Java tutorial
/* * 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.inference; import org.apache.commons.math3.distribution.ChiSquaredDistribution; import org.apache.commons.math3.exception.DimensionMismatchException; import org.apache.commons.math3.exception.MaxCountExceededException; import org.apache.commons.math3.exception.NotPositiveException; import org.apache.commons.math3.exception.NotStrictlyPositiveException; import org.apache.commons.math3.exception.NullArgumentException; import org.apache.commons.math3.exception.OutOfRangeException; import org.apache.commons.math3.exception.ZeroException; import org.apache.commons.math3.exception.util.LocalizedFormats; import org.apache.commons.math3.util.FastMath; import org.apache.commons.math3.util.MathArrays; /** * Implements Chi-Square test statistics. * * <p>This implementation handles both known and unknown distributions.</p> * * <p>Two samples tests can be used when the distribution is unknown <i>a priori</i> * but provided by one sample, or when the hypothesis under test is that the two * samples come from the same underlying distribution.</p> * */ public class ChiSquareTest { /** * Construct a ChiSquareTest */ public ChiSquareTest() { super(); } /** * Computes the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda35f.htm"> * Chi-Square statistic</a> comparing <code>observed</code> and <code>expected</code> * frequency counts. * <p> * This statistic can be used to perform a Chi-Square test evaluating the null * hypothesis that the observed counts follow the expected distribution.</p> * <p> * <strong>Preconditions</strong>: <ul> * <li>Expected counts must all be positive. * </li> * <li>Observed counts must all be ≥ 0. * </li> * <li>The observed and expected arrays must have the same length and * their common length must be at least 2. * </li></ul></p><p> * If any of the preconditions are not met, an * <code>IllegalArgumentException</code> is thrown.</p> * <p><strong>Note: </strong>This implementation rescales the * <code>expected</code> array if necessary to ensure that the sum of the * expected and observed counts are equal.</p> * * @param observed array of observed frequency counts * @param expected array of expected frequency counts * @return chiSquare test statistic * @throws NotPositiveException if <code>observed</code> has negative entries * @throws NotStrictlyPositiveException if <code>expected</code> has entries that are * not strictly positive * @throws DimensionMismatchException if the arrays length is less than 2 */ public double chiSquare(final double[] expected, final long[] observed) throws NotPositiveException, NotStrictlyPositiveException, DimensionMismatchException { if (expected.length < 2) { throw new DimensionMismatchException(expected.length, 2); } if (expected.length != observed.length) { throw new DimensionMismatchException(expected.length, observed.length); } MathArrays.checkPositive(expected); MathArrays.checkNonNegative(observed); double sumExpected = 0d; double sumObserved = 0d; for (int i = 0; i < observed.length; i++) { sumExpected += expected[i]; sumObserved += observed[i]; } double ratio = 1.0d; boolean rescale = false; if (FastMath.abs(sumExpected - sumObserved) > 10E-6) { ratio = sumObserved / sumExpected; rescale = true; } double sumSq = 0.0d; for (int i = 0; i < observed.length; i++) { if (rescale) { final double dev = observed[i] - ratio * expected[i]; sumSq += dev * dev / (ratio * expected[i]); } else { final double dev = observed[i] - expected[i]; sumSq += dev * dev / expected[i]; } } return sumSq; } /** * Returns the <i>observed significance level</i>, or <a href= * "http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue"> * p-value</a>, associated with a * <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda35f.htm"> * Chi-square goodness of fit test</a> comparing the <code>observed</code> * frequency counts to those in the <code>expected</code> array. * <p> * The number returned is the smallest significance level at which one can reject * the null hypothesis that the observed counts conform to the frequency distribution * described by the expected counts.</p> * <p> * <strong>Preconditions</strong>: <ul> * <li>Expected counts must all be positive. * </li> * <li>Observed counts must all be ≥ 0. * </li> * <li>The observed and expected arrays must have the same length and * their common length must be at least 2. * </li></ul></p><p> * If any of the preconditions are not met, an * <code>IllegalArgumentException</code> is thrown.</p> * <p><strong>Note: </strong>This implementation rescales the * <code>expected</code> array if necessary to ensure that the sum of the * expected and observed counts are equal.</p> * * @param observed array of observed frequency counts * @param expected array of expected frequency counts * @return p-value * @throws NotPositiveException if <code>observed</code> has negative entries * @throws NotStrictlyPositiveException if <code>expected</code> has entries that are * not strictly positive * @throws DimensionMismatchException if the arrays length is less than 2 * @throws MaxCountExceededException if an error occurs computing the p-value */ public double chiSquareTest(final double[] expected, final long[] observed) throws NotPositiveException, NotStrictlyPositiveException, DimensionMismatchException, MaxCountExceededException { // pass a null rng to avoid unneeded overhead as we will not sample from this distribution final ChiSquaredDistribution distribution = new ChiSquaredDistribution(null, expected.length - 1.0); return 1.0 - distribution.cumulativeProbability(chiSquare(expected, observed)); } /** * Performs a <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda35f.htm"> * Chi-square goodness of fit test</a> evaluating the null hypothesis that the * observed counts conform to the frequency distribution described by the expected * counts, with significance level <code>alpha</code>. Returns true iff the null * hypothesis can be rejected with 100 * (1 - alpha) percent confidence. * <p> * <strong>Example:</strong><br> * To test the hypothesis that <code>observed</code> follows * <code>expected</code> at the 99% level, use </p><p> * <code>chiSquareTest(expected, observed, 0.01) </code></p> * <p> * <strong>Preconditions</strong>: <ul> * <li>Expected counts must all be positive. * </li> * <li>Observed counts must all be ≥ 0. * </li> * <li>The observed and expected arrays must have the same length and * their common length must be at least 2. * <li> <code> 0 < alpha < 0.5 </code> * </li></ul></p><p> * If any of the preconditions are not met, an * <code>IllegalArgumentException</code> is thrown.</p> * <p><strong>Note: </strong>This implementation rescales the * <code>expected</code> array if necessary to ensure that the sum of the * expected and observed counts are equal.</p> * * @param observed array of observed frequency counts * @param expected array of expected frequency counts * @param alpha significance level of the test * @return true iff null hypothesis can be rejected with confidence * 1 - alpha * @throws NotPositiveException if <code>observed</code> has negative entries * @throws NotStrictlyPositiveException if <code>expected</code> has entries that are * not strictly positive * @throws DimensionMismatchException if the arrays length is less than 2 * @throws OutOfRangeException if <code>alpha</code> is not in the range (0, 0.5] * @throws MaxCountExceededException if an error occurs computing the p-value */ public boolean chiSquareTest(final double[] expected, final long[] observed, final double alpha) throws NotPositiveException, NotStrictlyPositiveException, DimensionMismatchException, OutOfRangeException, MaxCountExceededException { if ((alpha <= 0) || (alpha > 0.5)) { throw new OutOfRangeException(LocalizedFormats.OUT_OF_BOUND_SIGNIFICANCE_LEVEL, alpha, 0, 0.5); } return chiSquareTest(expected, observed) < alpha; } /** * Computes the Chi-Square statistic associated with a * <a href="http://www.itl.nist.gov/div898/handbook/prc/section4/prc45.htm"> * chi-square test of independence</a> based on the input <code>counts</code> * array, viewed as a two-way table. * <p> * The rows of the 2-way table are * <code>count[0], ... , count[count.length - 1] </code></p> * <p> * <strong>Preconditions</strong>: <ul> * <li>All counts must be ≥ 0. * </li> * <li>The count array must be rectangular (i.e. all count[i] subarrays * must have the same length). * </li> * <li>The 2-way table represented by <code>counts</code> must have at * least 2 columns and at least 2 rows. * </li> * </li></ul></p><p> * If any of the preconditions are not met, an * <code>IllegalArgumentException</code> is thrown.</p> * * @param counts array representation of 2-way table * @return chiSquare test statistic * @throws NullArgumentException if the array is null * @throws DimensionMismatchException if the array is not rectangular * @throws NotPositiveException if {@code counts} has negative entries */ public double chiSquare(final long[][] counts) throws NullArgumentException, NotPositiveException, DimensionMismatchException { checkArray(counts); int nRows = counts.length; int nCols = counts[0].length; // compute row, column and total sums double[] rowSum = new double[nRows]; double[] colSum = new double[nCols]; double total = 0.0d; for (int row = 0; row < nRows; row++) { for (int col = 0; col < nCols; col++) { rowSum[row] += counts[row][col]; colSum[col] += counts[row][col]; total += counts[row][col]; } } // compute expected counts and chi-square double sumSq = 0.0d; double expected = 0.0d; for (int row = 0; row < nRows; row++) { for (int col = 0; col < nCols; col++) { expected = (rowSum[row] * colSum[col]) / total; sumSq += ((counts[row][col] - expected) * (counts[row][col] - expected)) / expected; } } return sumSq; } /** * Returns the <i>observed significance level</i>, or <a href= * "http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue"> * p-value</a>, associated with a * <a href="http://www.itl.nist.gov/div898/handbook/prc/section4/prc45.htm"> * chi-square test of independence</a> based on the input <code>counts</code> * array, viewed as a two-way table. * <p> * The rows of the 2-way table are * <code>count[0], ... , count[count.length - 1] </code></p> * <p> * <strong>Preconditions</strong>: <ul> * <li>All counts must be ≥ 0. * </li> * <li>The count array must be rectangular (i.e. all count[i] subarrays must have * the same length). * </li> * <li>The 2-way table represented by <code>counts</code> must have at least 2 * columns and at least 2 rows. * </li> * </li></ul></p><p> * If any of the preconditions are not met, an * <code>IllegalArgumentException</code> is thrown.</p> * * @param counts array representation of 2-way table * @return p-value * @throws NullArgumentException if the array is null * @throws DimensionMismatchException if the array is not rectangular * @throws NotPositiveException if {@code counts} has negative entries * @throws MaxCountExceededException if an error occurs computing the p-value */ public double chiSquareTest(final long[][] counts) throws NullArgumentException, DimensionMismatchException, NotPositiveException, MaxCountExceededException { checkArray(counts); double df = ((double) counts.length - 1) * ((double) counts[0].length - 1); // pass a null rng to avoid unneeded overhead as we will not sample from this distribution final ChiSquaredDistribution distribution = new ChiSquaredDistribution(df); return 1 - distribution.cumulativeProbability(chiSquare(counts)); } /** * Performs a <a href="http://www.itl.nist.gov/div898/handbook/prc/section4/prc45.htm"> * chi-square test of independence</a> evaluating the null hypothesis that the * classifications represented by the counts in the columns of the input 2-way table * are independent of the rows, with significance level <code>alpha</code>. * Returns true iff the null hypothesis can be rejected with 100 * (1 - alpha) percent * confidence. * <p> * The rows of the 2-way table are * <code>count[0], ... , count[count.length - 1] </code></p> * <p> * <strong>Example:</strong><br> * To test the null hypothesis that the counts in * <code>count[0], ... , count[count.length - 1] </code> * all correspond to the same underlying probability distribution at the 99% level, use</p> * <p><code>chiSquareTest(counts, 0.01)</code></p> * <p> * <strong>Preconditions</strong>: <ul> * <li>All counts must be ≥ 0. * </li> * <li>The count array must be rectangular (i.e. all count[i] subarrays must have the * same length).</li> * <li>The 2-way table represented by <code>counts</code> must have at least 2 columns and * at least 2 rows.</li> * </li></ul></p><p> * If any of the preconditions are not met, an * <code>IllegalArgumentException</code> is thrown.</p> * * @param counts array representation of 2-way table * @param alpha significance level of the test * @return true iff null hypothesis can be rejected with confidence * 1 - alpha * @throws NullArgumentException if the array is null * @throws DimensionMismatchException if the array is not rectangular * @throws NotPositiveException if {@code counts} has any negative entries * @throws OutOfRangeException if <code>alpha</code> is not in the range (0, 0.5] * @throws MaxCountExceededException if an error occurs computing the p-value */ public boolean chiSquareTest(final long[][] counts, final double alpha) throws NullArgumentException, DimensionMismatchException, NotPositiveException, OutOfRangeException, MaxCountExceededException { if ((alpha <= 0) || (alpha > 0.5)) { throw new OutOfRangeException(LocalizedFormats.OUT_OF_BOUND_SIGNIFICANCE_LEVEL, alpha, 0, 0.5); } return chiSquareTest(counts) < alpha; } /** * <p>Computes a * <a href="http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/chi2samp.htm"> * Chi-Square two sample test statistic</a> comparing bin frequency counts * in <code>observed1</code> and <code>observed2</code>. The * sums of frequency counts in the two samples are not required to be the * same. The formula used to compute the test statistic is</p> * <code> * ∑[(K * observed1[i] - observed2[i]/K)<sup>2</sup> / (observed1[i] + observed2[i])] * </code> where * <br/><code>K = &sqrt;[&sum(observed2 / ∑(observed1)]</code> * </p> * <p>This statistic can be used to perform a Chi-Square test evaluating the * null hypothesis that both observed counts follow the same distribution.</p> * <p> * <strong>Preconditions</strong>: <ul> * <li>Observed counts must be non-negative. * </li> * <li>Observed counts for a specific bin must not both be zero. * </li> * <li>Observed counts for a specific sample must not all be 0. * </li> * <li>The arrays <code>observed1</code> and <code>observed2</code> must have * the same length and their common length must be at least 2. * </li></ul></p><p> * If any of the preconditions are not met, an * <code>IllegalArgumentException</code> is thrown.</p> * * @param observed1 array of observed frequency counts of the first data set * @param observed2 array of observed frequency counts of the second data set * @return chiSquare test statistic * @throws DimensionMismatchException the the length of the arrays does not match * @throws NotPositiveException if any entries in <code>observed1</code> or * <code>observed2</code> are negative * @throws ZeroException if either all counts of <code>observed1</code> or * <code>observed2</code> are zero, or if the count at some index is zero * for both arrays * @since 1.2 */ public double chiSquareDataSetsComparison(long[] observed1, long[] observed2) throws DimensionMismatchException, NotPositiveException, ZeroException { // Make sure lengths are same if (observed1.length < 2) { throw new DimensionMismatchException(observed1.length, 2); } if (observed1.length != observed2.length) { throw new DimensionMismatchException(observed1.length, observed2.length); } // Ensure non-negative counts MathArrays.checkNonNegative(observed1); MathArrays.checkNonNegative(observed2); // Compute and compare count sums long countSum1 = 0; long countSum2 = 0; boolean unequalCounts = false; double weight = 0.0; for (int i = 0; i < observed1.length; i++) { countSum1 += observed1[i]; countSum2 += observed2[i]; } // Ensure neither sample is uniformly 0 if (countSum1 == 0 || countSum2 == 0) { throw new ZeroException(); } // Compare and compute weight only if different unequalCounts = countSum1 != countSum2; if (unequalCounts) { weight = FastMath.sqrt((double) countSum1 / (double) countSum2); } // Compute ChiSquare statistic double sumSq = 0.0d; double dev = 0.0d; double obs1 = 0.0d; double obs2 = 0.0d; for (int i = 0; i < observed1.length; i++) { if (observed1[i] == 0 && observed2[i] == 0) { throw new ZeroException(LocalizedFormats.OBSERVED_COUNTS_BOTTH_ZERO_FOR_ENTRY, i); } else { obs1 = observed1[i]; obs2 = observed2[i]; if (unequalCounts) { // apply weights dev = obs1 / weight - obs2 * weight; } else { dev = obs1 - obs2; } sumSq += (dev * dev) / (obs1 + obs2); } } return sumSq; } /** * <p>Returns the <i>observed significance level</i>, or <a href= * "http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue"> * p-value</a>, associated with a Chi-Square two sample test comparing * bin frequency counts in <code>observed1</code> and * <code>observed2</code>. * </p> * <p>The number returned is the smallest significance level at which one * can reject the null hypothesis that the observed counts conform to the * same distribution. * </p> * <p>See {@link #chiSquareDataSetsComparison(long[], long[])} for details * on the formula used to compute the test statistic. The degrees of * of freedom used to perform the test is one less than the common length * of the input observed count arrays. * </p> * <strong>Preconditions</strong>: <ul> * <li>Observed counts must be non-negative. * </li> * <li>Observed counts for a specific bin must not both be zero. * </li> * <li>Observed counts for a specific sample must not all be 0. * </li> * <li>The arrays <code>observed1</code> and <code>observed2</code> must * have the same length and * their common length must be at least 2. * </li></ul><p> * If any of the preconditions are not met, an * <code>IllegalArgumentException</code> is thrown.</p> * * @param observed1 array of observed frequency counts of the first data set * @param observed2 array of observed frequency counts of the second data set * @return p-value * @throws DimensionMismatchException the the length of the arrays does not match * @throws NotPositiveException if any entries in <code>observed1</code> or * <code>observed2</code> are negative * @throws ZeroException if either all counts of <code>observed1</code> or * <code>observed2</code> are zero, or if the count at the same index is zero * for both arrays * @throws MaxCountExceededException if an error occurs computing the p-value * @since 1.2 */ public double chiSquareTestDataSetsComparison(long[] observed1, long[] observed2) throws DimensionMismatchException, NotPositiveException, ZeroException, MaxCountExceededException { // pass a null rng to avoid unneeded overhead as we will not sample from this distribution final ChiSquaredDistribution distribution = new ChiSquaredDistribution(null, (double) observed1.length - 1); return 1 - distribution.cumulativeProbability(chiSquareDataSetsComparison(observed1, observed2)); } /** * <p>Performs a Chi-Square two sample test comparing two binned data * sets. The test evaluates the null hypothesis that the two lists of * observed counts conform to the same frequency distribution, with * significance level <code>alpha</code>. Returns true iff the null * hypothesis can be rejected with 100 * (1 - alpha) percent confidence. * </p> * <p>See {@link #chiSquareDataSetsComparison(long[], long[])} for * details on the formula used to compute the Chisquare statistic used * in the test. The degrees of of freedom used to perform the test is * one less than the common length of the input observed count arrays. * </p> * <strong>Preconditions</strong>: <ul> * <li>Observed counts must be non-negative. * </li> * <li>Observed counts for a specific bin must not both be zero. * </li> * <li>Observed counts for a specific sample must not all be 0. * </li> * <li>The arrays <code>observed1</code> and <code>observed2</code> must * have the same length and their common length must be at least 2. * </li> * <li> <code> 0 < alpha < 0.5 </code> * </li></ul><p> * If any of the preconditions are not met, an * <code>IllegalArgumentException</code> is thrown.</p> * * @param observed1 array of observed frequency counts of the first data set * @param observed2 array of observed frequency counts of the second data set * @param alpha significance level of the test * @return true iff null hypothesis can be rejected with confidence * 1 - alpha * @throws DimensionMismatchException the the length of the arrays does not match * @throws NotPositiveException if any entries in <code>observed1</code> or * <code>observed2</code> are negative * @throws ZeroException if either all counts of <code>observed1</code> or * <code>observed2</code> are zero, or if the count at the same index is zero * for both arrays * @throws OutOfRangeException if <code>alpha</code> is not in the range (0, 0.5] * @throws MaxCountExceededException if an error occurs performing the test * @since 1.2 */ public boolean chiSquareTestDataSetsComparison(final long[] observed1, final long[] observed2, final double alpha) throws DimensionMismatchException, NotPositiveException, ZeroException, OutOfRangeException, MaxCountExceededException { if (alpha <= 0 || alpha > 0.5) { throw new OutOfRangeException(LocalizedFormats.OUT_OF_BOUND_SIGNIFICANCE_LEVEL, alpha, 0, 0.5); } return chiSquareTestDataSetsComparison(observed1, observed2) < alpha; } /** * Checks to make sure that the input long[][] array is rectangular, * has at least 2 rows and 2 columns, and has all non-negative entries. * * @param in input 2-way table to check * @throws NullArgumentException if the array is null * @throws DimensionMismatchException if the array is not valid * @throws NotPositiveException if the array contains any negative entries */ private void checkArray(final long[][] in) throws NullArgumentException, DimensionMismatchException, NotPositiveException { if (in.length < 2) { throw new DimensionMismatchException(in.length, 2); } if (in[0].length < 2) { throw new DimensionMismatchException(in[0].length, 2); } MathArrays.checkRectangular(in); MathArrays.checkNonNegative(in); } }