LevyDistribution.java

  1. /*
  2.  * Licensed to the Apache Software Foundation (ASF) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * The ASF licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *      http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */
  17. package org.apache.commons.statistics.distribution;

  19. import org.apache.commons.numbers.gamma.Erf;
  20. import org.apache.commons.numbers.gamma.Erfc;
  21. import org.apache.commons.numbers.gamma.InverseErf;
  22. import org.apache.commons.numbers.gamma.InverseErfc;
  23. import org.apache.commons.rng.UniformRandomProvider;
  24. import org.apache.commons.rng.sampling.distribution.LevySampler;

  26. /**
  27.  * Implementation of the Lévy distribution.
  28.  *
  29.  * <p>The probability density function of \( X \) is:
  30.  *
  31.  * <p>\[ f(x; \mu, c) = \sqrt{\frac{c}{2\pi}}~~\frac{e^{ -\frac{c}{2(x-\mu)}}} {(x-\mu)^{3/2}} \]
  32.  *
  33.  * <p>for \( \mu \) the location,
  34.  * \( c &gt; 0 \) the scale, and
  35.  * \( x \in [\mu, \infty) \).
  36.  *
  37.  * @see <a href="https://en.wikipedia.org/wiki/L%C3%A9vy_distribution">L&eacute;vy distribution (Wikipedia)</a>
  38.  * @see <a href="https://mathworld.wolfram.com/LevyDistribution.html">L&eacute;vy distribution (MathWorld)</a>
  39.  */
  40. public final class LevyDistribution extends AbstractContinuousDistribution {
  41.     /** 1 / 2(erfc^-1 (0.5))^2. Computed using Matlab's VPA to 30 digits. */
  42.     private static final double HALF_OVER_ERFCINV_HALF_SQUARED = 2.1981093383177324039996779530797;
  43.     /** Location parameter. */
  44.     private final double mu;
  45.     /** Scale parameter. */
  46.     private final double c;
  47.     /** Half of c (for calculations). */
  48.     private final double halfC;

  50.     /**
  51.      * @param mu Location parameter.
  52.      * @param c Scale parameter.
  53.      */
  54.     private LevyDistribution(double mu,
  55.                              double c) {
  56.         this.mu = mu;
  57.         this.c = c;
  58.         this.halfC = 0.5 * c;
  59.     }

  61.     /**
  62.      * Creates a Levy distribution.
  63.      *
  64.      * @param mu Location parameter.
  65.      * @param c Scale parameter.
  66.      * @return the distribution
  67.      * @throws IllegalArgumentException if {@code c <= 0}.
  68.      */
  69.     public static LevyDistribution of(double mu,
  70.                                       double c) {
  71.         if (c <= 0) {
  72.             throw new DistributionException(DistributionException.NOT_STRICTLY_POSITIVE,
  73.                                             c);
  74.         }
  75.         return new LevyDistribution(mu, c);
  76.     }

  78.     /**
  79.      * Gets the location parameter of this distribution.
  80.      *
  81.      * @return the location parameter.
  82.      */
  83.     public double getLocation() {
  84.         return mu;
  85.     }

  87.     /**
  88.      * Gets the scale parameter of this distribution.
  89.      *
  90.      * @return the scale parameter.
  91.      */
  92.     public double getScale() {
  93.         return c;
  94.     }

  96.     /**
  97.      * {@inheritDoc}
  98.      *
  99.      * <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.         }

  112.         final double delta = x - mu;
  113.         final double f = halfC / delta;
  114.         return Math.sqrt(f / Math.PI) * Math.exp(-f) / delta;
  115.     }

  117.     /** {@inheritDoc} */
  118.     @Override
  119.     public double logDensity(double x) {
  120.         if (x <= mu) {
  121.             return Double.NEGATIVE_INFINITY;
  122.         }

  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.     }

  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.     }

  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.     }

  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.     }

  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.     }

  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.     }

  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.     }

  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.     }

  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.     }

  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.     }

  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. }
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