001/* 002 * Licensed to the Apache Software Foundation (ASF) under one or more 003 * contributor license agreements. See the NOTICE file distributed with 004 * this work for additional information regarding copyright ownership. 005 * The ASF licenses this file to You under the Apache License, Version 2.0 006 * (the "License"); you may not use this file except in compliance with 007 * the License. You may obtain a copy of the License at 008 * 009 * http://www.apache.org/licenses/LICENSE-2.0 010 * 011 * Unless required by applicable law or agreed to in writing, software 012 * distributed under the License is distributed on an "AS IS" BASIS, 013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 014 * See the License for the specific language governing permissions and 015 * limitations under the License. 016 */ 017package org.apache.commons.statistics.distribution; 018 019/** 020 * Implementation of the Gumbel distribution. 021 * 022 * <p>The probability density function of \( X \) is: 023 * 024 * <p>\[ f(x; \mu, \beta) = \frac{1}{\beta} e^{-(z+e^{-z})} \] 025 * 026 * <p>where \[ z = \frac{x - \mu}{\beta} \] 027 * 028 * <p>for \( \mu \) the location, 029 * \( \beta > 0 \) the scale, and 030 * \( x \in (-\infty, \infty) \). 031 * 032 * @see <a href="https://en.wikipedia.org/wiki/Gumbel_distribution">Gumbel distribution (Wikipedia)</a> 033 * @see <a href="https://mathworld.wolfram.com/GumbelDistribution.html">Gumbel distribution (MathWorld)</a> 034 */ 035public final class GumbelDistribution extends AbstractContinuousDistribution { 036 /** Support lower bound. */ 037 private static final double SUPPORT_LO = Double.NEGATIVE_INFINITY; 038 /** Support upper bound. */ 039 private static final double SUPPORT_HI = Double.POSITIVE_INFINITY; 040 /** π<sup>2</sup>/6. */ 041 private static final double PI_SQUARED_OVER_SIX = Math.PI * Math.PI / 6; 042 /** 043 * <a href="https://en.wikipedia.org/wiki/Euler%27s_constant"> 044 * Approximation of Euler's constant</a>. 045 */ 046 private static final double EULER = 0.57721566490153286060; 047 /** ln(ln(2)). */ 048 private static final double LN_LN_2 = -0.3665129205816643270124; 049 /** Location parameter. */ 050 private final double mu; 051 /** Scale parameter. */ 052 private final double beta; 053 054 /** 055 * @param mu Location parameter. 056 * @param beta Scale parameter (must be positive). 057 */ 058 private GumbelDistribution(double mu, 059 double beta) { 060 this.beta = beta; 061 this.mu = mu; 062 } 063 064 /** 065 * Creates a Gumbel distribution. 066 * 067 * @param mu Location parameter. 068 * @param beta Scale parameter (must be positive). 069 * @return the distribution 070 * @throws IllegalArgumentException if {@code beta <= 0} 071 */ 072 public static GumbelDistribution of(double mu, 073 double beta) { 074 if (beta <= 0) { 075 throw new DistributionException(DistributionException.NOT_STRICTLY_POSITIVE, beta); 076 } 077 return new GumbelDistribution(mu, beta); 078 } 079 080 /** 081 * Gets the location parameter of this distribution. 082 * 083 * @return the location parameter. 084 */ 085 public double getLocation() { 086 return mu; 087 } 088 089 /** 090 * Gets the scale parameter of this distribution. 091 * 092 * @return the scale parameter. 093 */ 094 public double getScale() { 095 return beta; 096 } 097 098 /** {@inheritDoc} */ 099 @Override 100 public double density(double x) { 101 if (x <= SUPPORT_LO) { 102 return 0; 103 } 104 105 final double z = (x - mu) / beta; 106 final double t = Math.exp(-z); 107 return Math.exp(-z - t) / beta; 108 } 109 110 /** {@inheritDoc} */ 111 @Override 112 public double logDensity(double x) { 113 if (x <= SUPPORT_LO) { 114 return Double.NEGATIVE_INFINITY; 115 } 116 117 final double z = (x - mu) / beta; 118 final double t = Math.exp(-z); 119 return -z - t - Math.log(beta); 120 } 121 122 /** {@inheritDoc} */ 123 @Override 124 public double cumulativeProbability(double x) { 125 final double z = (x - mu) / beta; 126 return Math.exp(-Math.exp(-z)); 127 } 128 129 /** {@inheritDoc} */ 130 @Override 131 public double survivalProbability(double x) { 132 final double z = (x - mu) / beta; 133 return -Math.expm1(-Math.exp(-z)); 134 } 135 136 /** {@inheritDoc} */ 137 @Override 138 public double inverseCumulativeProbability(double p) { 139 ArgumentUtils.checkProbability(p); 140 if (p == 0) { 141 return Double.NEGATIVE_INFINITY; 142 } else if (p == 1) { 143 return Double.POSITIVE_INFINITY; 144 } 145 return mu - Math.log(-Math.log(p)) * beta; 146 } 147 148 /** {@inheritDoc} */ 149 @Override 150 public double inverseSurvivalProbability(double p) { 151 ArgumentUtils.checkProbability(p); 152 if (p == 1) { 153 return Double.NEGATIVE_INFINITY; 154 } else if (p == 0) { 155 return Double.POSITIVE_INFINITY; 156 } 157 return mu - Math.log(-Math.log1p(-p)) * beta; 158 } 159 160 /** 161 * {@inheritDoc} 162 * 163 * <p>For location parameter \( \mu \) and scale parameter \( \beta \), the mean is: 164 * 165 * <p>\[ \mu + \beta \gamma \] 166 * 167 * <p>where \( \gamma \) is the 168 * <a href="https://mathworld.wolfram.com/Euler-MascheroniConstantApproximations.html"> 169 * Euler-Mascheroni constant</a>. 170 */ 171 @Override 172 public double getMean() { 173 return mu + EULER * beta; 174 } 175 176 /** 177 * {@inheritDoc} 178 * 179 * <p>For scale parameter \( \beta \), the variance is: 180 * 181 * <p>\[ \frac{\pi^2}{6} \beta^2 \] 182 */ 183 @Override 184 public double getVariance() { 185 return PI_SQUARED_OVER_SIX * beta * beta; 186 } 187 188 /** 189 * {@inheritDoc} 190 * 191 * <p>The lower bound of the support is always negative infinity. 192 * 193 * @return {@link Double#NEGATIVE_INFINITY negative infinity}. 194 */ 195 @Override 196 public double getSupportLowerBound() { 197 return SUPPORT_LO; 198 } 199 200 /** 201 * {@inheritDoc} 202 * 203 * <p>The upper bound of the support is always positive infinity. 204 * 205 * @return {@link Double#POSITIVE_INFINITY positive infinity}. 206 */ 207 @Override 208 public double getSupportUpperBound() { 209 return SUPPORT_HI; 210 } 211 212 /** {@inheritDoc} */ 213 @Override 214 double getMedian() { 215 // Overridden for the probability(double, double) method. 216 // This is intentionally not a public method. 217 // u - beta * ln(ln(2)) 218 return mu - beta * LN_LN_2; 219 } 220}