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* 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,
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* See the License for the specific language governing permissions and
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package org.apache.commons.math3.stat.regression;
import org.apache.commons.math3.exception.MathIllegalArgumentException;
import org.apache.commons.math3.linear.Array2DRowRealMatrix;
import org.apache.commons.math3.linear.LUDecomposition;
import org.apache.commons.math3.linear.QRDecomposition;
import org.apache.commons.math3.linear.RealMatrix;
import org.apache.commons.math3.linear.RealVector;
import org.apache.commons.math3.stat.StatUtils;
import org.apache.commons.math3.stat.descriptive.moment.SecondMoment;
/**
* <p>Implements ordinary least squares (OLS) to estimate the parameters of a
* multiple linear regression model.</p>
*
* <p>The regression coefficients, <code>b</code>, satisfy the normal equations:
* <pre><code> X<sup>T</sup> X b = X<sup>T</sup> y </code></pre></p>
*
* <p>To solve the normal equations, this implementation uses QR decomposition
* of the <code>X</code> matrix. (See {@link QRDecomposition} for details on the
* decomposition algorithm.) The <code>X</code> matrix, also known as the <i>design matrix,</i>
* has rows corresponding to sample observations and columns corresponding to independent
* variables. When the model is estimated using an intercept term (i.e. when
* {@link #isNoIntercept() isNoIntercept} is false as it is by default), the <code>X</code>
* matrix includes an initial column identically equal to 1. We solve the normal equations
* as follows:
* <pre><code> X<sup>T</sup>X b = X<sup>T</sup> y
* (QR)<sup>T</sup> (QR) b = (QR)<sup>T</sup>y
* R<sup>T</sup> (Q<sup>T</sup>Q) R b = R<sup>T</sup> Q<sup>T</sup> y
* R<sup>T</sup> R b = R<sup>T</sup> Q<sup>T</sup> y
* (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
* R b = Q<sup>T</sup> y </code></pre></p>
*
* <p>Given <code>Q</code> and <code>R</code>, the last equation is solved by back-substitution.</p>
*
* @since 2.0
*/
public class OLSMultipleLinearRegression extends AbstractMultipleLinearRegression {
/** Cached QR decomposition of X matrix */
private QRDecomposition qr = null;
/** Singularity threshold for QR decomposition */
private final double threshold;
/**
* Create an empty OLSMultipleLinearRegression instance.
*/
public OLSMultipleLinearRegression() {
this(0d);
}
/**
* Create an empty OLSMultipleLinearRegression instance, using the given
* singularity threshold for the QR decomposition.
*
* @param threshold the singularity threshold
* @since 3.3
*/
public OLSMultipleLinearRegression(final double threshold) {
this.threshold = threshold;
}
/**
* Loads model x and y sample data, overriding any previous sample.
*
* Computes and caches QR decomposition of the X matrix.
* @param y the [n,1] array representing the y sample
* @param x the [n,k] array representing the x sample
* @throws MathIllegalArgumentException if the x and y array data are not
* compatible for the regression
*/
public void newSampleData(double[] y, double[][] x) throws MathIllegalArgumentException {
validateSampleData(x, y);
newYSampleData(y);
newXSampleData(x);
}
/**
* {@inheritDoc}
* <p>This implementation computes and caches the QR decomposition of the X matrix.</p>
*/
@Override
public void newSampleData(double[] data, int nobs, int nvars) {
super.newSampleData(data, nobs, nvars);
qr = new QRDecomposition(getX(), threshold);
}
/**
* <p>Compute the "hat" matrix.
* </p>
* <p>The hat matrix is defined in terms of the design matrix X
* by X(X<sup>T</sup>X)<sup>-1</sup>X<sup>T</sup>
* </p>
* <p>The implementation here uses the QR decomposition to compute the
* hat matrix as Q I<sub>p</sub>Q<sup>T</sup> where I<sub>p</sub> is the
* p-dimensional identity matrix augmented by 0's. This computational
* formula is from "The Hat Matrix in Regression and ANOVA",
* David C. Hoaglin and Roy E. Welsch,
* <i>The American Statistician</i>, Vol. 32, No. 1 (Feb., 1978), pp. 17-22.
* </p>
* <p>Data for the model must have been successfully loaded using one of
* the {@code newSampleData} methods before invoking this method; otherwise
* a {@code NullPointerException} will be thrown.</p>
*
* @return the hat matrix
* @throws NullPointerException unless method {@code newSampleData} has been
* called beforehand.
*/
public RealMatrix calculateHat() {
// Create augmented identity matrix
RealMatrix Q = qr.getQ();
final int p = qr.getR().getColumnDimension();
final int n = Q.getColumnDimension();
// No try-catch or advertised NotStrictlyPositiveException - NPE above if n < 3
Array2DRowRealMatrix augI = new Array2DRowRealMatrix(n, n);
double[][] augIData = augI.getDataRef();
for (int i = 0; i < n; i++) {
for (int j =0; j < n; j++) {
if (i == j && i < p) {
augIData[i][j] = 1d;
} else {
augIData[i][j] = 0d;
}
}
}
// Compute and return Hat matrix
// No DME advertised - args valid if we get here
return Q.multiply(augI).multiply(Q.transpose());
}
/**
* <p>Returns the sum of squared deviations of Y from its mean.</p>
*
* <p>If the model has no intercept term, <code>0</code> is used for the
* mean of Y - i.e., what is returned is the sum of the squared Y values.</p>
*
* <p>The value returned by this method is the SSTO value used in
* the {@link #calculateRSquared() R-squared} computation.</p>
*
* @return SSTO - the total sum of squares
* @throws NullPointerException if the sample has not been set
* @see #isNoIntercept()
* @since 2.2
*/
public double calculateTotalSumOfSquares() {
if (isNoIntercept()) {
return StatUtils.sumSq(getY().toArray());
} else {
return new SecondMoment().evaluate(getY().toArray());
}
}
/**
* Returns the sum of squared residuals.
*
* @return residual sum of squares
* @since 2.2
* @throws org.apache.commons.math3.linear.SingularMatrixException if the design matrix is singular
* @throws NullPointerException if the data for the model have not been loaded
*/
public double calculateResidualSumOfSquares() {
final RealVector residuals = calculateResiduals();
// No advertised DME, args are valid
return residuals.dotProduct(residuals);
}
/**
* Returns the R-Squared statistic, defined by the formula <pre>
* R<sup>2</sup> = 1 - SSR / SSTO
* </pre>
* where SSR is the {@link #calculateResidualSumOfSquares() sum of squared residuals}
* and SSTO is the {@link #calculateTotalSumOfSquares() total sum of squares}
*
* <p>If there is no variance in y, i.e., SSTO = 0, NaN is returned.</p>
*
* @return R-square statistic
* @throws NullPointerException if the sample has not been set
* @throws org.apache.commons.math3.linear.SingularMatrixException if the design matrix is singular
* @since 2.2
*/
public double calculateRSquared() {
return 1 - calculateResidualSumOfSquares() / calculateTotalSumOfSquares();
}
/**
* <p>Returns the adjusted R-squared statistic, defined by the formula <pre>
* R<sup>2</sup><sub>adj</sub> = 1 - [SSR (n - 1)] / [SSTO (n - p)]
* </pre>
* where SSR is the {@link #calculateResidualSumOfSquares() sum of squared residuals},
* SSTO is the {@link #calculateTotalSumOfSquares() total sum of squares}, n is the number
* of observations and p is the number of parameters estimated (including the intercept).</p>
*
* <p>If the regression is estimated without an intercept term, what is returned is <pre>
* <code> 1 - (1 - {@link #calculateRSquared()}) * (n / (n - p)) </code>
* </pre></p>
*
* <p>If there is no variance in y, i.e., SSTO = 0, NaN is returned.</p>
*
* @return adjusted R-Squared statistic
* @throws NullPointerException if the sample has not been set
* @throws org.apache.commons.math3.linear.SingularMatrixException if the design matrix is singular
* @see #isNoIntercept()
* @since 2.2
*/
public double calculateAdjustedRSquared() {
final double n = getX().getRowDimension();
if (isNoIntercept()) {
return 1 - (1 - calculateRSquared()) * (n / (n - getX().getColumnDimension()));
} else {
return 1 - (calculateResidualSumOfSquares() * (n - 1)) /
(calculateTotalSumOfSquares() * (n - getX().getColumnDimension()));
}
}
/**
* {@inheritDoc}
* <p>This implementation computes and caches the QR decomposition of the X matrix
* once it is successfully loaded.</p>
*/
@Override
protected void newXSampleData(double[][] x) {
super.newXSampleData(x);
qr = new QRDecomposition(getX(), threshold);
}
/**
* Calculates the regression coefficients using OLS.
*
* <p>Data for the model must have been successfully loaded using one of
* the {@code newSampleData} methods before invoking this method; otherwise
* a {@code NullPointerException} will be thrown.</p>
*
* @return beta
* @throws org.apache.commons.math3.linear.SingularMatrixException if the design matrix is singular
* @throws NullPointerException if the data for the model have not been loaded
*/
@Override
protected RealVector calculateBeta() {
return qr.getSolver().solve(getY());
}
/**
* <p>Calculates the variance-covariance matrix of the regression parameters.
* </p>
* <p>Var(b) = (X<sup>T</sup>X)<sup>-1</sup>
* </p>
* <p>Uses QR decomposition to reduce (X<sup>T</sup>X)<sup>-1</sup>
* to (R<sup>T</sup>R)<sup>-1</sup>, with only the top p rows of
* R included, where p = the length of the beta vector.</p>
*
* <p>Data for the model must have been successfully loaded using one of
* the {@code newSampleData} methods before invoking this method; otherwise
* a {@code NullPointerException} will be thrown.</p>
*
* @return The beta variance-covariance matrix
* @throws org.apache.commons.math3.linear.SingularMatrixException if the design matrix is singular
* @throws NullPointerException if the data for the model have not been loaded
*/
@Override
protected RealMatrix calculateBetaVariance() {
int p = getX().getColumnDimension();
RealMatrix Raug = qr.getR().getSubMatrix(0, p - 1 , 0, p - 1);
RealMatrix Rinv = new LUDecomposition(Raug).getSolver().getInverse();
return Rinv.multiply(Rinv.transpose());
}
}