/*
* 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 java.math.BigDecimal;
import java.util.Arrays;
import java.util.Iterator;
import org.apache.commons.math3.distribution.RealDistribution;
import org.apache.commons.math3.exception.InsufficientDataException;
import org.apache.commons.math3.exception.MathArithmeticException;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.exception.NumberIsTooLargeException;
import org.apache.commons.math3.exception.OutOfRangeException;
import org.apache.commons.math3.exception.TooManyIterationsException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.fraction.BigFraction;
import org.apache.commons.math3.fraction.BigFractionField;
import org.apache.commons.math3.fraction.FractionConversionException;
import org.apache.commons.math3.linear.Array2DRowFieldMatrix;
import org.apache.commons.math3.linear.FieldMatrix;
import org.apache.commons.math3.linear.MatrixUtils;
import org.apache.commons.math3.linear.RealMatrix;
import org.apache.commons.math3.random.RandomGenerator;
import org.apache.commons.math3.random.Well19937c;
import org.apache.commons.math3.util.CombinatoricsUtils;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathArrays;
import static org.apache.commons.math3.util.MathUtils.PI_SQUARED;
import static org.apache.commons.math3.util.FastMath.PI;
/**
* 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>For very small samples (where the product of the sample sizes is less than
* {@value #SMALL_SAMPLE_PRODUCT}), the exact distribution is used to compute the p-value for the
* 2-sample test.</li>
* <li>For mid-size samples (product of sample sizes greater than or equal to
* {@value #SMALL_SAMPLE_PRODUCT} but less than {@value #LARGE_SAMPLE_PRODUCT}), Monte Carlo
* simulation is used to compute the p-value. The simulation randomly generates partitions of \(m +
* n\) into an \(m\)-set and an \(n\)-set and reports the proportion that give \(D\) values
* exceeding the observed value.</li>
* <li>When the product of the sample sizes exceeds {@value #LARGE_SAMPLE_PRODUCT}, the asymptotic
* distribution of \(D_{n,m}\) is used. See {@link #approximateP(double, int, int)} for details on
* the approximation.</li>
* </ul>
* </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 \(H_0 : D_{n,m} > d \)
* 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 #monteCarloP(double, int, int, boolean, int)} computes 2-sample p-values by Monte
* Carlo simulation</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>
* <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>
* </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.
* </p>
*
* @since 3.3
*/
public class KolmogorovSmirnovTest {
/**
* Bound on the number of partial sums in {@link #ksSum(double, double, int)}
*/
protected static final int MAXIMUM_PARTIAL_SUM_COUNT = 100000;
/** Convergence criterion for {@link #ksSum(double, double, int)} */
protected static final double KS_SUM_CAUCHY_CRITERION = 1E-20;
/** Convergence criterion for the sums in #pelzGood(double, double, int)} */
protected static final double PG_SUM_RELATIVE_ERROR = 1.0e-10;
/** When product of sample sizes is less than this value, 2-sample K-S test is exact */
protected static final int SMALL_SAMPLE_PRODUCT = 200;
/**
* When product of sample sizes exceeds this value, 2-sample K-S test uses asymptotic
* distribution for strict inequality p-value.
*/
protected static final int LARGE_SAMPLE_PRODUCT = 10000;
/** Default number of iterations used by {@link #monteCarloP(double, int, int, boolean, int)} */
protected static final int MONTE_CARLO_ITERATIONS = 1000000;
/** Random data generator used by {@link #monteCarloP(double, int, int, boolean, int)} */
private final RandomGenerator rng;
/**
* Construct a KolmogorovSmirnovTest instance with a default random data generator.
*/
public KolmogorovSmirnovTest() {
rng = new Well19937c();
}
/**
* Construct a KolmogorovSmirnovTest with the provided random data generator.
*
* @param rng random data generator used by {@link #monteCarloP(double, int, int, boolean, int)}
*/
public KolmogorovSmirnovTest(RandomGenerator rng) {
this.rng = rng;
}
/**
* 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(RealDistribution 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(RealDistribution 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 = FastMath.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 {@code strict = false}) as {@code kolmogorovSmirnovStatistic(x, y)}.
* <ul>
* <li>For very small samples (where the product of the sample sizes is less than
* {@value #SMALL_SAMPLE_PRODUCT}), the exact distribution is used to compute the p-value. This
* is accomplished by enumerating all partitions of the combined sample into two subsamples of
* the respective sample sizes, computing \(D_{n,m}\) for each partition and returning the
* proportion of partitions that give \(D\) values exceeding the observed value.</li>
* <li>For mid-size samples (product of sample sizes greater than or equal to
* {@value #SMALL_SAMPLE_PRODUCT} but less than {@value #LARGE_SAMPLE_PRODUCT}), Monte Carlo
* simulation is used to compute the p-value. The simulation randomly generates partitions and
* reports the proportion that give \(D\) values exceeding the observed value.</li>
* <li>When the product of the sample sizes exceeds {@value #LARGE_SAMPLE_PRODUCT}, the
* asymptotic distribution of \(D_{n,m}\) is used. See {@link #approximateP(double, int, int)}
* for details on the approximation.</li>
* </ul>
*
* @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
*/
public double kolmogorovSmirnovTest(double[] x, double[] y, boolean strict) {
if (x.length * y.length < SMALL_SAMPLE_PRODUCT) {
return exactP(kolmogorovSmirnovStatistic(x, y), x.length, y.length, strict);
}
if (x.length * y.length < LARGE_SAMPLE_PRODUCT) {
return monteCarloP(kolmogorovSmirnovStatistic(x, y), x.length, y.length, strict, MONTE_CARLO_ITERATIONS);
}
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(RealDistribution, 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) {
checkArray(x);
checkArray(y);
// Copy and sort the sample arrays
final double[] sx = MathArrays.copyOf(x);
final double[] sy = MathArrays.copyOf(y);
Arrays.sort(sx);
Arrays.sort(sy);
final int n = sx.length;
final int m = sy.length;
// Find the max difference between cdf_x and cdf_y
double supD = 0d;
// First walk x points
for (int i = 0; i < n; i++) {
final double cdf_x = (i + 1d) / n;
final int yIndex = Arrays.binarySearch(sy, sx[i]);
final double cdf_y = yIndex >= 0 ? (yIndex + 1d) / m : (-yIndex - 1d) / m;
final double curD = FastMath.abs(cdf_x - cdf_y);
if (curD > supD) {
supD = curD;
}
}
// Now look at y
for (int i = 0; i < m; i++) {
final double cdf_y = (i + 1d) / m;
final int xIndex = Arrays.binarySearch(sx, sy[i]);
final double cdf_x = xIndex >= 0 ? (xIndex + 1d) / n : (-xIndex - 1d) / n;
final double curD = FastMath.abs(cdf_x - cdf_y);
if (curD > supD) {
supD = curD;
}
}
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(RealDistribution 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(RealDistribution 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;
}
/**
* 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 org.apache.commons.math3.fraction.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 org.apache.commons.math3.fraction.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)
throws MathArithmeticException {
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 org.apache.commons.math3.fraction.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)
throws MathArithmeticException {
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 org.apache.commons.math3.fraction.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 org.apache.commons.math3.fraction.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)
throws MathArithmeticException {
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 org.apache.commons.math3.fraction.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 org.apache.commons.math3.fraction.BigFraction} in expressing {@code d} as \((k
* - h) / m\) for integer {@code k, m} and \(0 \le h < 1\).
*/
private double exactK(double d, int n)
throws MathArithmeticException {
final int k = (int) Math.ceil(n * d);
final FieldMatrix<BigFraction> H = this.createExactH(d, n);
final FieldMatrix<BigFraction> Hpower = H.power(n);
BigFraction pFrac = Hpower.getEntry(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, BigDecimal.ROUND_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 = FastMath.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 = FastMath.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 * FastMath.sqrt(2 * FastMath.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) * FastMath.exp(-PI_SQUARED * kTerm2 / twoZ2);
sum += increment;
if (FastMath.abs(increment) < PG_SUM_RELATIVE_ERROR * FastMath.abs(sum)) {
break;
}
}
if (k == MAXIMUM_PARTIAL_SUM_COUNT) {
throw new TooManyIterationsException(MAXIMUM_PARTIAL_SUM_COUNT);
}
final double sqrtHalfPi = FastMath.sqrt(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) * FastMath.exp(-PI_SQUARED * kTerm2 / twoZ2);
sum += increment;
if (FastMath.abs(increment) < PG_SUM_RELATIVE_ERROR * FastMath.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 = k * k;
increment = PI_SQUARED * kTerm2 * FastMath.exp(-PI_SQUARED * kTerm2 / twoZ2);
sum2 += increment;
if (FastMath.abs(increment) < PG_SUM_RELATIVE_ERROR * FastMath.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) *
FastMath.exp(-PI_SQUARED * kTerm2 / twoZ2);
sum += increment;
if (FastMath.abs(increment) < PG_SUM_RELATIVE_ERROR * FastMath.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 = k * k;
kTerm4 = kTerm2 * kTerm2;
increment = (-pi4 * kTerm4 + 3 * PI_SQUARED * kTerm2 * z2) *
FastMath.exp(-PI_SQUARED * kTerm2 / twoZ2);
sum2 += increment;
if (FastMath.abs(increment) < PG_SUM_RELATIVE_ERROR * FastMath.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 FractionConversionException if algorithm fails to convert {@code h} to a
* {@link org.apache.commons.math3.fraction.BigFraction} in expressing {@code d} as \((k
* - h) / m\) for integer {@code k, m} and \(0 <= h < 1\).
*/
private FieldMatrix<BigFraction> createExactH(double d, int n)
throws NumberIsTooLargeException, FractionConversionException {
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 = new BigFraction(hDouble, 1.0e-20, 10000);
} catch (final FractionConversionException e1) {
try {
h = new BigFraction(hDouble, 1.0e-10, 10000);
} catch (final FractionConversionException e2) {
h = new BigFraction(hDouble, 1.0e-5, 10000);
}
}
final BigFraction[][] Hdata = new BigFraction[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] = BigFraction.ZERO;
} else {
Hdata[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[i][0] = Hdata[i][0].subtract(hPowers[i]);
Hdata[m - 1][i] = Hdata[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(BigFraction.ONE_HALF) == 1) {
Hdata[m - 1][0] = Hdata[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[i][j] = Hdata[i][j].divide(g);
}
}
}
}
return new Array2DRowFieldMatrix<BigFraction>(BigFractionField.getInstance(), 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] += FastMath.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) {
// 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 = FastMath.exp(x * i * i);
partialSum += sign * delta;
sign *= -1;
i++;
}
if (i == maxIterations) {
throw new TooManyIterationsException(maxIterations);
}
return partialSum * 2;
}
/**
* 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, obtained by enumerating all partitions of {@code m + n}
* into {@code m} and {@code n} sets, computing \(D_{n,m}\) for each partition and counting the
* number of partitions that yield \(D_{n,m}\) values exceeding (resp. greater than or equal to)
* {@code d}.
* </p>
* <p>
* <strong>USAGE NOTE</strong>: Since this method enumerates all combinations in \({m+n} \choose
* {n}\), it is very slow if called for large {@code m, n}. For this reason,
* {@link #kolmogorovSmirnovTest(double[], double[])} uses this only for {@code m * n < }
* {@value #SMALL_SAMPLE_PRODUCT}.
* </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) {
Iterator<int[]> combinationsIterator = CombinatoricsUtils.combinationsIterator(n + m, n);
long tail = 0;
final double[] nSet = new double[n];
final double[] mSet = new double[m];
while (combinationsIterator.hasNext()) {
// Generate an n-set
final int[] nSetI = combinationsIterator.next();
// Copy the n-set to nSet and its complement to mSet
int j = 0;
int k = 0;
for (int i = 0; i < n + m; i++) {
if (j < n && nSetI[j] == i) {
nSet[j++] = i;
} else {
mSet[k++] = i;
}
}
final double curD = kolmogorovSmirnovStatistic(nSet, mSet);
if (curD > d) {
tail++;
} else if (curD == d && !strict) {
tail++;
}
}
return (double) tail / (double) CombinatoricsUtils.binomialCoefficient(n + m, n);
}
/**
* 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. This implementation passes {@code ksSum}
* {@value #KS_SUM_CAUCHY_CRITERION} as {@code tolerance} and
* {@value #MAXIMUM_PARTIAL_SUM_COUNT} as {@code maxIterations}.
* </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) {
final double dm = m;
final double dn = n;
return 1 - ksSum(d * FastMath.sqrt((dm * dn) / (dm + dn)), KS_SUM_CAUCHY_CRITERION, MAXIMUM_PARTIAL_SUM_COUNT);
}
/**
* 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
* @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(double d, int n, int m, boolean strict, int iterations) {
final int[] nPlusMSet = MathArrays.natural(m + n);
final double[] nSet = new double[n];
final double[] mSet = new double[m];
int tail = 0;
for (int i = 0; i < iterations; i++) {
copyPartition(nSet, mSet, nPlusMSet, n, m);
final double curD = kolmogorovSmirnovStatistic(nSet, mSet);
if (curD > d) {
tail++;
} else if (curD == d && !strict) {
tail++;
}
MathArrays.shuffle(nPlusMSet, rng);
Arrays.sort(nPlusMSet, 0, n);
}
return (double) tail / iterations;
}
/**
* Copies the first {@code n} elements of {@code nSetI} into {@code nSet} and its complement
* relative to {@code m + n} into {@code mSet}. For example, if {@code m = 3}, {@code n = 3} and
* {@code nSetI = [1,4,5,2,3,0]} then after this method returns, we will have
* {@code nSet = [1,4,5], mSet = [0,2,3]}.
* <p>
* <strong>Precondition:</strong> The first {@code n} elements of {@code nSetI} must be sorted
* in ascending order.
* </p>
*
* @param nSet array to fill with the first {@code n} elements of {@code nSetI}
* @param mSet array to fill with the {@code m} complementary elements of {@code nSet} relative
* to {@code m + n}
* @param nSetI array whose first {@code n} elements specify the members of {@code nSet}
* @param n number of elements in the first output array
* @param m number of elements in the second output array
*/
private void copyPartition(double[] nSet, double[] mSet, int[] nSetI, int n, int m) {
int j = 0;
int k = 0;
for (int i = 0; i < n + m; i++) {
if (j < n && nSetI[j] == i) {
nSet[j++] = i;
} else {
mSet[k++] = i;
}
}
}
}