/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.stat.inference;
import org.apache.commons.math3.distribution.NormalDistribution;
import org.apache.commons.math3.distribution.UniformRealDistribution;
import org.apache.commons.math3.random.Well19937c;
import org.apache.commons.math3.util.FastMath;
import org.junit.Assert;
import org.junit.Test;
/**
* Test cases for {@link KolmogorovSmirnovTest}.
*
* @since 3.3
*/
public class KolmogorovSmirnovTestTest {
protected static final double TOLERANCE = 10e-10;
// Random N(0,1) values generated using R rnorm
protected static final double[] gaussian = {
0.26055895, -0.63665233, 1.51221323, 0.61246988, -0.03013003, -1.73025682, -0.51435805, 0.70494168, 0.18242945,
0.94734336, -0.04286604, -0.37931719, -1.07026403, -2.05861425, 0.11201862, 0.71400136, -0.52122185,
-0.02478725, -1.86811649, -1.79907688, 0.15046279, 1.32390193, 1.55889719, 1.83149171, -0.03948003,
-0.98579207, -0.76790540, 0.89080682, 0.19532153, 0.40692841, 0.15047336, -0.58546562, -0.39865469, 0.77604271,
-0.65188221, -1.80368554, 0.65273365, -0.75283102, -1.91022150, -0.07640869, -1.08681188, -0.89270600,
2.09017508, 0.43907981, 0.10744033, -0.70961218, 1.15707300, 0.44560525, -2.04593349, 0.53816843, -0.08366640,
0.24652218, 1.80549401, -0.99220707, -1.14589408, -0.27170290, -0.49696855, 0.00968353, -1.87113545,
-1.91116529, 0.97151891, -0.73576115, -0.59437029, 0.72148436, 0.01747695, -0.62601157, -1.00971538,
-1.42691397, 1.03250131, -0.30672627, -0.15353992, -1.19976069, -0.68364218, 0.37525652, -0.46592881,
-0.52116168, -0.17162202, 1.04679215, 0.25165971, -0.04125231, -0.23756244, -0.93389975, 0.75551407,
0.08347445, -0.27482228, -0.4717632, -0.1867746, -0.1166976, 0.5763333, 0.1307952, 0.7630584, -0.3616248,
2.1383790, -0.7946630, 0.0231885, 0.7919195, 1.6057144, -0.3802508, 0.1229078, 1.5252901, -0.8543149, 0.3025040
};
// Random N(0, 1.6) values generated using R rnorm
protected static final double[] gaussian2 = {
2.88041498038308, -0.632349445671017, 0.402121295225571, 0.692626364613243, 1.30693446815426,
-0.714176317131286, -0.233169206599583, 1.09113298322107, -1.53149079994305, 1.23259966205809,
1.01389927412503, 0.0143898711497477, -0.512813545447559, 2.79364360835469, 0.662008875538092,
1.04861546834788, -0.321280099931466, 0.250296656278743, 1.75820367603736, -2.31433523590905,
-0.462694696086403, 0.187725700950191, -2.24410950019152, 2.83473751105445, 0.252460174391016,
1.39051945380281, -1.56270144203134, 0.998522814471644, -1.50147469080896, 0.145307533554146,
0.469089457043406, -0.0914780723809334, -0.123446939266548, -0.610513388160565, -3.71548343891957,
-0.329577317349478, -0.312973794075871, 2.02051909758923, 2.85214308266271, 0.0193222002327237,
-0.0322422268266562, 0.514736012106768, 0.231484953375887, -2.22468798953629, 1.42197716075595,
2.69988043856357, 0.0443757119128293, 0.721536984407798, -0.0445688839903234, -0.294372724550705,
0.234041580912698, -0.868973119365727, 1.3524893453845, -0.931054600134503, -0.263514296006792,
0.540949457402918, -0.882544288773685, -0.34148675747989, 1.56664494810034, 2.19850536566584,
-0.667972122928022, -0.70889669526203, -0.00251758193079668, 2.39527162977682, -2.7559594317269,
-0.547393502656671, -2.62144031572617, 2.81504147017922, -1.02036850201042, -1.00713927602786,
-0.520197775122254, 1.00625480138649, 2.46756916531313, 1.64364743727799, 0.704545210648595,
-0.425885789416992, -1.78387854908546, -0.286783886710481, 0.404183648369076, -0.369324280845769,
-0.0391185138840443, 2.41257787857293, 2.49744281317859, -0.826964496939021, -0.792555379958975,
1.81097685787403, -0.475014580016638, 1.23387615291805, 0.646615294802053, 1.88496377454523, 1.20390698380814,
-0.27812153371728, 2.50149494533101, 0.406964323253817, -1.72253451309982, 1.98432494184332, 2.2223658560333,
0.393086362404685, -0.504073151377089, -0.0484610869883821
};
// Random uniform (0, 1) generated using R runif
protected static final double[] uniform = {
0.7930305, 0.6424382, 0.8747699, 0.7156518, 0.1845909, 0.2022326, 0.4877206, 0.8928752, 0.2293062, 0.4222006,
0.1610459, 0.2830535, 0.9946345, 0.7329499, 0.26411126, 0.87958133, 0.29827437, 0.39185988, 0.38351185,
0.36359611, 0.48646472, 0.05577866, 0.56152250, 0.52672013, 0.13171783, 0.95864085, 0.03060207, 0.33514887,
0.72508148, 0.38901437, 0.9978665, 0.5981300, 0.1065388, 0.7036991, 0.1071584, 0.4423963, 0.1107071, 0.6437221,
0.58523872, 0.05044634, 0.65999539, 0.37367260, 0.73270024, 0.47473755, 0.74661163, 0.50765549, 0.05377347,
0.40998009, 0.55235182, 0.21361998, 0.63117971, 0.18109222, 0.89153510, 0.23203248, 0.6177106, 0.6856418,
0.2158557, 0.9870501, 0.2036914, 0.2100311, 0.9065020, 0.7459159, 0.56631790, 0.06753629, 0.39684629,
0.52504615, 0.14199103, 0.78551120, 0.90503321, 0.80452362, 0.9960115, 0.8172592, 0.5831134, 0.8794187,
0.2021501, 0.2923505, 0.9561824, 0.8792248, 0.85201008, 0.02945562, 0.26200374, 0.11382818, 0.17238856,
0.36449473, 0.69688273, 0.96216330, 0.4859432, 0.4503438, 0.1917656, 0.8357845, 0.9957812, 0.4633570,
0.8654599, 0.4597996, 0.68190289, 0.58887855, 0.09359396, 0.98081979, 0.73659533, 0.89344777, 0.18903099,
0.97660425
};
/** Unit normal distribution, unit normal data */
@Test
public void testOneSampleGaussianGaussian() {
final KolmogorovSmirnovTest test = new KolmogorovSmirnovTest();
final NormalDistribution unitNormal = new NormalDistribution(0d, 1d);
// Uncomment to run exact test - takes about a minute. Same value is used in R tests and for
// approx.
// Assert.assertEquals(0.3172069207622391, test.kolmogorovSmirnovTest(unitNormal, gaussian,
// true), TOLERANCE);
Assert.assertEquals(0.3172069207622391, test.kolmogorovSmirnovTest(unitNormal, gaussian, false), TOLERANCE);
Assert.assertFalse(test.kolmogorovSmirnovTest(unitNormal, gaussian, 0.05));
Assert.assertEquals(0.0932947561266756, test.kolmogorovSmirnovStatistic(unitNormal, gaussian), TOLERANCE);
}
/** Unit normal distribution, unit normal data, small dataset */
@Test
public void testOneSampleGaussianGaussianSmallSample() {
final KolmogorovSmirnovTest test = new KolmogorovSmirnovTest();
final NormalDistribution unitNormal = new NormalDistribution(0d, 1d);
final double[] shortGaussian = new double[50];
System.arraycopy(gaussian, 0, shortGaussian, 0, 50);
Assert.assertEquals(0.683736463728347, test.kolmogorovSmirnovTest(unitNormal, shortGaussian, false), TOLERANCE);
Assert.assertFalse(test.kolmogorovSmirnovTest(unitNormal, gaussian, 0.05));
Assert.assertEquals(0.09820779969463278, test.kolmogorovSmirnovStatistic(unitNormal, shortGaussian), TOLERANCE);
}
/** Unit normal distribution, uniform data */
@Test
public void testOneSampleGaussianUniform() {
final KolmogorovSmirnovTest test = new KolmogorovSmirnovTest();
final NormalDistribution unitNormal = new NormalDistribution(0d, 1d);
// Uncomment to run exact test - takes a long time. Same value is used in R tests and for
// approx.
// Assert.assertEquals(0.3172069207622391, test.kolmogorovSmirnovTest(unitNormal, uniform,
// true), TOLERANCE);
Assert.assertEquals(8.881784197001252E-16, test.kolmogorovSmirnovTest(unitNormal, uniform, false), TOLERANCE);
Assert.assertFalse(test.kolmogorovSmirnovTest(unitNormal, gaussian, 0.05));
Assert.assertEquals(0.5117493931609258, test.kolmogorovSmirnovStatistic(unitNormal, uniform), TOLERANCE);
}
/** Uniform distribution, uniform data */
// @Test - takes about 6 seconds, uncomment for
public void testOneSampleUniformUniform() {
final KolmogorovSmirnovTest test = new KolmogorovSmirnovTest();
final UniformRealDistribution unif = new UniformRealDistribution(-0.5, 0.5);
Assert.assertEquals(8.881784197001252E-16, test.kolmogorovSmirnovTest(unif, uniform, false), TOLERANCE);
Assert.assertTrue(test.kolmogorovSmirnovTest(unif, uniform, 0.05));
Assert.assertEquals(0.5400666982352942, test.kolmogorovSmirnovStatistic(unif, uniform), TOLERANCE);
}
/** Uniform distribution, uniform data, small dataset */
@Test
public void testOneSampleUniformUniformSmallSample() {
final KolmogorovSmirnovTest test = new KolmogorovSmirnovTest();
final UniformRealDistribution unif = new UniformRealDistribution(-0.5, 0.5);
final double[] shortUniform = new double[20];
System.arraycopy(uniform, 0, shortUniform, 0, 20);
Assert.assertEquals(4.117594598618268E-9, test.kolmogorovSmirnovTest(unif, shortUniform, false), TOLERANCE);
Assert.assertTrue(test.kolmogorovSmirnovTest(unif, shortUniform, 0.05));
Assert.assertEquals(0.6610459, test.kolmogorovSmirnovStatistic(unif, shortUniform), TOLERANCE);
}
/** Uniform distribution, unit normal dataset */
@Test
public void testOneSampleUniformGaussian() {
final KolmogorovSmirnovTest test = new KolmogorovSmirnovTest();
final UniformRealDistribution unif = new UniformRealDistribution(-0.5, 0.5);
// Value was obtained via exact test, validated against R. Running exact test takes a long
// time.
Assert.assertEquals(4.9405812774239166E-11, test.kolmogorovSmirnovTest(unif, gaussian, false), TOLERANCE);
Assert.assertTrue(test.kolmogorovSmirnovTest(unif, gaussian, 0.05));
Assert.assertEquals(0.3401058049019608, test.kolmogorovSmirnovStatistic(unif, gaussian), TOLERANCE);
}
/** Small samples - exact p-value, checked against R */
@Test
public void testTwoSampleSmallSampleExact() {
final KolmogorovSmirnovTest test = new KolmogorovSmirnovTest();
final double[] smallSample1 = {
6, 7, 9, 13, 19, 21, 22, 23, 24
};
final double[] smallSample2 = {
10, 11, 12, 16, 20, 27, 28, 32, 44, 54
};
// Reference values from R, version 2.15.3 - R uses non-strict inequality in null hypothesis
Assert
.assertEquals(0.105577085453247, test.kolmogorovSmirnovTest(smallSample1, smallSample2, false), TOLERANCE);
Assert.assertEquals(0.5, test.kolmogorovSmirnovStatistic(smallSample1, smallSample2), TOLERANCE);
}
/**
* Checks exact p-value computations using critical values from Table 9 in V.K Rohatgi, An
* Introduction to Probability and Mathematical Statistics, Wiley, 1976, ISBN 0-471-73135-8.
*/
@Test
public void testTwoSampleExactP() {
checkExactTable(4, 6, 5d / 6d, 0.01d);
checkExactTable(4, 7, 17d / 28d, 0.2d);
checkExactTable(6, 7, 29d / 42d, 0.05d);
checkExactTable(4, 10, 7d / 10d, 0.05d);
checkExactTable(5, 15, 11d / 15d, 0.02d);
checkExactTable(9, 10, 31d / 45d, 0.01d);
checkExactTable(7, 10, 43d / 70d, 0.05d);
}
@Test
public void testTwoSampleApproximateCritialValues() {
final double tol = .01;
final double[] alpha = {
0.10, 0.05, 0.025, 0.01, 0.005, 0.001
};
// From Wikipedia KS article - TODO: get (and test) more precise values
final double[] c = {
1.22, 1.36, 1.48, 1.63, 1.73, 1.95
};
final int k[] = {
60, 100, 500
};
double n;
double m;
for (int i = 0; i < k.length; i++) {
for (int j = 0; j < i; j++) {
n = k[i];
m = k[j];
for (int l = 0; l < alpha.length; l++) {
final double dCrit = c[l] * FastMath.sqrt((n + m) / (n * m));
checkApproximateTable(k[i], k[j], dCrit, alpha[l], tol);
}
}
}
}
@Test
public void testPelzGoodApproximation() {
KolmogorovSmirnovTest ksTest = new KolmogorovSmirnovTest();
final double d[] = {0.15, 0.20, 0.25, 0.3, 0.35, 0.4};
final int n[] = {141, 150, 180, 220, 1000};
// Reference values computed using the Pelz method from
// http://simul.iro.umontreal.ca/ksdir/KolmogorovSmirnovDist.java
final double ref[] = {
0.9968940168727819, 0.9979326624184857, 0.9994677598604506, 0.9999128354780209, 0.9999999999998661,
0.9999797514476236, 0.9999902122242081, 0.9999991327060908, 0.9999999657681911, 0.9999999999977929,
0.9999999706444976, 0.9999999906571532, 0.9999999997949596, 0.999999999998745, 0.9999999999993876,
0.9999999999916627, 0.9999999999984447, 0.9999999999999936, 0.999999999999341, 0.9999999999971508,
0.9999999999999877, 0.9999999999999191, 0.9999999999999254, 0.9999999999998178, 0.9999999999917788,
0.9999999999998556, 0.9999999999992014, 0.9999999999988859, 0.9999999999999325, 0.9999999999821726
};
final double tol = 10e-15;
int k = 0;
for (int i = 0; i < 6; i++) {
for (int j = 0; j < 5; j++, k++) {
Assert.assertEquals(ref[k], ksTest.pelzGood(d[i], n[j]), tol);
}
}
}
/** Verifies large sample approximate p values against R */
@Test
public void testTwoSampleApproximateP() {
final KolmogorovSmirnovTest test = new KolmogorovSmirnovTest();
// Reference values from R, version 2.15.3
Assert.assertEquals(0.0319983962391632, test.kolmogorovSmirnovTest(gaussian, gaussian2), TOLERANCE);
Assert.assertEquals(0.202352941176471, test.kolmogorovSmirnovStatistic(gaussian, gaussian2), TOLERANCE);
}
/**
* Verifies that Monte Carlo simulation gives results close to exact p values. This test is a
* little long-running (more than two minutes on a fast machine), so is disabled by default.
*/
// @Test
public void testTwoSampleMonteCarlo() {
final KolmogorovSmirnovTest test = new KolmogorovSmirnovTest(new Well19937c(1000));
final int sampleSize = 14;
final double tol = .001;
final double[] shortUniform = new double[sampleSize];
System.arraycopy(uniform, 0, shortUniform, 0, sampleSize);
final double[] shortGaussian = new double[sampleSize];
final double[] shortGaussian2 = new double[sampleSize];
System.arraycopy(gaussian, 0, shortGaussian, 0, sampleSize);
System.arraycopy(gaussian, 10, shortGaussian2, 0, sampleSize);
final double[] d = {
test.kolmogorovSmirnovStatistic(shortGaussian, shortUniform),
test.kolmogorovSmirnovStatistic(shortGaussian2, shortGaussian)
};
for (double dv : d) {
double exactPStrict = test.exactP(dv, sampleSize, sampleSize, true);
double exactPNonStrict = test.exactP(dv, sampleSize, sampleSize, false);
double montePStrict = test.monteCarloP(dv, sampleSize, sampleSize, true,
KolmogorovSmirnovTest.MONTE_CARLO_ITERATIONS);
double montePNonStrict = test.monteCarloP(dv, sampleSize, sampleSize, false,
KolmogorovSmirnovTest.MONTE_CARLO_ITERATIONS);
Assert.assertEquals(exactPStrict, montePStrict, tol);
Assert.assertEquals(exactPNonStrict, montePNonStrict, tol);
}
}
/**
* Verifies the inequality exactP(criticalValue, n, m, true) < alpha < exactP(criticalValue, n,
* m, false).
*
* Note that the validity of this check depends on the fact that alpha lies strictly between two
* attained values of the distribution and that criticalValue is one of the attained values. The
* critical value table (reference below) uses attained values. This test therefore also
* verifies that criticalValue is attained.
*
* @param n first sample size
* @param m second sample size
* @param criticalValue critical value
* @param alpha significance level
*/
private void checkExactTable(int n, int m, double criticalValue, double alpha) {
final KolmogorovSmirnovTest test = new KolmogorovSmirnovTest();
Assert.assertTrue(test.exactP(criticalValue, n, m, true) < alpha);
Assert.assertTrue(test.exactP(criticalValue, n, m, false) > alpha);
}
/**
* Verifies that approximateP(criticalValue, n, m) is within epsilon of alpha.
*
* @param n first sample size
* @param m second sample size
* @param criticalValue critical value (from table)
* @param alpha significance level
* @param epsilon tolerance
*/
private void checkApproximateTable(int n, int m, double criticalValue, double alpha, double epsilon) {
final KolmogorovSmirnovTest test = new KolmogorovSmirnovTest();
Assert.assertEquals(alpha, test.approximateP(criticalValue, n, m), epsilon);
}
}