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 */
017package org.apache.commons.math3.stat.regression;
018
019import org.apache.commons.math3.exception.MathIllegalArgumentException;
020import org.apache.commons.math3.linear.Array2DRowRealMatrix;
021import org.apache.commons.math3.linear.LUDecomposition;
022import org.apache.commons.math3.linear.QRDecomposition;
023import org.apache.commons.math3.linear.RealMatrix;
024import org.apache.commons.math3.linear.RealVector;
025import org.apache.commons.math3.stat.StatUtils;
026import org.apache.commons.math3.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 QRDecomposition} 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 $Id: OLSMultipleLinearRegression.java 1416643 2012-12-03 19:37:14Z tn $
053 * @since 2.0
054 */
055public 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 MathIllegalArgumentException if the x and y array data are not
067     *             compatible for the regression
068     */
069    public void newSampleData(double[] y, double[][] x) throws MathIllegalArgumentException {
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 QRDecomposition(getX());
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     * </p>
098     * <p>Data for the model must have been successfully loaded using one of
099     * the {@code newSampleData} methods before invoking this method; otherwise
100     * a {@code NullPointerException} will be thrown.</p>
101     *
102     * @return the hat matrix
103     */
104    public RealMatrix calculateHat() {
105        // Create augmented identity matrix
106        RealMatrix Q = qr.getQ();
107        final int p = qr.getR().getColumnDimension();
108        final int n = Q.getColumnDimension();
109        // No try-catch or advertised NotStrictlyPositiveException - NPE above if n < 3
110        Array2DRowRealMatrix augI = new Array2DRowRealMatrix(n, n);
111        double[][] augIData = augI.getDataRef();
112        for (int i = 0; i < n; i++) {
113            for (int j =0; j < n; j++) {
114                if (i == j && i < p) {
115                    augIData[i][j] = 1d;
116                } else {
117                    augIData[i][j] = 0d;
118                }
119            }
120        }
121
122        // Compute and return Hat matrix
123        // No DME advertised - args valid if we get here
124        return Q.multiply(augI).multiply(Q.transpose());
125    }
126
127    /**
128     * <p>Returns the sum of squared deviations of Y from its mean.</p>
129     *
130     * <p>If the model has no intercept term, <code>0</code> is used for the
131     * mean of Y - i.e., what is returned is the sum of the squared Y values.</p>
132     *
133     * <p>The value returned by this method is the SSTO value used in
134     * the {@link #calculateRSquared() R-squared} computation.</p>
135     *
136     * @return SSTO - the total sum of squares
137     * @throws MathIllegalArgumentException if the sample has not been set or does
138     * not contain at least 3 observations
139     * @see #isNoIntercept()
140     * @since 2.2
141     */
142    public double calculateTotalSumOfSquares() throws MathIllegalArgumentException {
143        if (isNoIntercept()) {
144            return StatUtils.sumSq(getY().toArray());
145        } else {
146            return new SecondMoment().evaluate(getY().toArray());
147        }
148    }
149
150    /**
151     * Returns the sum of squared residuals.
152     *
153     * @return residual sum of squares
154     * @since 2.2
155     */
156    public double calculateResidualSumOfSquares() {
157        final RealVector residuals = calculateResiduals();
158        // No advertised DME, args are valid
159        return residuals.dotProduct(residuals);
160    }
161
162    /**
163     * Returns the R-Squared statistic, defined by the formula <pre>
164     * R<sup>2</sup> = 1 - SSR / SSTO
165     * </pre>
166     * where SSR is the {@link #calculateResidualSumOfSquares() sum of squared residuals}
167     * and SSTO is the {@link #calculateTotalSumOfSquares() total sum of squares}
168     *
169     * @return R-square statistic
170     * @throws MathIllegalArgumentException if the sample has not been set or does
171     * not contain at least 3 observations
172     * @since 2.2
173     */
174    public double calculateRSquared() throws MathIllegalArgumentException {
175        return 1 - calculateResidualSumOfSquares() / calculateTotalSumOfSquares();
176    }
177
178    /**
179     * <p>Returns the adjusted R-squared statistic, defined by the formula <pre>
180     * R<sup>2</sup><sub>adj</sub> = 1 - [SSR (n - 1)] / [SSTO (n - p)]
181     * </pre>
182     * where SSR is the {@link #calculateResidualSumOfSquares() sum of squared residuals},
183     * SSTO is the {@link #calculateTotalSumOfSquares() total sum of squares}, n is the number
184     * of observations and p is the number of parameters estimated (including the intercept).</p>
185     *
186     * <p>If the regression is estimated without an intercept term, what is returned is <pre>
187     * <code> 1 - (1 - {@link #calculateRSquared()}) * (n / (n - p)) </code>
188     * </pre></p>
189     *
190     * @return adjusted R-Squared statistic
191     * @throws MathIllegalArgumentException if the sample has not been set or does
192     * not contain at least 3 observations
193     * @see #isNoIntercept()
194     * @since 2.2
195     */
196    public double calculateAdjustedRSquared() throws MathIllegalArgumentException {
197        final double n = getX().getRowDimension();
198        if (isNoIntercept()) {
199            return 1 - (1 - calculateRSquared()) * (n / (n - getX().getColumnDimension()));
200        } else {
201            return 1 - (calculateResidualSumOfSquares() * (n - 1)) /
202                (calculateTotalSumOfSquares() * (n - getX().getColumnDimension()));
203        }
204    }
205
206    /**
207     * {@inheritDoc}
208     * <p>This implementation computes and caches the QR decomposition of the X matrix
209     * once it is successfully loaded.</p>
210     */
211    @Override
212    protected void newXSampleData(double[][] x) {
213        super.newXSampleData(x);
214        qr = new QRDecomposition(getX());
215    }
216
217    /**
218     * Calculates the regression coefficients using OLS.
219     *
220     * <p>Data for the model must have been successfully loaded using one of
221     * the {@code newSampleData} methods before invoking this method; otherwise
222     * a {@code NullPointerException} will be thrown.</p>
223     *
224     * @return beta
225     */
226    @Override
227    protected RealVector calculateBeta() {
228        return qr.getSolver().solve(getY());
229    }
230
231    /**
232     * <p>Calculates the variance-covariance matrix of the regression parameters.
233     * </p>
234     * <p>Var(b) = (X<sup>T</sup>X)<sup>-1</sup>
235     * </p>
236     * <p>Uses QR decomposition to reduce (X<sup>T</sup>X)<sup>-1</sup>
237     * to (R<sup>T</sup>R)<sup>-1</sup>, with only the top p rows of
238     * R included, where p = the length of the beta vector.</p>
239     *
240     * <p>Data for the model must have been successfully loaded using one of
241     * the {@code newSampleData} methods before invoking this method; otherwise
242     * a {@code NullPointerException} will be thrown.</p>
243     *
244     * @return The beta variance-covariance matrix
245     */
246    @Override
247    protected RealMatrix calculateBetaVariance() {
248        int p = getX().getColumnDimension();
249        RealMatrix Raug = qr.getR().getSubMatrix(0, p - 1 , 0, p - 1);
250        RealMatrix Rinv = new LUDecomposition(Raug).getSolver().getInverse();
251        return Rinv.multiply(Rinv.transpose());
252    }
253
254}