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.math4.stat.correlation;
018
019import org.apache.commons.statistics.distribution.TDistribution;
020import org.apache.commons.math4.exception.DimensionMismatchException;
021import org.apache.commons.math4.exception.MathIllegalArgumentException;
022import org.apache.commons.math4.exception.NullArgumentException;
023import org.apache.commons.math4.exception.util.LocalizedFormats;
024import org.apache.commons.math4.linear.BlockRealMatrix;
025import org.apache.commons.math4.linear.RealMatrix;
026import org.apache.commons.math4.stat.regression.SimpleRegression;
027import org.apache.commons.math4.util.FastMath;
028
029/**
030 * Computes Pearson's product-moment correlation coefficients for pairs of arrays
031 * or columns of a matrix.
032 *
033 * <p>The constructors that take <code>RealMatrix</code> or
034 * <code>double[][]</code> arguments generate correlation matrices.  The
035 * columns of the input matrices are assumed to represent variable values.
036 * Correlations are given by the formula</p>
037 *
038 * <p><code>cor(X, Y) = &Sigma;[(x<sub>i</sub> - E(X))(y<sub>i</sub> - E(Y))] / [(n - 1)s(X)s(Y)]</code>
039 * where <code>E(X)</code> is the mean of <code>X</code>, <code>E(Y)</code>
040 * is the mean of the <code>Y</code> values and s(X), s(Y) are standard deviations.</p>
041 *
042 * <p>To compute the correlation coefficient for a single pair of arrays, use {@link #PearsonsCorrelation()}
043 * to construct an instance with no data and then {@link #correlation(double[], double[])}.
044 * Correlation matrices can also be computed directly from an instance with no data using
045 * {@link #computeCorrelationMatrix(double[][])}. In order to use {@link #getCorrelationMatrix()},
046 * {@link #getCorrelationPValues()},  or {@link #getCorrelationStandardErrors()}; however, one of the
047 * constructors supplying data or a covariance matrix must be used to create the instance.</p>
048 *
049 * @since 2.0
050 */
051public class PearsonsCorrelation {
052
053    /** correlation matrix */
054    private final RealMatrix correlationMatrix;
055
056    /** number of observations */
057    private final int nObs;
058
059    /**
060     * Create a PearsonsCorrelation instance without data.
061     */
062    public PearsonsCorrelation() {
063        super();
064        correlationMatrix = null;
065        nObs = 0;
066    }
067
068    /**
069     * Create a PearsonsCorrelation from a rectangular array
070     * whose columns represent values of variables to be correlated.
071     *
072     * Throws MathIllegalArgumentException if the input array does not have at least
073     * two columns and two rows.  Pairwise correlations are set to NaN if one
074     * of the correlates has zero variance.
075     *
076     * @param data rectangular array with columns representing variables
077     * @throws MathIllegalArgumentException if the input data array is not
078     * rectangular with at least two rows and two columns.
079     * @see #correlation(double[], double[])
080     */
081    public PearsonsCorrelation(double[][] data) {
082        this(new BlockRealMatrix(data));
083    }
084
085    /**
086     * Create a PearsonsCorrelation from a RealMatrix whose columns
087     * represent variables to be correlated.
088     *
089     * Throws MathIllegalArgumentException if the matrix does not have at least
090     * two columns and two rows.  Pairwise correlations are set to NaN if one
091     * of the correlates has zero variance.
092     *
093     * @param matrix matrix with columns representing variables to correlate
094     * @throws MathIllegalArgumentException if the matrix does not contain sufficient data
095     * @see #correlation(double[], double[])
096     */
097    public PearsonsCorrelation(RealMatrix matrix) {
098        nObs = matrix.getRowDimension();
099        correlationMatrix = computeCorrelationMatrix(matrix);
100    }
101
102    /**
103     * Create a PearsonsCorrelation from a {@link Covariance}.  The correlation
104     * matrix is computed by scaling the Covariance's covariance matrix.
105     * The Covariance instance must have been created from a data matrix with
106     * columns representing variable values.
107     *
108     * @param covariance Covariance instance
109     */
110    public PearsonsCorrelation(Covariance covariance) {
111        RealMatrix covarianceMatrix = covariance.getCovarianceMatrix();
112        if (covarianceMatrix == null) {
113            throw new NullArgumentException(LocalizedFormats.COVARIANCE_MATRIX);
114        }
115        nObs = covariance.getN();
116        correlationMatrix = covarianceToCorrelation(covarianceMatrix);
117    }
118
119    /**
120     * Create a PearsonsCorrelation from a covariance matrix. The correlation
121     * matrix is computed by scaling the covariance matrix.
122     *
123     * @param covarianceMatrix covariance matrix
124     * @param numberOfObservations the number of observations in the dataset used to compute
125     * the covariance matrix
126     */
127    public PearsonsCorrelation(RealMatrix covarianceMatrix, int numberOfObservations) {
128        nObs = numberOfObservations;
129        correlationMatrix = covarianceToCorrelation(covarianceMatrix);
130    }
131
132    /**
133     * Returns the correlation matrix.
134     *
135     * <p>This method will return null if the non-argument constructor was used
136     * to create this instance, even if {@link #computeCorrelationMatrix(double[][])}
137     * has been called before it is activated.</p>
138     *
139     * @return correlation matrix
140     */
141    public RealMatrix getCorrelationMatrix() {
142        return correlationMatrix;
143    }
144
145    /**
146     * Returns a matrix of standard errors associated with the estimates
147     * in the correlation matrix.<br>
148     * <code>getCorrelationStandardErrors().getEntry(i,j)</code> is the standard
149     * error associated with <code>getCorrelationMatrix.getEntry(i,j)</code>
150     *
151     * <p>The formula used to compute the standard error is <br>
152     * <code>SE<sub>r</sub> = ((1 - r<sup>2</sup>) / (n - 2))<sup>1/2</sup></code>
153     * where <code>r</code> is the estimated correlation coefficient and
154     * <code>n</code> is the number of observations in the source dataset.</p>
155     *
156     * <p>To use this method, one of the constructors that supply an input
157     * matrix must have been used to create this instance.</p>
158     *
159     * @return matrix of correlation standard errors
160     * @throws NullPointerException if this instance was created with no data
161     */
162    public RealMatrix getCorrelationStandardErrors() {
163        int nVars = correlationMatrix.getColumnDimension();
164        double[][] out = new double[nVars][nVars];
165        for (int i = 0; i < nVars; i++) {
166            for (int j = 0; j < nVars; j++) {
167                double r = correlationMatrix.getEntry(i, j);
168                out[i][j] = FastMath.sqrt((1 - r * r) /(nObs - 2));
169            }
170        }
171        return new BlockRealMatrix(out);
172    }
173
174    /**
175     * Returns a matrix of p-values associated with the (two-sided) null
176     * hypothesis that the corresponding correlation coefficient is zero.
177     *
178     * <p><code>getCorrelationPValues().getEntry(i,j)</code> is the probability
179     * that a random variable distributed as <code>t<sub>n-2</sub></code> takes
180     * a value with absolute value greater than or equal to <br>
181     * <code>|r|((n - 2) / (1 - r<sup>2</sup>))<sup>1/2</sup></code></p>
182     *
183     * <p>The values in the matrix are sometimes referred to as the
184     * <i>significance</i> of the corresponding correlation coefficients.</p>
185     *
186     * <p>To use this method, one of the constructors that supply an input
187     * matrix must have been used to create this instance.</p>
188     *
189     * @return matrix of p-values
190     * @throws org.apache.commons.math4.exception.MaxCountExceededException
191     * if an error occurs estimating probabilities
192     * @throws NullPointerException if this instance was created with no data
193     */
194    public RealMatrix getCorrelationPValues() {
195        TDistribution tDistribution = new TDistribution(nObs - 2);
196        int nVars = correlationMatrix.getColumnDimension();
197        double[][] out = new double[nVars][nVars];
198        for (int i = 0; i < nVars; i++) {
199            for (int j = 0; j < nVars; j++) {
200                if (i == j) {
201                    out[i][j] = 0d;
202                } else {
203                    double r = correlationMatrix.getEntry(i, j);
204                    double t = FastMath.abs(r * FastMath.sqrt((nObs - 2)/(1 - r * r)));
205                    out[i][j] = 2 * tDistribution.cumulativeProbability(-t);
206                }
207            }
208        }
209        return new BlockRealMatrix(out);
210    }
211
212
213    /**
214     * Computes the correlation matrix for the columns of the
215     * input matrix, using {@link #correlation(double[], double[])}.
216     *
217     * Throws MathIllegalArgumentException if the matrix does not have at least
218     * two columns and two rows.  Pairwise correlations are set to NaN if one
219     * of the correlates has zero variance.
220     *
221     * @param matrix matrix with columns representing variables to correlate
222     * @return correlation matrix
223     * @throws MathIllegalArgumentException if the matrix does not contain sufficient data
224     * @see #correlation(double[], double[])
225     */
226    public RealMatrix computeCorrelationMatrix(RealMatrix matrix) {
227        checkSufficientData(matrix);
228        int nVars = matrix.getColumnDimension();
229        RealMatrix outMatrix = new BlockRealMatrix(nVars, nVars);
230        for (int i = 0; i < nVars; i++) {
231            for (int j = 0; j < i; j++) {
232              double corr = correlation(matrix.getColumn(i), matrix.getColumn(j));
233              outMatrix.setEntry(i, j, corr);
234              outMatrix.setEntry(j, i, corr);
235            }
236            outMatrix.setEntry(i, i, 1d);
237        }
238        return outMatrix;
239    }
240
241    /**
242     * Computes the correlation matrix for the columns of the
243     * input rectangular array.  The columns of the array represent values
244     * of variables to be correlated.
245     *
246     * Throws MathIllegalArgumentException if the matrix does not have at least
247     * two columns and two rows or if the array is not rectangular. Pairwise
248     * correlations are set to NaN if one of the correlates has zero variance.
249     *
250     * @param data matrix with columns representing variables to correlate
251     * @return correlation matrix
252     * @throws MathIllegalArgumentException if the array does not contain sufficient data
253     * @see #correlation(double[], double[])
254     */
255    public RealMatrix computeCorrelationMatrix(double[][] data) {
256       return computeCorrelationMatrix(new BlockRealMatrix(data));
257    }
258
259    /**
260     * Computes the Pearson's product-moment correlation coefficient between two arrays.
261     *
262     * <p>Throws MathIllegalArgumentException if the arrays do not have the same length
263     * or their common length is less than 2.  Returns {@code NaN} if either of the arrays
264     * has zero variance (i.e., if one of the arrays does not contain at least two distinct
265     * values).</p>
266     *
267     * @param xArray first data array
268     * @param yArray second data array
269     * @return Returns Pearson's correlation coefficient for the two arrays
270     * @throws DimensionMismatchException if the arrays lengths do not match
271     * @throws MathIllegalArgumentException if there is insufficient data
272     */
273    public double correlation(final double[] xArray, final double[] yArray) {
274        SimpleRegression regression = new SimpleRegression();
275        if (xArray.length != yArray.length) {
276            throw new DimensionMismatchException(xArray.length, yArray.length);
277        } else if (xArray.length < 2) {
278            throw new MathIllegalArgumentException(LocalizedFormats.INSUFFICIENT_DIMENSION,
279                                                   xArray.length, 2);
280        } else {
281            for(int i=0; i<xArray.length; i++) {
282                regression.addData(xArray[i], yArray[i]);
283            }
284            return regression.getR();
285        }
286    }
287
288    /**
289     * Derives a correlation matrix from a covariance matrix.
290     *
291     * <p>Uses the formula <br>
292     * <code>r(X,Y) = cov(X,Y)/s(X)s(Y)</code> where
293     * <code>r(&middot;,&middot;)</code> is the correlation coefficient and
294     * <code>s(&middot;)</code> means standard deviation.</p>
295     *
296     * @param covarianceMatrix the covariance matrix
297     * @return correlation matrix
298     */
299    public RealMatrix covarianceToCorrelation(RealMatrix covarianceMatrix) {
300        int nVars = covarianceMatrix.getColumnDimension();
301        RealMatrix outMatrix = new BlockRealMatrix(nVars, nVars);
302        for (int i = 0; i < nVars; i++) {
303            double sigma = FastMath.sqrt(covarianceMatrix.getEntry(i, i));
304            outMatrix.setEntry(i, i, 1d);
305            for (int j = 0; j < i; j++) {
306                double entry = covarianceMatrix.getEntry(i, j) /
307                       (sigma * FastMath.sqrt(covarianceMatrix.getEntry(j, j)));
308                outMatrix.setEntry(i, j, entry);
309                outMatrix.setEntry(j, i, entry);
310            }
311        }
312        return outMatrix;
313    }
314
315    /**
316     * Throws MathIllegalArgumentException if the matrix does not have at least
317     * two columns and two rows.
318     *
319     * @param matrix matrix to check for sufficiency
320     * @throws MathIllegalArgumentException if there is insufficient data
321     */
322    private void checkSufficientData(final RealMatrix matrix) {
323        int nRows = matrix.getRowDimension();
324        int nCols = matrix.getColumnDimension();
325        if (nRows < 2 || nCols < 2) {
326            throw new MathIllegalArgumentException(LocalizedFormats.INSUFFICIENT_ROWS_AND_COLUMNS,
327                                                   nRows, nCols);
328        }
329    }
330}