/*
    * 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.statistics.distribution;
  
  import org.apache.commons.numbers.gamma.Erf;
  import org.apache.commons.numbers.gamma.Erfc;
  import org.apache.commons.numbers.gamma.InverseErf;
  import org.apache.commons.numbers.gamma.InverseErfc;
  import org.apache.commons.rng.UniformRandomProvider;
  import org.apache.commons.rng.sampling.distribution.LevySampler;
  
  /**
   * Implementation of the Lévy distribution.
   *
   * <p>The probability density function of \( X \) is:
   *
   * <p>\[ f(x; \mu, c) = \sqrt{\frac{c}{2\pi}}~~\frac{e^{ -\frac{c}{2(x-\mu)}}} {(x-\mu)^{3/2}} \]
   *
   * <p>for \( \mu \) the location,
   * \( c &gt; 0 \) the scale, and
   * \( x \in [\mu, \infty) \).
   *
   * @see <a href="https://en.wikipedia.org/wiki/L%C3%A9vy_distribution">L&eacute;vy distribution (Wikipedia)</a>
   * @see <a href="https://mathworld.wolfram.com/LevyDistribution.html">L&eacute;vy distribution (MathWorld)</a>
   */
  public final class LevyDistribution extends AbstractContinuousDistribution {
      /** 1 / 2(erfc^-1 (0.5))^2. Computed using Matlab's VPA to 30 digits. */
      private static final double HALF_OVER_ERFCINV_HALF_SQUARED = 2.1981093383177324039996779530797;
      /** Location parameter. */
      private final double mu;
      /** Scale parameter. */
      private final double c;
      /** Half of c (for calculations). */
      private final double halfC;
  
      /**
       * @param mu Location parameter.
       * @param c Scale parameter.
       */
      private LevyDistribution(double mu,
                               double c) {
          this.mu = mu;
          this.c = c;
          this.halfC = 0.5 * c;
      }
  
      /**
       * Creates a Levy distribution.
       *
       * @param mu Location parameter.
       * @param c Scale parameter.
       * @return the distribution
       * @throws IllegalArgumentException if {@code c <= 0}.
       */
      public static LevyDistribution of(double mu,
                                        double c) {
          if (c <= 0) {
              throw new DistributionException(DistributionException.NOT_STRICTLY_POSITIVE,
                                              c);
          }
          return new LevyDistribution(mu, c);
      }
  
      /**
       * Gets the location parameter of this distribution.
       *
       * @return the location parameter.
       */
      public double getLocation() {
          return mu;
      }
  
      /**
       * Gets the scale parameter of this distribution.
       *
       * @return the scale parameter.
       */
      public double getScale() {
          return c;
      }
  
      /**
       * {@inheritDoc}
       *
       * <p>If {@code x} is less than the location parameter then {@code 0} is
      * returned, as in these cases the distribution is not defined.
      */
     @Override
     public double density(final double x) {
         if (x <= mu) {
             // x=mu creates NaN:
             // sqrt(c / 2pi) * exp(-c / 2(x-mu)) / (x-mu)^1.5
             // = F * exp(-inf) * (x-mu)^-1.5 = F * 0 * inf
             // Return 0 for this case.
             return 0;
         }
 
         final double delta = x - mu;
         final double f = halfC / delta;
         return Math.sqrt(f / Math.PI) * Math.exp(-f) / delta;
     }
 
     /** {@inheritDoc} */
     @Override
     public double logDensity(double x) {
         if (x <= mu) {
             return Double.NEGATIVE_INFINITY;
         }
 
         final double delta = x - mu;
         final double f     = halfC / delta;
         return 0.5 * Math.log(f / Math.PI) - f - Math.log(delta);
     }
 
     /** {@inheritDoc} */
     @Override
     public double cumulativeProbability(final double x) {
         if (x <= mu) {
             return 0;
         }
         return Erfc.value(Math.sqrt(halfC / (x - mu)));
     }
 
     /** {@inheritDoc} */
     @Override
     public double survivalProbability(final double x) {
         if (x <= mu) {
             return 1;
         }
         return Erf.value(Math.sqrt(halfC / (x - mu)));
     }
 
     /** {@inheritDoc} */
     @Override
     public double inverseCumulativeProbability(double p) {
         ArgumentUtils.checkProbability(p);
         final double t = InverseErfc.value(p);
         return mu + halfC / (t * t);
     }
 
     /** {@inheritDoc} */
     @Override
     public double inverseSurvivalProbability(double p) {
         ArgumentUtils.checkProbability(p);
         final double t = InverseErf.value(p);
         return mu + halfC / (t * t);
     }
 
     /**
      * {@inheritDoc}
      *
      * <p>The mean is equal to positive infinity.
      *
      * @return {@linkplain Double#POSITIVE_INFINITY positive infinity}.
      */
     @Override
     public double getMean() {
         return Double.POSITIVE_INFINITY;
     }
 
     /**
      * {@inheritDoc}
      *
      * <p>The variance is equal to positive infinity.
      *
      * @return {@linkplain Double#POSITIVE_INFINITY positive infinity}.
      */
     @Override
     public double getVariance() {
         return Double.POSITIVE_INFINITY;
     }
 
     /**
      * {@inheritDoc}
      *
      * <p>The lower bound of the support is the {@linkplain #getLocation() location}.
      *
      * @return location.
      */
     @Override
     public double getSupportLowerBound() {
         return getLocation();
     }
 
     /**
      * {@inheritDoc}
      *
      * <p>The upper bound of the support is always positive infinity.
      *
      * @return {@linkplain Double#POSITIVE_INFINITY positive infinity}.
      */
     @Override
     public double getSupportUpperBound() {
         return Double.POSITIVE_INFINITY;
     }
 
     /** {@inheritDoc} */
     @Override
     double getMedian() {
         // Overridden for the probability(double, double) method.
         // This is intentionally not a public method.
         // u + c / 2(erfc^-1 (0.5))^2
         return mu + c * HALF_OVER_ERFCINV_HALF_SQUARED;
     }
 
     /** {@inheritDoc} */
     @Override
     public ContinuousDistribution.Sampler createSampler(final UniformRandomProvider rng) {
         // Levy distribution sampler.
         return LevySampler.of(rng, getLocation(), getScale())::sample;
     }
 }