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Java example source code file (KendallsCorrelation.java)

This example Java source code file (KendallsCorrelation.java) is included in the alvinalexander.com "Java Source Code Warehouse" project. The intent of this project is to help you "Learn Java by Example" TM.

Learn more about this Java project at its project page.

Java - Java tags/keywords

blockrealmatrix, comparator, dimensionmismatchexception, double, kendallscorrelation, pair, realmatrix, suppresswarnings, util

The KendallsCorrelation.java Java example source code

/*
 * 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.stat.correlation;

import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.linear.BlockRealMatrix;
import org.apache.commons.math3.linear.MatrixUtils;
import org.apache.commons.math3.linear.RealMatrix;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.Pair;

import java.util.Arrays;
import java.util.Comparator;

/**
 * Implementation of Kendall's Tau-b rank correlation</a>.
 * <p>
 * A pair of observations (x<sub>1, y1) and
 * (x<sub>2, y2) are considered concordant if
 * x<sub>1 < x2 and y1 < y2
 * or x<sub>2 < x1 and y2 < y1.
 * The pair is <i>discordant if x1 < x2 and
 * y<sub>2 < y1 or x2 < x1 and
 * y<sub>1 < y2.  If either x1 = x2
 * or y<sub>1 = y2, the pair is neither concordant nor
 * discordant.
 * <p>
 * Kendall's Tau-b is defined as:
 * <pre>
 * tau<sub>b = (nc - nd) / sqrt((n0 - n1) * (n0 - n2))
 * </pre>
 * <p>
 * where:
 * <ul>
 *     <li>n0 = n * (n - 1) / 2
 *     <li>nc = Number of concordant pairs
 *     <li>nd = Number of discordant pairs
 *     <li>n1 = sum of ti * (ti - 1) / 2 for all i
 *     <li>n2 = sum of uj * (uj - 1) / 2 for all j
 *     <li>ti = Number of tied values in the ith group of ties in x
 *     <li>uj = Number of tied values in the jth group of ties in y
 * </ul>
 * <p>
 * This implementation uses the O(n log n) algorithm described in
 * William R. Knight's 1966 paper "A Computer Method for Calculating
 * Kendall's Tau with Ungrouped Data" in the Journal of the American
 * Statistical Association.
 *
 * @see <a href="http://en.wikipedia.org/wiki/Kendall_tau_rank_correlation_coefficient">
 * Kendall tau rank correlation coefficient (Wikipedia)</a>
 * @see <a href="http://www.jstor.org/stable/2282833">A Computer
 * Method for Calculating Kendall's Tau with Ungrouped Data</a>
 *
 * @since 3.3
 */
public class KendallsCorrelation {

    /** correlation matrix */
    private final RealMatrix correlationMatrix;

    /**
     * Create a KendallsCorrelation instance without data.
     */
    public KendallsCorrelation() {
        correlationMatrix = null;
    }

    /**
     * Create a KendallsCorrelation from a rectangular array
     * whose columns represent values of variables to be correlated.
     *
     * @param data rectangular array with columns representing variables
     * @throws IllegalArgumentException if the input data array is not
     * rectangular with at least two rows and two columns.
     */
    public KendallsCorrelation(double[][] data) {
        this(MatrixUtils.createRealMatrix(data));
    }

    /**
     * Create a KendallsCorrelation from a RealMatrix whose columns
     * represent variables to be correlated.
     *
     * @param matrix matrix with columns representing variables to correlate
     */
    public KendallsCorrelation(RealMatrix matrix) {
        correlationMatrix = computeCorrelationMatrix(matrix);
    }

    /**
     * Returns the correlation matrix.
     *
     * @return correlation matrix
     */
    public RealMatrix getCorrelationMatrix() {
        return correlationMatrix;
    }

    /**
     * Computes the Kendall's Tau rank correlation matrix for the columns of
     * the input matrix.
     *
     * @param matrix matrix with columns representing variables to correlate
     * @return correlation matrix
     */
    public RealMatrix computeCorrelationMatrix(final RealMatrix matrix) {
        int nVars = matrix.getColumnDimension();
        RealMatrix outMatrix = new BlockRealMatrix(nVars, nVars);
        for (int i = 0; i < nVars; i++) {
            for (int j = 0; j < i; j++) {
                double corr = correlation(matrix.getColumn(i), matrix.getColumn(j));
                outMatrix.setEntry(i, j, corr);
                outMatrix.setEntry(j, i, corr);
            }
            outMatrix.setEntry(i, i, 1d);
        }
        return outMatrix;
    }

    /**
     * Computes the Kendall's Tau rank correlation matrix for the columns of
     * the input rectangular array.  The columns of the array represent values
     * of variables to be correlated.
     *
     * @param matrix matrix with columns representing variables to correlate
     * @return correlation matrix
     */
    public RealMatrix computeCorrelationMatrix(final double[][] matrix) {
       return computeCorrelationMatrix(new BlockRealMatrix(matrix));
    }

    /**
     * Computes the Kendall's Tau rank correlation coefficient between the two arrays.
     *
     * @param xArray first data array
     * @param yArray second data array
     * @return Returns Kendall's Tau rank correlation coefficient for the two arrays
     * @throws DimensionMismatchException if the arrays lengths do not match
     */
    public double correlation(final double[] xArray, final double[] yArray)
            throws DimensionMismatchException {

        if (xArray.length != yArray.length) {
            throw new DimensionMismatchException(xArray.length, yArray.length);
        }

        final int n = xArray.length;
        final long numPairs = sum(n - 1);

        @SuppressWarnings("unchecked")
        Pair<Double, Double>[] pairs = new Pair[n];
        for (int i = 0; i < n; i++) {
            pairs[i] = new Pair<Double, Double>(xArray[i], yArray[i]);
        }

        Arrays.sort(pairs, new Comparator<Pair() {
            /** {@inheritDoc} */
            public int compare(Pair<Double, Double> pair1, Pair pair2) {
                int compareFirst = pair1.getFirst().compareTo(pair2.getFirst());
                return compareFirst != 0 ? compareFirst : pair1.getSecond().compareTo(pair2.getSecond());
            }
        });

        long tiedXPairs = 0;
        long tiedXYPairs = 0;
        long consecutiveXTies = 1;
        long consecutiveXYTies = 1;
        Pair<Double, Double> prev = pairs[0];
        for (int i = 1; i < n; i++) {
            final Pair<Double, Double> curr = pairs[i];
            if (curr.getFirst().equals(prev.getFirst())) {
                consecutiveXTies++;
                if (curr.getSecond().equals(prev.getSecond())) {
                    consecutiveXYTies++;
                } else {
                    tiedXYPairs += sum(consecutiveXYTies - 1);
                    consecutiveXYTies = 1;
                }
            } else {
                tiedXPairs += sum(consecutiveXTies - 1);
                consecutiveXTies = 1;
                tiedXYPairs += sum(consecutiveXYTies - 1);
                consecutiveXYTies = 1;
            }
            prev = curr;
        }
        tiedXPairs += sum(consecutiveXTies - 1);
        tiedXYPairs += sum(consecutiveXYTies - 1);

        long swaps = 0;
        @SuppressWarnings("unchecked")
        Pair<Double, Double>[] pairsDestination = new Pair[n];
        for (int segmentSize = 1; segmentSize < n; segmentSize <<= 1) {
            for (int offset = 0; offset < n; offset += 2 * segmentSize) {
                int i = offset;
                final int iEnd = FastMath.min(i + segmentSize, n);
                int j = iEnd;
                final int jEnd = FastMath.min(j + segmentSize, n);

                int copyLocation = offset;
                while (i < iEnd || j < jEnd) {
                    if (i < iEnd) {
                        if (j < jEnd) {
                            if (pairs[i].getSecond().compareTo(pairs[j].getSecond()) <= 0) {
                                pairsDestination[copyLocation] = pairs[i];
                                i++;
                            } else {
                                pairsDestination[copyLocation] = pairs[j];
                                j++;
                                swaps += iEnd - i;
                            }
                        } else {
                            pairsDestination[copyLocation] = pairs[i];
                            i++;
                        }
                    } else {
                        pairsDestination[copyLocation] = pairs[j];
                        j++;
                    }
                    copyLocation++;
                }
            }
            final Pair<Double, Double>[] pairsTemp = pairs;
            pairs = pairsDestination;
            pairsDestination = pairsTemp;
        }

        long tiedYPairs = 0;
        long consecutiveYTies = 1;
        prev = pairs[0];
        for (int i = 1; i < n; i++) {
            final Pair<Double, Double> curr = pairs[i];
            if (curr.getSecond().equals(prev.getSecond())) {
                consecutiveYTies++;
            } else {
                tiedYPairs += sum(consecutiveYTies - 1);
                consecutiveYTies = 1;
            }
            prev = curr;
        }
        tiedYPairs += sum(consecutiveYTies - 1);

        final long concordantMinusDiscordant = numPairs - tiedXPairs - tiedYPairs + tiedXYPairs - 2 * swaps;
        final double nonTiedPairsMultiplied = (numPairs - tiedXPairs) * (double) (numPairs - tiedYPairs);
        return concordantMinusDiscordant / FastMath.sqrt(nonTiedPairsMultiplied);
    }

    /**
     * Returns the sum of the number from 1 .. n according to Gauss' summation formula:
     * \[ \sum\limits_{k=1}^n k = \frac{n(n + 1)}{2} \]
     *
     * @param n the summation end
     * @return the sum of the number from 1 to n
     */
    private static long sum(long n) {
        return n * (n + 1) / 2l;
    }
}

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