KolmogorovSmirnovTest.java
/*
* 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.
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package org.apache.commons.math4.legacy.stat.inference;
import java.math.RoundingMode;
import java.util.Arrays;
import org.apache.commons.rng.simple.RandomSource;
import org.apache.commons.rng.UniformRandomProvider;
import org.apache.commons.statistics.distribution.ContinuousDistribution;
import org.apache.commons.numbers.combinatorics.BinomialCoefficientDouble;
import org.apache.commons.numbers.fraction.BigFraction;
import org.apache.commons.numbers.field.BigFractionField;
import org.apache.commons.math4.legacy.distribution.EnumeratedRealDistribution;
import org.apache.commons.math4.legacy.distribution.AbstractRealDistribution;
import org.apache.commons.math4.legacy.exception.InsufficientDataException;
import org.apache.commons.math4.legacy.exception.MathArithmeticException;
import org.apache.commons.math4.legacy.exception.MaxCountExceededException;
import org.apache.commons.math4.legacy.exception.NullArgumentException;
import org.apache.commons.math4.legacy.exception.NumberIsTooLargeException;
import org.apache.commons.math4.legacy.exception.OutOfRangeException;
import org.apache.commons.math4.legacy.exception.TooManyIterationsException;
import org.apache.commons.math4.legacy.exception.NotANumberException;
import org.apache.commons.math4.legacy.exception.util.LocalizedFormats;
import org.apache.commons.math4.legacy.linear.MatrixUtils;
import org.apache.commons.math4.legacy.linear.RealMatrix;
import org.apache.commons.math4.core.jdkmath.JdkMath;
import org.apache.commons.math4.legacy.core.MathArrays;
import org.apache.commons.math4.legacy.field.linalg.FieldDenseMatrix;
/**
* Implementation of the <a href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test">
* Kolmogorov-Smirnov (K-S) test</a> for equality of continuous distributions.
* <p>
* The K-S test uses a statistic based on the maximum deviation of the empirical distribution of
* sample data points from the distribution expected under the null hypothesis. For one-sample tests
* evaluating the null hypothesis that a set of sample data points follow a given distribution, the
* test statistic is \(D_n=\sup_x |F_n(x)-F(x)|\), where \(F\) is the expected distribution and
* \(F_n\) is the empirical distribution of the \(n\) sample data points. The distribution of
* \(D_n\) is estimated using a method based on [1] with certain quick decisions for extreme values
* given in [2].
* </p>
* <p>
* Two-sample tests are also supported, evaluating the null hypothesis that the two samples
* {@code x} and {@code y} come from the same underlying distribution. In this case, the test
* statistic is \(D_{n,m}=\sup_t | F_n(t)-F_m(t)|\) where \(n\) is the length of {@code x}, \(m\) is
* the length of {@code y}, \(F_n\) is the empirical distribution that puts mass \(1/n\) at each of
* the values in {@code x} and \(F_m\) is the empirical distribution of the {@code y} values. The
* default 2-sample test method, {@link #kolmogorovSmirnovTest(double[], double[])} works as
* follows:
* <ul>
* <li>When the product of the sample sizes is less than 10000, the method presented in [4]
* is used to compute the exact p-value for the 2-sample test.</li>
* <li>When the product of the sample sizes is larger, the asymptotic
* distribution of \(D_{n,m}\) is used. See {@link #approximateP(double, int, int)} for details on
* the approximation.</li>
* </ul><p>
* For small samples (former case), if the data contains ties, random jitter is added
* to the sample data to break ties before applying the algorithm above. Alternatively,
* the {@link #bootstrap(double[],double[],int,boolean,UniformRandomProvider)}
* method, modeled after <a href="http://sekhon.berkeley.edu/matching/ks.boot.html">ks.boot</a>
* in the R Matching package [3], can be used if ties are known to be present in the data.
* </p>
* <p>
* In the two-sample case, \(D_{n,m}\) has a discrete distribution. This makes the p-value
* associated with the null hypothesis \(H_0 : D_{n,m} \ge d \) differ from \(H_0 : D_{n,m} \ge d \)
* by the mass of the observed value \(d\). To distinguish these, the two-sample tests use a boolean
* {@code strict} parameter. This parameter is ignored for large samples.
* </p>
* <p>
* The methods used by the 2-sample default implementation are also exposed directly:
* <ul>
* <li>{@link #exactP(double, int, int, boolean)} computes exact 2-sample p-values</li>
* <li>{@link #approximateP(double, int, int)} uses the asymptotic distribution The {@code boolean}
* arguments in the first two methods allow the probability used to estimate the p-value to be
* expressed using strict or non-strict inequality. See
* {@link #kolmogorovSmirnovTest(double[], double[], boolean)}.</li>
* </ul>
* <p>
* References:
* <ul>
* <li>[1] <a href="http://www.jstatsoft.org/v08/i18/"> Evaluating Kolmogorov's Distribution</a> by
* George Marsaglia, Wai Wan Tsang, and Jingbo Wang</li>
* <li>[2] <a href="http://www.jstatsoft.org/v39/i11/"> Computing the Two-Sided Kolmogorov-Smirnov
* Distribution</a> by Richard Simard and Pierre L'Ecuyer</li>
* <li>[3] Jasjeet S. Sekhon. 2011. <a href="http://www.jstatsoft.org/article/view/v042i07">
* Multivariate and Propensity Score Matching Software with Automated Balance Optimization:
* The Matching package for R</a> Journal of Statistical Software, 42(7): 1-52.</li>
* <li>[4] Wilcox, Rand. 2012. Introduction to Robust Estimation and Hypothesis Testing,
* Chapter 5, 3rd Ed. Academic Press.</li>
* </ul>
* <br>
* Note that [1] contains an error in computing h, refer to <a
* href="https://issues.apache.org/jira/browse/MATH-437">MATH-437</a> for details.
*
* @since 3.3
*/
public class KolmogorovSmirnovTest {
/** pi^2. */
private static final double PI_SQUARED = JdkMath.PI * JdkMath.PI;
/**
* Bound on the number of partial sums in {@link #ksSum(double, double, int)}.
*/
private static final int MAXIMUM_PARTIAL_SUM_COUNT = 100000;
/** Convergence criterion for {@link #ksSum(double, double, int)}. */
private static final double KS_SUM_CAUCHY_CRITERION = 1e-20;
/** Convergence criterion for the sums in {@link #pelzGood(double, int)}. */
private static final double PG_SUM_RELATIVE_ERROR = 1e-10;
/** 1/2. */
private static final BigFraction ONE_HALF = BigFraction.of(1, 2);
/**
* When product of sample sizes exceeds this value, 2-sample K-S test uses asymptotic
* distribution to compute the p-value.
*/
private static final int LARGE_SAMPLE_PRODUCT = 10000;
/**
* Computes the <i>p-value</i>, or <i>observed significance level</i>, of a one-sample <a
* href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov test</a>
* evaluating the null hypothesis that {@code data} conforms to {@code distribution}. If
* {@code exact} is true, the distribution used to compute the p-value is computed using
* extended precision. See {@link #cdfExact(double, int)}.
*
* @param distribution reference distribution
* @param data sample being being evaluated
* @param exact whether or not to force exact computation of the p-value
* @return the p-value associated with the null hypothesis that {@code data} is a sample from
* {@code distribution}
* @throws InsufficientDataException if {@code data} does not have length at least 2
* @throws NullArgumentException if {@code data} is null
*/
public double kolmogorovSmirnovTest(ContinuousDistribution distribution, double[] data, boolean exact) {
return 1d - cdf(kolmogorovSmirnovStatistic(distribution, data), data.length, exact);
}
/**
* Computes the one-sample Kolmogorov-Smirnov test statistic, \(D_n=\sup_x |F_n(x)-F(x)|\) where
* \(F\) is the distribution (cdf) function associated with {@code distribution}, \(n\) is the
* length of {@code data} and \(F_n\) is the empirical distribution that puts mass \(1/n\) at
* each of the values in {@code data}.
*
* @param distribution reference distribution
* @param data sample being evaluated
* @return Kolmogorov-Smirnov statistic \(D_n\)
* @throws InsufficientDataException if {@code data} does not have length at least 2
* @throws NullArgumentException if {@code data} is null
*/
public double kolmogorovSmirnovStatistic(ContinuousDistribution distribution, double[] data) {
checkArray(data);
final int n = data.length;
final double nd = n;
final double[] dataCopy = new double[n];
System.arraycopy(data, 0, dataCopy, 0, n);
Arrays.sort(dataCopy);
double d = 0d;
for (int i = 1; i <= n; i++) {
final double yi = distribution.cumulativeProbability(dataCopy[i - 1]);
final double currD = JdkMath.max(yi - (i - 1) / nd, i / nd - yi);
if (currD > d) {
d = currD;
}
}
return d;
}
/**
* Computes the <i>p-value</i>, or <i>observed significance level</i>, of a two-sample <a
* href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov test</a>
* evaluating the null hypothesis that {@code x} and {@code y} are samples drawn from the same
* probability distribution. Specifically, what is returned is an estimate of the probability
* that the {@link #kolmogorovSmirnovStatistic(double[], double[])} associated with a randomly
* selected partition of the combined sample into subsamples of sizes {@code x.length} and
* {@code y.length} will strictly exceed (if {@code strict} is {@code true}) or be at least as
* large as (if {@code strict} is {@code false}) as {@code kolmogorovSmirnovStatistic(x, y)}.
*
* @param x first sample dataset.
* @param y second sample dataset.
* @param strict whether or not the probability to compute is expressed as
* a strict inequality (ignored for large samples).
* @return p-value associated with the null hypothesis that {@code x} and
* {@code y} represent samples from the same distribution.
* @throws InsufficientDataException if either {@code x} or {@code y} does
* not have length at least 2.
* @throws NullArgumentException if either {@code x} or {@code y} is null.
* @throws NotANumberException if the input arrays contain NaN values.
*
* @see #bootstrap(double[],double[],int,boolean,UniformRandomProvider)
*/
public double kolmogorovSmirnovTest(double[] x, double[] y, boolean strict) {
final long lengthProduct = (long) x.length * y.length;
double[] xa = null;
double[] ya = null;
if (lengthProduct < LARGE_SAMPLE_PRODUCT && hasTies(x,y)) {
xa = Arrays.copyOf(x, x.length);
ya = Arrays.copyOf(y, y.length);
fixTies(xa, ya);
} else {
xa = x;
ya = y;
}
if (lengthProduct < LARGE_SAMPLE_PRODUCT) {
return exactP(kolmogorovSmirnovStatistic(xa, ya), x.length, y.length, strict);
}
return approximateP(kolmogorovSmirnovStatistic(x, y), x.length, y.length);
}
/**
* Computes the <i>p-value</i>, or <i>observed significance level</i>, of a two-sample <a
* href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov test</a>
* evaluating the null hypothesis that {@code x} and {@code y} are samples drawn from the same
* probability distribution. Assumes the strict form of the inequality used to compute the
* p-value. See {@link #kolmogorovSmirnovTest(ContinuousDistribution, double[], boolean)}.
*
* @param x first sample dataset
* @param y second sample dataset
* @return p-value associated with the null hypothesis that {@code x} and {@code y} represent
* samples from the same distribution
* @throws InsufficientDataException if either {@code x} or {@code y} does not have length at
* least 2
* @throws NullArgumentException if either {@code x} or {@code y} is null
*/
public double kolmogorovSmirnovTest(double[] x, double[] y) {
return kolmogorovSmirnovTest(x, y, true);
}
/**
* Computes the two-sample Kolmogorov-Smirnov test statistic, \(D_{n,m}=\sup_x |F_n(x)-F_m(x)|\)
* where \(n\) is the length of {@code x}, \(m\) is the length of {@code y}, \(F_n\) is the
* empirical distribution that puts mass \(1/n\) at each of the values in {@code x} and \(F_m\)
* is the empirical distribution of the {@code y} values.
*
* @param x first sample
* @param y second sample
* @return test statistic \(D_{n,m}\) used to evaluate the null hypothesis that {@code x} and
* {@code y} represent samples from the same underlying distribution
* @throws InsufficientDataException if either {@code x} or {@code y} does not have length at
* least 2
* @throws NullArgumentException if either {@code x} or {@code y} is null
*/
public double kolmogorovSmirnovStatistic(double[] x, double[] y) {
return integralKolmogorovSmirnovStatistic(x, y)/((double)(x.length * (long)y.length));
}
/**
* Computes the two-sample Kolmogorov-Smirnov test statistic, \(D_{n,m}=\sup_x |F_n(x)-F_m(x)|\)
* where \(n\) is the length of {@code x}, \(m\) is the length of {@code y}, \(F_n\) is the
* empirical distribution that puts mass \(1/n\) at each of the values in {@code x} and \(F_m\)
* is the empirical distribution of the {@code y} values. Finally \(n m D_{n,m}\) is returned
* as long value.
*
* @param x first sample
* @param y second sample
* @return test statistic \(n m D_{n,m}\) used to evaluate the null hypothesis that {@code x} and
* {@code y} represent samples from the same underlying distribution
* @throws InsufficientDataException if either {@code x} or {@code y} does not have length at
* least 2
* @throws NullArgumentException if either {@code x} or {@code y} is null
*/
private long integralKolmogorovSmirnovStatistic(double[] x, double[] y) {
checkArray(x);
checkArray(y);
// Copy and sort the sample arrays
final double[] sx = Arrays.copyOf(x, x.length);
final double[] sy = Arrays.copyOf(y, y.length);
Arrays.sort(sx);
Arrays.sort(sy);
final int n = sx.length;
final int m = sy.length;
int rankX = 0;
int rankY = 0;
long curD = 0L;
// Find the max difference between cdf_x and cdf_y
long supD = 0L;
do {
double z = Double.compare(sx[rankX], sy[rankY]) <= 0 ? sx[rankX] : sy[rankY];
while(rankX < n && Double.compare(sx[rankX], z) == 0) {
rankX += 1;
curD += m;
}
while(rankY < m && Double.compare(sy[rankY], z) == 0) {
rankY += 1;
curD -= n;
}
if (curD > supD) {
supD = curD;
} else if (-curD > supD) {
supD = -curD;
}
} while(rankX < n && rankY < m);
return supD;
}
/**
* Computes the <i>p-value</i>, or <i>observed significance level</i>, of a one-sample <a
* href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov test</a>
* evaluating the null hypothesis that {@code data} conforms to {@code distribution}.
*
* @param distribution reference distribution
* @param data sample being being evaluated
* @return the p-value associated with the null hypothesis that {@code data} is a sample from
* {@code distribution}
* @throws InsufficientDataException if {@code data} does not have length at least 2
* @throws NullArgumentException if {@code data} is null
*/
public double kolmogorovSmirnovTest(ContinuousDistribution distribution, double[] data) {
return kolmogorovSmirnovTest(distribution, data, false);
}
/**
* Performs a <a href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov
* test</a> evaluating the null hypothesis that {@code data} conforms to {@code distribution}.
*
* @param distribution reference distribution
* @param data sample being being evaluated
* @param alpha significance level of the test
* @return true iff the null hypothesis that {@code data} is a sample from {@code distribution}
* can be rejected with confidence 1 - {@code alpha}
* @throws InsufficientDataException if {@code data} does not have length at least 2
* @throws NullArgumentException if {@code data} is null
*/
public boolean kolmogorovSmirnovTest(ContinuousDistribution distribution, double[] data, double alpha) {
if (alpha <= 0 || alpha > 0.5) {
throw new OutOfRangeException(LocalizedFormats.OUT_OF_BOUND_SIGNIFICANCE_LEVEL, alpha, 0, 0.5);
}
return kolmogorovSmirnovTest(distribution, data) < alpha;
}
/**
* Estimates the <i>p-value</i> of a two-sample
* <a href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test">Kolmogorov-Smirnov test</a>
* evaluating the null hypothesis that {@code x} and {@code y} are samples
* drawn from the same probability distribution.
* This method estimates the p-value by repeatedly sampling sets of size
* {@code x.length} and {@code y.length} from the empirical distribution
* of the combined sample.
* When {@code strict} is true, this is equivalent to the algorithm implemented
* in the R function {@code ks.boot}, described in <pre>
* Jasjeet S. Sekhon. 2011. 'Multivariate and Propensity Score Matching
* Software with Automated Balance Optimization: The Matching package for R.'
* Journal of Statistical Software, 42(7): 1-52.
* </pre>
*
* @param x First sample.
* @param y Second sample.
* @param iterations Number of bootstrap resampling iterations.
* @param strict Whether or not the null hypothesis is expressed as a strict inequality.
* @param rng RNG for creating the sampling sets.
* @return the estimated p-value.
*/
public double bootstrap(double[] x,
double[] y,
int iterations,
boolean strict,
UniformRandomProvider rng) {
final int xLength = x.length;
final int yLength = y.length;
final double[] combined = new double[xLength + yLength];
System.arraycopy(x, 0, combined, 0, xLength);
System.arraycopy(y, 0, combined, xLength, yLength);
final ContinuousDistribution.Sampler sampler = new EnumeratedRealDistribution(combined).createSampler(rng);
final long d = integralKolmogorovSmirnovStatistic(x, y);
int greaterCount = 0;
int equalCount = 0;
double[] curX;
double[] curY;
long curD;
for (int i = 0; i < iterations; i++) {
curX = AbstractRealDistribution.sample(xLength, sampler);
curY = AbstractRealDistribution.sample(yLength, sampler);
curD = integralKolmogorovSmirnovStatistic(curX, curY);
if (curD > d) {
greaterCount++;
} else if (curD == d) {
equalCount++;
}
}
return strict ? greaterCount / (double) iterations :
(greaterCount + equalCount) / (double) iterations;
}
/**
* Calculates \(P(D_n < d)\) using the method described in [1] with quick decisions for extreme
* values given in [2] (see above). The result is not exact as with
* {@link #cdfExact(double, int)} because calculations are based on
* {@code double} rather than {@link BigFraction}.
*
* @param d statistic
* @param n sample size
* @return \(P(D_n < d)\)
* @throws MathArithmeticException if algorithm fails to convert {@code h} to a
* {@link BigFraction} in expressing {@code d} as
* \((k - h) / m\) for integer {@code k, m} and \(0 \le h < 1\)
*/
public double cdf(double d, int n) {
return cdf(d, n, false);
}
/**
* Calculates {@code P(D_n < d)}. The result is exact in the sense that BigFraction/BigReal is
* used everywhere at the expense of very slow execution time. Almost never choose this in real
* applications unless you are very sure; this is almost solely for verification purposes.
* Normally, you would choose {@link #cdf(double, int)}. See the class
* javadoc for definitions and algorithm description.
*
* @param d statistic
* @param n sample size
* @return \(P(D_n < d)\)
* @throws MathArithmeticException if the algorithm fails to convert {@code h} to a
* {@link BigFraction} in expressing {@code d} as
* \((k - h) / m\) for integer {@code k, m} and \(0 \le h < 1\)
*/
public double cdfExact(double d, int n) {
return cdf(d, n, true);
}
/**
* Calculates {@code P(D_n < d)} using method described in [1] with quick decisions for extreme
* values given in [2] (see above).
*
* @param d statistic
* @param n sample size
* @param exact whether the probability should be calculated exact using
* {@link BigFraction} everywhere at the expense of
* very slow execution time, or if {@code double} should be used convenient places to
* gain speed. Almost never choose {@code true} in real applications unless you are very
* sure; {@code true} is almost solely for verification purposes.
* @return \(P(D_n < d)\)
* @throws MathArithmeticException if algorithm fails to convert {@code h} to a
* {@link BigFraction} in expressing {@code d} as
* \((k - h) / m\) for integer {@code k, m} and \(0 \le h < 1\).
*/
public double cdf(double d, int n, boolean exact) {
final double ninv = 1 / ((double) n);
final double ninvhalf = 0.5 * ninv;
if (d <= ninvhalf) {
return 0;
} else if (ninvhalf < d && d <= ninv) {
double res = 1;
final double f = 2 * d - ninv;
// n! f^n = n*f * (n-1)*f * ... * 1*x
for (int i = 1; i <= n; ++i) {
res *= i * f;
}
return res;
} else if (1 - ninv <= d && d < 1) {
return 1 - 2 * Math.pow(1 - d, n);
} else if (1 <= d) {
return 1;
}
if (exact) {
return exactK(d, n);
}
if (n <= 140) {
return roundedK(d, n);
}
return pelzGood(d, n);
}
/**
* Calculates the exact value of {@code P(D_n < d)} using the method described in [1] (reference
* in class javadoc above) and {@link BigFraction} (see above).
*
* @param d statistic
* @param n sample size
* @return the two-sided probability of \(P(D_n < d)\)
* @throws MathArithmeticException if algorithm fails to convert {@code h} to a
* {@link BigFraction}.
*/
private double exactK(double d, int n) {
final int k = (int) Math.ceil(n * d);
final FieldDenseMatrix<BigFraction> h;
try {
h = createExactH(d, n);
} catch (ArithmeticException e) {
throw new MathArithmeticException(LocalizedFormats.FRACTION);
}
final FieldDenseMatrix<BigFraction> hPower = h.pow(n);
BigFraction pFrac = hPower.get(k - 1, k - 1);
for (int i = 1; i <= n; ++i) {
pFrac = pFrac.multiply(i).divide(n);
}
/*
* BigFraction.doubleValue converts numerator to double and the denominator to double and
* divides afterwards. That gives NaN quite easy. This does not (scale is the number of
* digits):
*/
return pFrac.bigDecimalValue(20, RoundingMode.HALF_UP).doubleValue();
}
/**
* Calculates {@code P(D_n < d)} using method described in [1] and doubles (see above).
*
* @param d statistic
* @param n sample size
* @return \(P(D_n < d)\)
*/
private double roundedK(double d, int n) {
final int k = (int) Math.ceil(n * d);
final RealMatrix h = this.createRoundedH(d, n);
final RealMatrix hPower = h.power(n);
double pFrac = hPower.getEntry(k - 1, k - 1);
for (int i = 1; i <= n; ++i) {
pFrac *= (double) i / (double) n;
}
return pFrac;
}
/**
* Computes the Pelz-Good approximation for \(P(D_n < d)\) as described in [2] in the class javadoc.
*
* @param d value of d-statistic (x in [2])
* @param n sample size
* @return \(P(D_n < d)\)
* @since 3.4
*/
public double pelzGood(double d, int n) {
// Change the variable since approximation is for the distribution evaluated at d / sqrt(n)
final double sqrtN = JdkMath.sqrt(n);
final double z = d * sqrtN;
final double z2 = d * d * n;
final double z4 = z2 * z2;
final double z6 = z4 * z2;
final double z8 = z4 * z4;
// Eventual return value
double ret = 0;
// Compute K_0(z)
double sum = 0;
double increment = 0;
double kTerm = 0;
double z2Term = PI_SQUARED / (8 * z2);
int k = 1;
for (; k < MAXIMUM_PARTIAL_SUM_COUNT; k++) {
kTerm = 2 * k - 1;
increment = JdkMath.exp(-z2Term * kTerm * kTerm);
sum += increment;
if (increment <= PG_SUM_RELATIVE_ERROR * sum) {
break;
}
}
if (k == MAXIMUM_PARTIAL_SUM_COUNT) {
throw new TooManyIterationsException(MAXIMUM_PARTIAL_SUM_COUNT);
}
ret = sum * JdkMath.sqrt(2 * JdkMath.PI) / z;
// K_1(z)
// Sum is -inf to inf, but k term is always (k + 1/2) ^ 2, so really have
// twice the sum from k = 0 to inf (k = -1 is same as 0, -2 same as 1, ...)
final double twoZ2 = 2 * z2;
sum = 0;
kTerm = 0;
double kTerm2 = 0;
for (k = 0; k < MAXIMUM_PARTIAL_SUM_COUNT; k++) {
kTerm = k + 0.5;
kTerm2 = kTerm * kTerm;
increment = (PI_SQUARED * kTerm2 - z2) * JdkMath.exp(-PI_SQUARED * kTerm2 / twoZ2);
sum += increment;
if (JdkMath.abs(increment) < PG_SUM_RELATIVE_ERROR * JdkMath.abs(sum)) {
break;
}
}
if (k == MAXIMUM_PARTIAL_SUM_COUNT) {
throw new TooManyIterationsException(MAXIMUM_PARTIAL_SUM_COUNT);
}
final double sqrtHalfPi = JdkMath.sqrt(JdkMath.PI / 2);
// Instead of doubling sum, divide by 3 instead of 6
ret += sum * sqrtHalfPi / (3 * z4 * sqrtN);
// K_2(z)
// Same drill as K_1, but with two doubly infinite sums, all k terms are even powers.
final double z4Term = 2 * z4;
final double z6Term = 6 * z6;
z2Term = 5 * z2;
final double pi4 = PI_SQUARED * PI_SQUARED;
sum = 0;
kTerm = 0;
kTerm2 = 0;
for (k = 0; k < MAXIMUM_PARTIAL_SUM_COUNT; k++) {
kTerm = k + 0.5;
kTerm2 = kTerm * kTerm;
increment = (z6Term + z4Term + PI_SQUARED * (z4Term - z2Term) * kTerm2 +
pi4 * (1 - twoZ2) * kTerm2 * kTerm2) * JdkMath.exp(-PI_SQUARED * kTerm2 / twoZ2);
sum += increment;
if (JdkMath.abs(increment) < PG_SUM_RELATIVE_ERROR * JdkMath.abs(sum)) {
break;
}
}
if (k == MAXIMUM_PARTIAL_SUM_COUNT) {
throw new TooManyIterationsException(MAXIMUM_PARTIAL_SUM_COUNT);
}
double sum2 = 0;
kTerm2 = 0;
for (k = 1; k < MAXIMUM_PARTIAL_SUM_COUNT; k++) {
kTerm2 = (double) k * k;
increment = PI_SQUARED * kTerm2 * JdkMath.exp(-PI_SQUARED * kTerm2 / twoZ2);
sum2 += increment;
if (JdkMath.abs(increment) < PG_SUM_RELATIVE_ERROR * JdkMath.abs(sum2)) {
break;
}
}
if (k == MAXIMUM_PARTIAL_SUM_COUNT) {
throw new TooManyIterationsException(MAXIMUM_PARTIAL_SUM_COUNT);
}
// Again, adjust coefficients instead of doubling sum, sum2
ret += (sqrtHalfPi / n) * (sum / (36 * z2 * z2 * z2 * z) - sum2 / (18 * z2 * z));
// K_3(z) One more time with feeling - two doubly infinite sums, all k powers even.
// Multiply coefficient denominators by 2, so omit doubling sums.
final double pi6 = pi4 * PI_SQUARED;
sum = 0;
double kTerm4 = 0;
double kTerm6 = 0;
for (k = 0; k < MAXIMUM_PARTIAL_SUM_COUNT; k++) {
kTerm = k + 0.5;
kTerm2 = kTerm * kTerm;
kTerm4 = kTerm2 * kTerm2;
kTerm6 = kTerm4 * kTerm2;
increment = (pi6 * kTerm6 * (5 - 30 * z2) + pi4 * kTerm4 * (-60 * z2 + 212 * z4) +
PI_SQUARED * kTerm2 * (135 * z4 - 96 * z6) - 30 * z6 - 90 * z8) *
JdkMath.exp(-PI_SQUARED * kTerm2 / twoZ2);
sum += increment;
if (JdkMath.abs(increment) < PG_SUM_RELATIVE_ERROR * JdkMath.abs(sum)) {
break;
}
}
if (k == MAXIMUM_PARTIAL_SUM_COUNT) {
throw new TooManyIterationsException(MAXIMUM_PARTIAL_SUM_COUNT);
}
sum2 = 0;
for (k = 1; k < MAXIMUM_PARTIAL_SUM_COUNT; k++) {
kTerm2 = (double) k * k;
kTerm4 = kTerm2 * kTerm2;
increment = (-pi4 * kTerm4 + 3 * PI_SQUARED * kTerm2 * z2) *
JdkMath.exp(-PI_SQUARED * kTerm2 / twoZ2);
sum2 += increment;
if (JdkMath.abs(increment) < PG_SUM_RELATIVE_ERROR * JdkMath.abs(sum2)) {
break;
}
}
if (k == MAXIMUM_PARTIAL_SUM_COUNT) {
throw new TooManyIterationsException(MAXIMUM_PARTIAL_SUM_COUNT);
}
return ret + (sqrtHalfPi / (sqrtN * n)) * (sum / (3240 * z6 * z4) +
sum2 / (108 * z6));
}
/**
* Creates {@code H} of size {@code m x m} as described in [1] (see above).
*
* @param d statistic
* @param n sample size
* @return H matrix
* @throws NumberIsTooLargeException if fractional part is greater than 1.
* @throws ArithmeticException if algorithm fails to convert {@code h} to a
* {@link BigFraction}.
*/
private FieldDenseMatrix<BigFraction> createExactH(double d,
int n) {
final int k = (int) Math.ceil(n * d);
final int m = 2 * k - 1;
final double hDouble = k - n * d;
if (hDouble >= 1) {
throw new NumberIsTooLargeException(hDouble, 1.0, false);
}
BigFraction h = null;
try {
h = BigFraction.from(hDouble, 1e-20, 10000);
} catch (final ArithmeticException e1) {
try {
h = BigFraction.from(hDouble, 1e-10, 10000);
} catch (final ArithmeticException e2) {
h = BigFraction.from(hDouble, 1e-5, 10000);
}
}
final FieldDenseMatrix<BigFraction> hData = FieldDenseMatrix.create(BigFractionField.get(), m, m);
/*
* Start by filling everything with either 0 or 1.
*/
for (int i = 0; i < m; ++i) {
for (int j = 0; j < m; ++j) {
if (i - j + 1 < 0) {
hData.set(i, j, BigFraction.ZERO);
} else {
hData.set(i, j, BigFraction.ONE);
}
}
}
/*
* Setting up power-array to avoid calculating the same value twice: hPowers[0] = h^1 ...
* hPowers[m-1] = h^m
*/
final BigFraction[] hPowers = new BigFraction[m];
hPowers[0] = h;
for (int i = 1; i < m; i++) {
hPowers[i] = h.multiply(hPowers[i - 1]);
}
/*
* First column and last row has special values (each other reversed).
*/
for (int i = 0; i < m; ++i) {
hData.set(i, 0,
hData.get(i, 0).subtract(hPowers[i]));
hData.set(m - 1, i,
hData.get(m - 1, i).subtract(hPowers[m - i - 1]));
}
/*
* [1] states: "For 1/2 < h < 1 the bottom left element of the matrix should be (1 - 2*h^m +
* (2h - 1)^m )/m!" Since 0 <= h < 1, then if h > 1/2 is sufficient to check:
*/
if (h.compareTo(ONE_HALF) > 0) {
hData.set(m - 1, 0,
hData.get(m - 1, 0).add(h.multiply(2).subtract(1).pow(m)));
}
/*
* Aside from the first column and last row, the (i, j)-th element is 1/(i - j + 1)! if i -
* j + 1 >= 0, else 0. 1's and 0's are already put, so only division with (i - j + 1)! is
* needed in the elements that have 1's. There is no need to calculate (i - j + 1)! and then
* divide - small steps avoid overflows. Note that i - j + 1 > 0 <=> i + 1 > j instead of
* j'ing all the way to m. Also note that it is started at g = 2 because dividing by 1 isn't
* really necessary.
*/
for (int i = 0; i < m; ++i) {
for (int j = 0; j < i + 1; ++j) {
if (i - j + 1 > 0) {
for (int g = 2; g <= i - j + 1; ++g) {
hData.set(i, j,
hData.get(i, j).divide(g));
}
}
}
}
return hData;
}
/***
* Creates {@code H} of size {@code m x m} as described in [1] (see above)
* using double-precision.
*
* @param d statistic
* @param n sample size
* @return H matrix
* @throws NumberIsTooLargeException if fractional part is greater than 1
*/
private RealMatrix createRoundedH(double d, int n)
throws NumberIsTooLargeException {
final int k = (int) Math.ceil(n * d);
final int m = 2 * k - 1;
final double h = k - n * d;
if (h >= 1) {
throw new NumberIsTooLargeException(h, 1.0, false);
}
final double[][] hData = new double[m][m];
/*
* Start by filling everything with either 0 or 1.
*/
for (int i = 0; i < m; ++i) {
for (int j = 0; j < m; ++j) {
if (i - j + 1 < 0) {
hData[i][j] = 0;
} else {
hData[i][j] = 1;
}
}
}
/*
* Setting up power-array to avoid calculating the same value twice: hPowers[0] = h^1 ...
* hPowers[m-1] = h^m
*/
final double[] hPowers = new double[m];
hPowers[0] = h;
for (int i = 1; i < m; ++i) {
hPowers[i] = h * hPowers[i - 1];
}
/*
* First column and last row has special values (each other reversed).
*/
for (int i = 0; i < m; ++i) {
hData[i][0] = hData[i][0] - hPowers[i];
hData[m - 1][i] -= hPowers[m - i - 1];
}
/*
* [1] states: "For 1/2 < h < 1 the bottom left element of the matrix should be (1 - 2*h^m +
* (2h - 1)^m )/m!" Since 0 <= h < 1, then if h > 1/2 is sufficient to check:
*/
if (Double.compare(h, 0.5) > 0) {
hData[m - 1][0] += JdkMath.pow(2 * h - 1, m);
}
/*
* Aside from the first column and last row, the (i, j)-th element is 1/(i - j + 1)! if i -
* j + 1 >= 0, else 0. 1's and 0's are already put, so only division with (i - j + 1)! is
* needed in the elements that have 1's. There is no need to calculate (i - j + 1)! and then
* divide - small steps avoid overflows. Note that i - j + 1 > 0 <=> i + 1 > j instead of
* j'ing all the way to m. Also note that it is started at g = 2 because dividing by 1 isn't
* really necessary.
*/
for (int i = 0; i < m; ++i) {
for (int j = 0; j < i + 1; ++j) {
if (i - j + 1 > 0) {
for (int g = 2; g <= i - j + 1; ++g) {
hData[i][j] /= g;
}
}
}
}
return MatrixUtils.createRealMatrix(hData);
}
/**
* Verifies that {@code array} has length at least 2.
*
* @param array array to test
* @throws NullArgumentException if array is null
* @throws InsufficientDataException if array is too short
*/
private void checkArray(double[] array) {
if (array == null) {
throw new NullArgumentException(LocalizedFormats.NULL_NOT_ALLOWED);
}
if (array.length < 2) {
throw new InsufficientDataException(LocalizedFormats.INSUFFICIENT_OBSERVED_POINTS_IN_SAMPLE, array.length,
2);
}
}
/**
* Computes \( 1 + 2 \sum_{i=1}^\infty (-1)^i e^{-2 i^2 t^2} \) stopping when successive partial
* sums are within {@code tolerance} of one another, or when {@code maxIterations} partial sums
* have been computed. If the sum does not converge before {@code maxIterations} iterations a
* {@link TooManyIterationsException} is thrown.
*
* @param t argument
* @param tolerance Cauchy criterion for partial sums
* @param maxIterations maximum number of partial sums to compute
* @return Kolmogorov sum evaluated at t
* @throws TooManyIterationsException if the series does not converge
*/
public double ksSum(double t, double tolerance, int maxIterations) {
if (t == 0.0) {
return 0.0;
}
// TODO: for small t (say less than 1), the alternative expansion in part 3 of [1]
// from class javadoc should be used.
final double x = -2 * t * t;
int sign = -1;
long i = 1;
double partialSum = 0.5d;
double delta = 1;
while (delta > tolerance && i < maxIterations) {
delta = JdkMath.exp(x * i * i);
partialSum += sign * delta;
sign *= -1;
i++;
}
if (i == maxIterations) {
throw new TooManyIterationsException(maxIterations);
}
return partialSum * 2;
}
/**
* Given a d-statistic in the range [0, 1] and the two sample sizes n and m,
* an integral d-statistic in the range [0, n*m] is calculated, that can be used for
* comparison with other integral d-statistics. Depending whether {@code strict} is
* {@code true} or not, the returned value divided by (n*m) is greater than
* (resp greater than or equal to) the given d value (allowing some tolerance).
*
* @param d a d-statistic in the range [0, 1]
* @param n first sample size
* @param m second sample size
* @param strict whether the returned value divided by (n*m) is allowed to be equal to d
* @return the integral d-statistic in the range [0, n*m]
*/
private static long calculateIntegralD(double d, int n, int m, boolean strict) {
final double tol = 1e-12; // d-values within tol of one another are considered equal
long nm = n * (long)m;
long upperBound = (long)JdkMath.ceil((d - tol) * nm);
long lowerBound = (long)JdkMath.floor((d + tol) * nm);
if (strict && lowerBound == upperBound) {
return upperBound + 1L;
} else {
return upperBound;
}
}
/**
* Computes \(P(D_{n,m} > d)\) if {@code strict} is {@code true}; otherwise \(P(D_{n,m} \ge
* d)\), where \(D_{n,m}\) is the 2-sample Kolmogorov-Smirnov statistic. See
* {@link #kolmogorovSmirnovStatistic(double[], double[])} for the definition of \(D_{n,m}\).
* <p>
* The returned probability is exact, implemented by unwinding the recursive function
* definitions presented in [4] (class javadoc).
* </p>
*
* @param d D-statistic value
* @param n first sample size
* @param m second sample size
* @param strict whether or not the probability to compute is expressed as a strict inequality
* @return probability that a randomly selected m-n partition of m + n generates \(D_{n,m}\)
* greater than (resp. greater than or equal to) {@code d}
*/
public double exactP(double d, int n, int m, boolean strict) {
return 1 - n(m, n, m, n, calculateIntegralD(d, m, n, strict), strict) /
BinomialCoefficientDouble.value(n + m, m);
}
/**
* Uses the Kolmogorov-Smirnov distribution to approximate \(P(D_{n,m} > d)\) where \(D_{n,m}\)
* is the 2-sample Kolmogorov-Smirnov statistic. See
* {@link #kolmogorovSmirnovStatistic(double[], double[])} for the definition of \(D_{n,m}\).
* <p>
* Specifically, what is returned is \(1 - k(d \sqrt{mn / (m + n)})\) where \(k(t) = 1 + 2
* \sum_{i=1}^\infty (-1)^i e^{-2 i^2 t^2}\). See {@link #ksSum(double, double, int)} for
* details on how convergence of the sum is determined.
* </p>
*
* @param d D-statistic value
* @param n first sample size
* @param m second sample size
* @return approximate probability that a randomly selected m-n partition of m + n generates
* \(D_{n,m}\) greater than {@code d}
*/
public double approximateP(double d, int n, int m) {
return 1 - ksSum(d * JdkMath.sqrt(((double) m * (double) n) / ((double) m + (double) n)),
KS_SUM_CAUCHY_CRITERION, MAXIMUM_PARTIAL_SUM_COUNT);
}
/**
* Fills a boolean array randomly with a fixed number of {@code true} values.
* The method uses a simplified version of the Fisher-Yates shuffle algorithm.
* By processing first the {@code true} values followed by the remaining {@code false} values
* less random numbers need to be generated. The method is optimized for the case
* that the number of {@code true} values is larger than or equal to the number of
* {@code false} values.
*
* @param b boolean array
* @param numberOfTrueValues number of {@code true} values the boolean array should finally have
* @param rng random data generator
*/
private static void fillBooleanArrayRandomlyWithFixedNumberTrueValues(final boolean[] b, final int numberOfTrueValues, final UniformRandomProvider rng) {
Arrays.fill(b, true);
for (int k = numberOfTrueValues; k < b.length; k++) {
final int r = rng.nextInt(k + 1);
b[(b[r]) ? r : k] = false;
}
}
/**
* Uses Monte Carlo simulation to approximate \(P(D_{n,m} > d)\) where \(D_{n,m}\) is the
* 2-sample Kolmogorov-Smirnov statistic. See
* {@link #kolmogorovSmirnovStatistic(double[], double[])} for the definition of \(D_{n,m}\).
* <p>
* The simulation generates {@code iterations} random partitions of {@code m + n} into an
* {@code n} set and an {@code m} set, computing \(D_{n,m}\) for each partition and returning
* the proportion of values that are greater than {@code d}, or greater than or equal to
* {@code d} if {@code strict} is {@code false}.
* </p>
*
* @param d D-statistic value.
* @param n First sample size.
* @param m Second sample size.
* @param iterations Number of random partitions to generate.
* @param strict whether or not the probability to compute is expressed as a strict inequality
* @param rng RNG used for generating the partitions.
* @return proportion of randomly generated m-n partitions of m + n that result in \(D_{n,m}\)
* greater than (resp. greater than or equal to) {@code d}.
*/
public double monteCarloP(final double d,
final int n,
final int m,
final boolean strict,
final int iterations,
UniformRandomProvider rng) {
return integralMonteCarloP(calculateIntegralD(d, n, m, strict), n, m, iterations, rng);
}
/**
* Uses Monte Carlo simulation to approximate \(P(D_{n,m} >= d / (n * m))\)
* where \(D_{n,m}\) is the 2-sample Kolmogorov-Smirnov statistic.
* <p>
* Here {@code d} is the D-statistic represented as long value.
* The real D-statistic is obtained by dividing {@code d} by {@code n * m}.
* See also {@link #monteCarloP(double,int,int,boolean,int,UniformRandomProvider)}.
*
* @param d Integral D-statistic.
* @param n First sample size.
* @param m Second sample size.
* @param iterations Number of random partitions to generate.
* @param rng RNG used for generating the partitions.
* @return proportion of randomly generated m-n partitions of m + n that result in \(D_{n,m}\)
* greater than or equal to {@code d / (n * m))}.
*/
private double integralMonteCarloP(final long d,
final int n,
final int m,
final int iterations,
UniformRandomProvider rng) {
// ensure that nn is always the max of (n, m) to require fewer random numbers
final int nn = JdkMath.max(n, m);
final int mm = JdkMath.min(n, m);
final int sum = nn + mm;
int tail = 0;
final boolean[] b = new boolean[sum];
for (int i = 0; i < iterations; i++) {
fillBooleanArrayRandomlyWithFixedNumberTrueValues(b, nn, rng);
long curD = 0L;
for(int j = 0; j < b.length; ++j) {
if (b[j]) {
curD += mm;
if (curD >= d) {
tail++;
break;
}
} else {
curD -= nn;
if (curD <= -d) {
tail++;
break;
}
}
}
}
return (double) tail / iterations;
}
/**
* If there are no ties in the combined dataset formed from x and y,
* this method is a no-op.
* If there are ties, a uniform random deviate in
* is added to each value in x and y, and this method overwrites the
* data in x and y with the jittered values.
*
* @param x First sample.
* @param y Second sample.
* @throw NotANumberException if any of the input arrays contain
* a NaN value.
*/
private static void fixTies(double[] x, double[] y) {
if (hasTies(x, y)) {
// Add jitter using a fixed seed (so same arguments always give same results),
// low-initialization-overhead generator.
final UniformRandomProvider rng = RandomSource.SPLIT_MIX_64.create(876543217L);
// It is theoretically possible that jitter does not break ties, so repeat
// until all ties are gone. Bound the loop and throw MIE if bound is exceeded.
int fixedRangeTrials = 10;
int jitterRange = 10;
int ct = 0;
boolean ties = true;
do {
// Nudge the data using a fixed range.
jitter(x, rng, jitterRange);
jitter(y, rng, jitterRange);
ties = hasTies(x, y);
++ct;
} while (ties && ct < fixedRangeTrials);
if (ties) {
throw new MaxCountExceededException(fixedRangeTrials);
}
}
}
/**
* Returns true iff there are ties in the combined sample formed from
* x and y.
*
* @param x First sample.
* @param y Second sample.
* @return true if x and y together contain ties.
* @throw InsufficientDataException if the input arrays only contain infinite values.
* @throw NotANumberException if any of the input arrays contain a NaN value.
*/
private static boolean hasTies(double[] x, double[] y) {
final double[] values = MathArrays.unique(MathArrays.concatenate(x, y));
// "unique" moves NaN to the head of the output array.
if (Double.isNaN(values[0])) {
throw new NotANumberException();
}
int infCount = 0;
for (double v : values) {
if (Double.isInfinite(v)) {
++infCount;
}
}
if (infCount == values.length) {
throw new InsufficientDataException();
}
return values.length != x.length + y.length; // There are no ties.
}
/**
* Adds random jitter to {@code data} using deviates sampled from {@code dist}.
* <p>
* Note that jitter is applied in-place - i.e., the array
* values are overwritten with the result of applying jitter.</p>
*
* @param data input/output data array - entries overwritten by the method
* @param rng probability distribution to sample for jitter values
* @param ulp ulp used when generating random numbers
* @throws NullPointerException if either of the parameters is null
*/
private static void jitter(double[] data,
UniformRandomProvider rng,
int ulp) {
final int range = ulp * 2;
for (int i = 0; i < data.length; i++) {
final double orig = data[i];
if (Double.isInfinite(orig)) {
continue; // Avoid NaN creation.
}
final int rand = rng.nextInt(range) - ulp;
final double ulps = Math.ulp(orig);
final double r = rand * ulps;
data[i] += r;
}
}
/**
* The function C(i, j) defined in [4] (class javadoc), formula (5.5).
* defined to return 1 if |i/n - j/m| <= c; 0 otherwise. Here c is scaled up
* and recoded as a long to avoid rounding errors in comparison tests, so what
* is actually tested is |im - jn| <= cmn.
*
* @param i first path parameter
* @param j second path paramter
* @param m first sample size
* @param n second sample size
* @param cmn integral D-statistic (see {@link #calculateIntegralD(double, int, int, boolean)})
* @param strict whether or not the null hypothesis uses strict inequality
* @return C(i,j) for given m, n, c
*/
private static int c(int i, int j, int m, int n, long cmn, boolean strict) {
if (strict) {
return JdkMath.abs(i*(long)n - j*(long)m) <= cmn ? 1 : 0;
}
return JdkMath.abs(i*(long)n - j*(long)m) < cmn ? 1 : 0;
}
/**
* The function N(i, j) defined in [4] (class javadoc).
* Returns the number of paths over the lattice {(i,j) : 0 <= i <= n, 0 <= j <= m}
* from (0,0) to (i,j) satisfying C(h,k, m, n, c) = 1 for each (h,k) on the path.
* The return value is integral, but subject to overflow, so it is maintained and
* returned as a double.
*
* @param i first path parameter
* @param j second path parameter
* @param m first sample size
* @param n second sample size
* @param cnm integral D-statistic (see {@link #calculateIntegralD(double, int, int, boolean)})
* @param strict whether or not the null hypothesis uses strict inequality
* @return number or paths to (i, j) from (0,0) representing D-values as large as c for given m, n
*/
private static double n(int i, int j, int m, int n, long cnm, boolean strict) {
/*
* Unwind the recursive definition given in [4].
* Compute n(1,1), n(1,2)...n(2,1), n(2,2)... up to n(i,j), one row at a time.
* When n(i,*) are being computed, lag[] holds the values of n(i - 1, *).
*/
final double[] lag = new double[n];
double last = 0;
for (int k = 0; k < n; k++) {
lag[k] = c(0, k + 1, m, n, cnm, strict);
}
for (int k = 1; k <= i; k++) {
last = c(k, 0, m, n, cnm, strict);
for (int l = 1; l <= j; l++) {
lag[l - 1] = c(k, l, m, n, cnm, strict) * (last + lag[l - 1]);
last = lag[l - 1];
}
}
return last;
}
}