreements. See the NOTICE file distributed with 004 * this work for additional information regarding copyright ownership. 005 * The ASF licenses this file to You under the Apache License, Version 2.0 006 * (the "License"); you may not use this file except in compliance with 007 * the License. You may obtain a copy of the License at 008 * 009 * http://www.apache.org/licenses/LICENSE-2.0 010 * 011 * Unless required by applicable law or agreed to in writing, software 012 * distributed under the License is distributed on an "AS IS" BASIS, 013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 014 * See the License for the specific language governing permissions and 015 * limitations under the License. 016 */ 017 package org.apache.commons.math.stat.regression; 018 019 import org.apache.commons.math.linear.Array2DRowRealMatrix; 020 import org.apache.commons.math.linear.LUDecompositionImpl; 021 import org.apache.commons.math.linear.QRDecomposition; 022 import org.apache.commons.math.linear.QRDecompositionImpl; 023 import org.apache.commons.math.linear.RealMatrix; 024 import org.apache.commons.math.linear.RealVector; 025 import org.apache.commons.math.stat.StatUtils; 026 import org.apache.commons.math.stat.descriptive.moment.SecondMoment; 027 028 /** 029 * <p>Implements ordinary least squares (OLS) to estimate the parameters of a 030 * multiple linear regression model.</p> 031 * 032 * <p>The regression coefficients, <code>b</code>, satisfy the normal equations: 033 * <pre><code> X<sup>T</sup> X b = X<sup>T</sup> y </code></pre></p> 034 * 035 * <p>To solve the normal equations, this implementation uses QR decomposition 036 * of the <code>X</code> matrix. (See {@link QRDecompositionImpl} for details on the 037 * decomposition algorithm.) The <code>X</code> matrix, also known as the <i>design matrix,</i> 038 * has rows corresponding to sample observations and columns corresponding to independent 039 * variables. When the model is estimated using an intercept term (i.e. when 040 * {@link #isNoIntercept() isNoIntercept} is false as it is by default), the <code>X</code> 041 * matrix includes an initial column identically equal to 1. We solve the normal equations 042 * as follows: 043 * <pre><code> X<sup>T</sup>X b = X<sup>T</sup> y 044 * (QR)<sup>T</sup> (QR) b = (QR)<sup>T</sup>y 045 * R<sup>T</sup> (Q<sup>T</sup>Q) R b = R<sup>T</sup> Q<sup>T</sup> y 046 * R<sup>T</sup> R b = R<sup>T</sup> Q<sup>T</sup> y 047 * (R<sup>T</sup>)<sup>-1</sup> R<sup>T</sup> R b = (R<sup>T</sup>)<sup>-1</sup> R<sup>T</sup> Q<sup>T</sup> y 048 * R b = Q<sup>T</sup> y </code></pre></p> 049 * 050 * <p>Given <code>Q</code> and <code>R</code>, the last equation is solved by back-substitution.</p> 051 * 052 * @version $Revision: 1073464 $ $Date: 2011-02-22 20:35:02 +0100 (mar. 22 f??vr. 2011) $ 053 * @since 2.0 054 */ 055 public class OLSMultipleLinearRegression extends AbstractMultipleLinearRegression { 056 057 /** Cached QR decomposition of X matrix */ 058 private QRDecomposition qr = null; 059 060 /** 061 * Loads model x and y sample data, overriding any previous sample. 062 * 063 * Computes and caches QR decomposition of the X matrix. 064 * @param y the [n,1] array representing the y sample 065 * @param x the [n,k] array representing the x sample 066 * @throws IllegalArgumentException if the x and y array data are not 067 * compatible for the regression 068 */ 069 public void newSampleData(double[] y, double[][] x) { 070 validateSampleData(x, y); 071 newYSampleData(y); 072 newXSampleData(x); 073 } 074 075 /** 076 * {@inheritDoc} 077 * <p>This implementation computes and caches the QR decomposition of the X matrix.</p> 078 */ 079 @Override 080 public void newSampleData(double[] data, int nobs, int nvars) { 081 super.newSampleData(data, nobs, nvars); 082 qr = new QRDecompositionImpl(X); 083 } 084 085 /** 086 * <p>Compute the "hat" matrix. 087 * </p> 088 * <p>The hat matrix is defined in terms of the design matrix X 089 * by X(X<sup>T</sup>X)<sup>-1</sup>X<sup>T</sup> 090 * </p> 091 * <p>The implementation here uses the QR decomposition to compute the 092 * hat matrix as Q I<sub>p</sub>Q<sup>T</sup> where I<sub>p</sub> is the 093 * p-dimensional identity matrix augmented by 0's. This computational 094 * formula is from "The Hat Matrix in Regression and ANOVA", 095 * David C. Hoaglin and Roy E. Welsch, 096 * <i>The American Statistician</i>, Vol. 32, No. 1 (Feb., 1978), pp. 17-22. 097 * 098 * @return the hat matrix 099 */ 100 public RealMatrix calculateHat() { 101 // Create augmented identity matrix 102 RealMatrix Q = qr.getQ(); 103 final int p = qr.getR().getColumnDimension(); 104 final int n = Q.getColumnDimension(); 105 Array2DRowRealMatrix augI = new Array2DRowRealMatrix(n, n); 106 double[][] augIData = augI.getDataRef(); 107 for (int i = 0; i < n; i++) { 108 for (int j =0; j < n; j++) { 109 if (i == j && i < p) { 110 augIData[i][j] = 1d; 111 } else { 112 augIData[i][j] = 0d; 113 } 114 } 115 } 116 117 // Compute and return Hat matrix 118 return Q.multiply(augI).multiply(Q.transpose()); 119 } 120 121 /** 122 * <p>Returns the sum of squared deviations of Y from its mean.</p> 123 * 124 * <p>If the model has no intercept term, <code>0</code> is used for the 125 * mean of Y - i.e., what is returned is the sum of the squared Y values.</p> 126 * 127 * <p>The value returned by this method is the SSTO value used in 128 * the {@link #calculateRSquared() R-squared} computation.</p> 129 * 130 * @return SSTO - the total sum of squares 131 * @see #isNoIntercept() 132 * @since 2.2 133 */ 134 public double calculateTotalSumOfSquares() { 135 if (isNoIntercept()) { 136 return StatUtils.sumSq(Y.getData()); 137 } else { 138 return new SecondMoment().evaluate(Y.getData()); 139 } 140 } 141 142 /** 143 * Returns the sum of squared residuals. 144 * 145 * @return residual sum of squares 146 * @since 2.2 147 */ 148 public double calculateResidualSumOfSquares() { 149 final RealVector residuals = calculateResiduals(); 150 return residuals.dotProduct(residuals); 151 } 152 153 /** 154 * Returns the R-Squared statistic, defined by the formula <pre> 155 * R<sup>2</sup> = 1 - SSR / SSTO 156 * </pre> 157 * where SSR is the {@link #calculateResidualSumOfSquares() sum of squared residuals} 158 * and SSTO is the {@link #calculateTotalSumOfSquares() total sum of squares} 159 * 160 * @return R-square statistic 161 * @since 2.2 162 */ 163 public double calculateRSquared() { 164 return 1 - calculateResidualSumOfSquares() / calculateTotalSumOfSquares(); 165 } 166 167 /** 168 * <p>Returns the adjusted R-squared statistic, defined by the formula <pre> 169 * R<sup>2</sup><sub>adj</sub> = 1 - [SSR (n - 1)] / [SSTO (n - p)] 170 * </pre> 171 * where SSR is the {@link #calculateResidualSumOfSquares() sum of squared residuals}, 172 * SSTO is the {@link #calculateTotalSumOfSquares() total sum of squares}, n is the number 173 * of observations and p is the number of parameters estimated (including the intercept).</p> 174 * 175 * <p>If the regression is estimated without an intercept term, what is returned is <pre> 176 * <code> 1 - (1 - {@link #calculateRSquared()}) * (n / (n - p)) </code> 177 * </pre></p> 178 * 179 * @return adjusted R-Squared statistic 180 * @see #isNoIntercept() 181 * @since 2.2 182 */ 183 public double calculateAdjustedRSquared() { 184 final double n = X.getRowDimension(); 185 if (isNoIntercept()) { 186 return 1 - (1 - calculateRSquared()) * (n / (n - X.getColumnDimension())); 187 } else { 188 return 1 - (calculateResidualSumOfSquares() * (n - 1)) / 189 (calculateTotalSumOfSquares() * (n - X.getColumnDimension())); 190 } 191 } 192 193 /** 194 * {@inheritDoc} 195 * <p>This implementation computes and caches the QR decomposition of the X matrix 196 * once it is successfully loaded.</p> 197 */ 198 @Override 199 protected void newXSampleData(double[][] x) { 200 super.newXSampleData(x); 201 qr = new QRDecompositionImpl(X); 202 } 203 204 /** 205 * Calculates the regression coefficients using OLS. 206 * 207 * @return beta 208 */ 209 @Override 210 protected RealVector calculateBeta() { 211 return qr.getSolver().solve(Y); 212 } 213 214 /** 215 * <p>Calculates the variance-covariance matrix of the regression parameters. 216 * </p> 217 * <p>Var(b) = (X<sup>T</sup>X)<sup>-1</sup> 218 * </p> 219 * <p>Uses QR decomposition to reduce (X<sup>T</sup>X)<sup>-1</sup> 220 * to (R<sup>T</sup>R)<sup>-1</sup>, with only the top p rows of 221 * R included, where p = the length of the beta vector.</p> 222 * 223 * @return The beta variance-covariance matrix 224 */ 225 @Override 226 protected RealMatrix calculateBetaVariance() { 227 int p = X.getColumnDimension(); 228 RealMatrix Raug = qr.getR().getSubMatrix(0, p - 1 , 0, p - 1); 229 RealMatrix Rinv = new LUDecompositionImpl(Raug).getSolver().getInverse(); 230 return Rinv.multiply(Rinv.transpose()); 231 } 232 233 }