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
019import org.apache.commons.numbers.gamma.Erf;
020import org.apache.commons.numbers.gamma.Erfc;
021import org.apache.commons.numbers.gamma.InverseErf;
022import org.apache.commons.numbers.gamma.InverseErfc;
023import org.apache.commons.rng.UniformRandomProvider;
024import org.apache.commons.rng.sampling.distribution.LevySampler;
025
026/**
027 * Implementation of the Lévy distribution.
028 *
029 * <p>The probability density function of \( X \) is:
030 *
031 * <p>\[ f(x; \mu, c) = \sqrt{\frac{c}{2\pi}}~~\frac{e^{ -\frac{c}{2(x-\mu)}}} {(x-\mu)^{3/2}} \]
032 *
033 * <p>for \( \mu \) the location,
034 * \( c &gt; 0 \) the scale, and
035 * \( x \in [\mu, \infty) \).
036 *
037 * @see <a href="https://en.wikipedia.org/wiki/L%C3%A9vy_distribution">L&eacute;vy distribution (Wikipedia)</a>
038 * @see <a href="https://mathworld.wolfram.com/LevyDistribution.html">L&eacute;vy distribution (MathWorld)</a>
039 */
040public final class LevyDistribution extends AbstractContinuousDistribution {
041    /** 1 / 2(erfc^-1 (0.5))^2. Computed using Matlab's VPA to 30 digits. */
042    private static final double HALF_OVER_ERFCINV_HALF_SQUARED = 2.1981093383177324039996779530797;
043    /** Location parameter. */
044    private final double mu;
045    /** Scale parameter. */
046    private final double c;
047    /** Half of c (for calculations). */
048    private final double halfC;
049
050    /**
051     * @param mu Location parameter.
052     * @param c Scale parameter.
053     */
054    private LevyDistribution(double mu,
055                             double c) {
056        this.mu = mu;
057        this.c = c;
058        this.halfC = 0.5 * c;
059    }
060
061    /**
062     * Creates a Levy distribution.
063     *
064     * @param mu Location parameter.
065     * @param c Scale parameter.
066     * @return the distribution
067     * @throws IllegalArgumentException if {@code c <= 0}.
068     */
069    public static LevyDistribution of(double mu,
070                                      double c) {
071        if (c <= 0) {
072            throw new DistributionException(DistributionException.NOT_STRICTLY_POSITIVE,
073                                            c);
074        }
075        return new LevyDistribution(mu, c);
076    }
077
078    /**
079     * Gets the location parameter of this distribution.
080     *
081     * @return the location parameter.
082     */
083    public double getLocation() {
084        return mu;
085    }
086
087    /**
088     * Gets the scale parameter of this distribution.
089     *
090     * @return the scale parameter.
091     */
092    public double getScale() {
093        return c;
094    }
095
096    /**
097     * {@inheritDoc}
098     *
099     * <p>If {@code x} is less than the location parameter then {@code 0} is
100     * returned, as in these cases the distribution is not defined.
101     */
102    @Override
103    public double density(final double x) {
104        if (x <= mu) {
105            // x=mu creates NaN:
106            // sqrt(c / 2pi) * exp(-c / 2(x-mu)) / (x-mu)^1.5
107            // = F * exp(-inf) * (x-mu)^-1.5 = F * 0 * inf
108            // Return 0 for this case.
109            return 0;
110        }
111
112        final double delta = x - mu;
113        final double f = halfC / delta;
114        return Math.sqrt(f / Math.PI) * Math.exp(-f) / delta;
115    }
116
117    /** {@inheritDoc} */
118    @Override
119    public double logDensity(double x) {
120        if (x <= mu) {
121            return Double.NEGATIVE_INFINITY;
122        }
123
124        final double delta = x - mu;
125        final double f     = halfC / delta;
126        return 0.5 * Math.log(f / Math.PI) - f - Math.log(delta);
127    }
128
129    /** {@inheritDoc} */
130    @Override
131    public double cumulativeProbability(final double x) {
132        if (x <= mu) {
133            return 0;
134        }
135        return Erfc.value(Math.sqrt(halfC / (x - mu)));
136    }
137
138    /** {@inheritDoc} */
139    @Override
140    public double survivalProbability(final double x) {
141        if (x <= mu) {
142            return 1;
143        }
144        return Erf.value(Math.sqrt(halfC / (x - mu)));
145    }
146
147    /** {@inheritDoc} */
148    @Override
149    public double inverseCumulativeProbability(double p) {
150        ArgumentUtils.checkProbability(p);
151        final double t = InverseErfc.value(p);
152        return mu + halfC / (t * t);
153    }
154
155    /** {@inheritDoc} */
156    @Override
157    public double inverseSurvivalProbability(double p) {
158        ArgumentUtils.checkProbability(p);
159        final double t = InverseErf.value(p);
160        return mu + halfC / (t * t);
161    }
162
163    /**
164     * {@inheritDoc}
165     *
166     * <p>The mean is equal to positive infinity.
167     *
168     * @return {@linkplain Double#POSITIVE_INFINITY positive infinity}.
169     */
170    @Override
171    public double getMean() {
172        return Double.POSITIVE_INFINITY;
173    }
174
175    /**
176     * {@inheritDoc}
177     *
178     * <p>The variance is equal to positive infinity.
179     *
180     * @return {@linkplain Double#POSITIVE_INFINITY positive infinity}.
181     */
182    @Override
183    public double getVariance() {
184        return Double.POSITIVE_INFINITY;
185    }
186
187    /**
188     * {@inheritDoc}
189     *
190     * <p>The lower bound of the support is the {@linkplain #getLocation() location}.
191     *
192     * @return location.
193     */
194    @Override
195    public double getSupportLowerBound() {
196        return getLocation();
197    }
198
199    /**
200     * {@inheritDoc}
201     *
202     * <p>The upper bound of the support is always positive infinity.
203     *
204     * @return {@linkplain Double#POSITIVE_INFINITY positive infinity}.
205     */
206    @Override
207    public double getSupportUpperBound() {
208        return Double.POSITIVE_INFINITY;
209    }
210
211    /** {@inheritDoc} */
212    @Override
213    double getMedian() {
214        // Overridden for the probability(double, double) method.
215        // This is intentionally not a public method.
216        // u + c / 2(erfc^-1 (0.5))^2
217        return mu + c * HALF_OVER_ERFCINV_HALF_SQUARED;
218    }
219
220    /** {@inheritDoc} */
221    @Override
222    public ContinuousDistribution.Sampler createSampler(final UniformRandomProvider rng) {
223        // Levy distribution sampler.
224        return LevySampler.of(rng, getLocation(), getScale())::sample;
225    }
226}