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 * \( \beta &gt; 0 \) 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    /** &pi;<sup>2</sup>/6. https://oeis.org/A013661. */
041    private static final double PI_SQUARED_OVER_SIX = 1.644934066848226436472415166646;
042    /**
043     * <a href="https://en.wikipedia.org/wiki/Euler%27s_constant">
044     * Approximation of Euler's constant</a>.
045     * https://oeis.org/A001620.
046     */
047    private static final double EULER = 0.5772156649015328606065;
048    /** ln(ln(2)). https://oeis.org/A074785. */
049    private static final double LN_LN_2 = -0.3665129205816643270124;
050    /** Location parameter. */
051    private final double mu;
052    /** Scale parameter. */
053    private final double beta;
054
055    /**
056     * @param mu Location parameter.
057     * @param beta Scale parameter (must be positive).
058     */
059    private GumbelDistribution(double mu,
060                               double beta) {
061        this.beta = beta;
062        this.mu = mu;
063    }
064
065    /**
066     * Creates a Gumbel distribution.
067     *
068     * @param mu Location parameter.
069     * @param beta Scale parameter (must be positive).
070     * @return the distribution
071     * @throws IllegalArgumentException if {@code beta <= 0}
072     */
073    public static GumbelDistribution of(double mu,
074                                        double beta) {
075        if (beta <= 0) {
076            throw new DistributionException(DistributionException.NOT_STRICTLY_POSITIVE, beta);
077        }
078        return new GumbelDistribution(mu, beta);
079    }
080
081    /**
082     * Gets the location parameter of this distribution.
083     *
084     * @return the location parameter.
085     */
086    public double getLocation() {
087        return mu;
088    }
089
090    /**
091     * Gets the scale parameter of this distribution.
092     *
093     * @return the scale parameter.
094     */
095    public double getScale() {
096        return beta;
097    }
098
099    /** {@inheritDoc} */
100    @Override
101    public double density(double x) {
102        if (x <= SUPPORT_LO) {
103            return 0;
104        }
105
106        final double z = (x - mu) / beta;
107        final double t = Math.exp(-z);
108        return Math.exp(-z - t) / beta;
109    }
110
111    /** {@inheritDoc} */
112    @Override
113    public double logDensity(double x) {
114        if (x <= SUPPORT_LO) {
115            return Double.NEGATIVE_INFINITY;
116        }
117
118        final double z = (x - mu) / beta;
119        final double t = Math.exp(-z);
120        return -z - t - Math.log(beta);
121    }
122
123    /** {@inheritDoc} */
124    @Override
125    public double cumulativeProbability(double x) {
126        final double z = (x - mu) / beta;
127        return Math.exp(-Math.exp(-z));
128    }
129
130    /** {@inheritDoc} */
131    @Override
132    public double survivalProbability(double x) {
133        final double z = (x - mu) / beta;
134        return -Math.expm1(-Math.exp(-z));
135    }
136
137    /** {@inheritDoc} */
138    @Override
139    public double inverseCumulativeProbability(double p) {
140        ArgumentUtils.checkProbability(p);
141        if (p == 0) {
142            return Double.NEGATIVE_INFINITY;
143        } else if (p == 1) {
144            return Double.POSITIVE_INFINITY;
145        }
146        return mu - Math.log(-Math.log(p)) * beta;
147    }
148
149    /** {@inheritDoc} */
150    @Override
151    public double inverseSurvivalProbability(double p) {
152        ArgumentUtils.checkProbability(p);
153        if (p == 1) {
154            return Double.NEGATIVE_INFINITY;
155        } else if (p == 0) {
156            return Double.POSITIVE_INFINITY;
157        }
158        return mu - Math.log(-Math.log1p(-p)) * beta;
159    }
160
161    /**
162     * {@inheritDoc}
163     *
164     * <p>For location parameter \( \mu \) and scale parameter \( \beta \), the mean is:
165     *
166     * <p>\[ \mu + \beta \gamma \]
167     *
168     * <p>where \( \gamma \) is the
169     * <a href="https://mathworld.wolfram.com/Euler-MascheroniConstantApproximations.html">
170     * Euler-Mascheroni constant</a>.
171     */
172    @Override
173    public double getMean() {
174        return mu + EULER * beta;
175    }
176
177    /**
178     * {@inheritDoc}
179     *
180     * <p>For scale parameter \( \beta \), the variance is:
181     *
182     * <p>\[ \frac{\pi^2}{6} \beta^2 \]
183     */
184    @Override
185    public double getVariance() {
186        return PI_SQUARED_OVER_SIX * beta * beta;
187    }
188
189    /**
190     * {@inheritDoc}
191     *
192     * <p>The lower bound of the support is always negative infinity.
193     *
194     * @return {@linkplain Double#NEGATIVE_INFINITY negative infinity}.
195     */
196    @Override
197    public double getSupportLowerBound() {
198        return SUPPORT_LO;
199    }
200
201    /**
202     * {@inheritDoc}
203     *
204     * <p>The upper bound of the support is always positive infinity.
205     *
206     * @return {@linkplain Double#POSITIVE_INFINITY positive infinity}.
207     */
208    @Override
209    public double getSupportUpperBound() {
210        return SUPPORT_HI;
211    }
212
213    /** {@inheritDoc} */
214    @Override
215    double getMedian() {
216        // Overridden for the probability(double, double) method.
217        // This is intentionally not a public method.
218        // u - beta * ln(ln(2))
219        return mu - beta * LN_LN_2;
220    }
221}