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* 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.math4.stat.inference;
import org.apache.commons.statistics.distribution.ChiSquaredDistribution;
import org.apache.commons.math4.exception.DimensionMismatchException;
import org.apache.commons.math4.exception.MaxCountExceededException;
import org.apache.commons.math4.exception.NotPositiveException;
import org.apache.commons.math4.exception.NotStrictlyPositiveException;
import org.apache.commons.math4.exception.NullArgumentException;
import org.apache.commons.math4.exception.OutOfRangeException;
import org.apache.commons.math4.exception.ZeroException;
import org.apache.commons.math4.exception.util.LocalizedFormats;
import org.apache.commons.math4.util.FastMath;
import org.apache.commons.math4.util.MathArrays;
/**
* Implements Chi-Square test statistics.
*
* This implementation handles both known and unknown distributions.
*
* Two samples tests can be used when the distribution is unknown a priori
* but provided by one sample, or when the hypothesis under test is that the two
* samples come from the same underlying distribution.
*
*/
public class ChiSquareTest {
/**
* Construct a ChiSquareTest
*/
public ChiSquareTest() {
super();
}
/**
* Computes the
* Chi-Square statistic comparing observed and expected
* frequency counts.
*
* This statistic can be used to perform a Chi-Square test evaluating the null
* hypothesis that the observed counts follow the expected distribution.
*
* Preconditions:
* - Expected counts must all be positive.
*
* - Observed counts must all be ≥ 0.
*
* - The observed and expected arrays must have the same length and
* their common length must be at least 2.
*
* If any of the preconditions are not met, an
* IllegalArgumentException is thrown.
* Note: This implementation rescales the
* expected array if necessary to ensure that the sum of the
* expected and observed counts are equal.
*
* @param observed array of observed frequency counts
* @param expected array of expected frequency counts
* @return chiSquare test statistic
* @throws NotPositiveException if observed has negative entries
* @throws NotStrictlyPositiveException if expected 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 observed significance level, or
* p-value, associated with a
*
* Chi-square goodness of fit test comparing the observed
* frequency counts to those in the expected array.
*
* 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.
*
* Preconditions:
* - Expected counts must all be positive.
*
* - Observed counts must all be ≥ 0.
*
* - The observed and expected arrays must have the same length and
* their common length must be at least 2.
*
* If any of the preconditions are not met, an
* IllegalArgumentException is thrown.
* Note: This implementation rescales the
* expected array if necessary to ensure that the sum of the
* expected and observed counts are equal.
*
* @param observed array of observed frequency counts
* @param expected array of expected frequency counts
* @return p-value
* @throws NotPositiveException if observed has negative entries
* @throws NotStrictlyPositiveException if expected 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(expected.length - 1.0);
return 1.0 - distribution.cumulativeProbability(chiSquare(expected, observed));
}
/**
* Performs a
* Chi-square goodness of fit test evaluating the null hypothesis that the
* observed counts conform to the frequency distribution described by the expected
* counts, with significance level alpha. Returns true iff the null
* hypothesis can be rejected with 100 * (1 - alpha) percent confidence.
*
* Example:
* To test the hypothesis that observed follows
* expected at the 99% level, use
* chiSquareTest(expected, observed, 0.01)
*
* Preconditions:
* - Expected counts must all be positive.
*
* - Observed counts must all be ≥ 0.
*
* - The observed and expected arrays must have the same length and
* their common length must be at least 2.
*
-
0 < alpha < 0.5
*
* If any of the preconditions are not met, an
* IllegalArgumentException is thrown.
* Note: This implementation rescales the
* expected array if necessary to ensure that the sum of the
* expected and observed counts are equal.
*
* @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 observed has negative entries
* @throws NotStrictlyPositiveException if expected has entries that are
* not strictly positive
* @throws DimensionMismatchException if the arrays length is less than 2
* @throws OutOfRangeException if alpha 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
*
* chi-square test of independence based on the input counts
* array, viewed as a two-way table.
*
* The rows of the 2-way table are
* count[0], ... , count[count.length - 1]
*
* Preconditions:
* - All counts must be ≥ 0.
*
* - The count array must be rectangular (i.e. all count[i] subarrays
* must have the same length).
*
* - The 2-way table represented by
counts must have at
* least 2 columns and at least 2 rows.
*
*
* If any of the preconditions are not met, an
* IllegalArgumentException is thrown.
*
* @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 observed significance level, or
* p-value, associated with a
*
* chi-square test of independence based on the input counts
* array, viewed as a two-way table.
*
* The rows of the 2-way table are
* count[0], ... , count[count.length - 1]
*
* Preconditions:
* - All counts must be ≥ 0.
*
* - The count array must be rectangular (i.e. all count[i] subarrays must have
* the same length).
*
* - The 2-way table represented by
counts must have at least 2
* columns and at least 2 rows.
*
*
* If any of the preconditions are not met, an
* IllegalArgumentException is thrown.
*
* @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
* chi-square test of independence 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 alpha.
* Returns true iff the null hypothesis can be rejected with 100 * (1 - alpha) percent
* confidence.
*
* The rows of the 2-way table are
* count[0], ... , count[count.length - 1]
*
* Example:
* To test the null hypothesis that the counts in
* count[0], ... , count[count.length - 1]
* all correspond to the same underlying probability distribution at the 99% level, use
* chiSquareTest(counts, 0.01)
*
* Preconditions:
* - All counts must be ≥ 0.
*
* - The count array must be rectangular (i.e. all count[i] subarrays must have the
* same length).
* - The 2-way table represented by
counts must have at least 2 columns and
* at least 2 rows.
*
* If any of the preconditions are not met, an
* IllegalArgumentException is thrown.
*
* @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 alpha 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;
}
/**
* Computes a
*
* Chi-Square two sample test statistic comparing bin frequency counts
* in observed1 and observed2. 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
*
* ∑[(K * observed1[i] - observed2[i]/K)2 / (observed1[i] + observed2[i])]
* where
*
K = √[∑(observed2 / ∑(observed1)]
*
* This statistic can be used to perform a Chi-Square test evaluating the
* null hypothesis that both observed counts follow the same distribution.
*
* Preconditions:
* - Observed counts must be non-negative.
*
* - Observed counts for a specific bin must not both be zero.
*
* - Observed counts for a specific sample must not all be 0.
*
* - The arrays
observed1 and observed2 must have
* the same length and their common length must be at least 2.
*
* If any of the preconditions are not met, an
* IllegalArgumentException is thrown.
*
* @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 length of the arrays does not match
* @throws NotPositiveException if any entries in observed1 or
* observed2 are negative
* @throws ZeroException if either all counts of observed1 or
* observed2 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;
}
/**
* Returns the observed significance level, or
* p-value, associated with a Chi-Square two sample test comparing
* bin frequency counts in observed1 and
* observed2.
*
* 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.
*
* 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.
*
* Preconditions:
* - Observed counts must be non-negative.
*
* - Observed counts for a specific bin must not both be zero.
*
* - Observed counts for a specific sample must not all be 0.
*
* - The arrays
observed1 and observed2 must
* have the same length and
* their common length must be at least 2.
*
* If any of the preconditions are not met, an
* IllegalArgumentException is thrown.
*
* @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 length of the arrays does not match
* @throws NotPositiveException if any entries in observed1 or
* observed2 are negative
* @throws ZeroException if either all counts of observed1 or
* observed2 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((double) observed1.length - 1);
return 1 - distribution.cumulativeProbability(
chiSquareDataSetsComparison(observed1, observed2));
}
/**
* 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 alpha. Returns true iff the null
* hypothesis can be rejected with 100 * (1 - alpha) percent confidence.
*
* 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.
*
* Preconditions:
* - Observed counts must be non-negative.
*
* - Observed counts for a specific bin must not both be zero.
*
* - Observed counts for a specific sample must not all be 0.
*
* - The arrays
observed1 and observed2 must
* have the same length and their common length must be at least 2.
*
* -
0 < alpha < 0.5
*
* If any of the preconditions are not met, an
* IllegalArgumentException is thrown.
*
* @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 length of the arrays does not match
* @throws NotPositiveException if any entries in observed1 or
* observed2 are negative
* @throws ZeroException if either all counts of observed1 or
* observed2 are zero, or if the count at the same index is zero
* for both arrays
* @throws OutOfRangeException if alpha 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);
}
}