/*
    * 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.rng.UniformRandomProvider;
  import org.apache.commons.rng.sampling.distribution.RejectionInversionZipfSampler;
  
  /**
   * Implementation of the Zipf distribution.
   *
   * <p>The probability mass function of \( X \) is:
   *
   * <p>\[ f(k; N, s) = \frac{1/k^s}{H_{N,s}} \]
   *
   * <p>for \( N \in \{1, 2, 3, \dots\} \) the number of elements,
   * \( s \gt 0 \) the exponent characterizing the distribution,
   * \( k \in \{1, 2, \dots, N\} \) the element rank, and
   * \( H_{N,s} \) is the normalizing constant which corresponds to the
   * <a href="https://en.wikipedia.org/wiki/Harmonic_number#Generalized_harmonic_numbers">
   * generalized harmonic number</a> of order N of s.
   *
   * @see <a href="https://en.wikipedia.org/wiki/Zipf's_law">Zipf distribution (Wikipedia)</a>
   */
  public final class ZipfDistribution extends AbstractDiscreteDistribution {
      /** Number of elements. */
      private final int numberOfElements;
      /** Exponent parameter of the distribution. */
      private final double exponent;
      /** Cached value of the nth generalized harmonic. */
      private final double nthHarmonic;
      /** Cached value of the log of the nth generalized harmonic. */
      private final double logNthHarmonic;
  
      /**
       * @param numberOfElements Number of elements.
       * @param exponent Exponent.
       */
      private ZipfDistribution(int numberOfElements,
                               double exponent) {
          this.numberOfElements = numberOfElements;
          this.exponent = exponent;
          this.nthHarmonic = generalizedHarmonic(numberOfElements, exponent);
          logNthHarmonic = Math.log(nthHarmonic);
      }
  
      /**
       * Creates a Zipf distribution.
       *
       * @param numberOfElements Number of elements.
       * @param exponent Exponent.
       * @return the distribution
       * @exception IllegalArgumentException if {@code numberOfElements <= 0}
       * or {@code exponent <= 0}.
       */
      public static ZipfDistribution of(int numberOfElements,
                                        double exponent) {
          if (numberOfElements <= 0) {
              throw new DistributionException(DistributionException.NOT_STRICTLY_POSITIVE,
                                              numberOfElements);
          }
          if (exponent < 0) {
              throw new DistributionException(DistributionException.NEGATIVE,
                                              exponent);
          }
          return new ZipfDistribution(numberOfElements, exponent);
      }
  
      /**
       * Gets the number of elements parameter of this distribution.
       *
       * @return the number of elements.
       */
      public int getNumberOfElements() {
          return numberOfElements;
      }
  
      /**
       * Gets the exponent parameter of this distribution.
       *
       * @return the exponent.
       */
      public double getExponent() {
          return exponent;
      }
 
     /** {@inheritDoc} */
     @Override
     public double probability(final int x) {
         if (x <= 0 || x > numberOfElements) {
             return 0;
         }
 
         return Math.pow(x, -exponent) / nthHarmonic;
     }
 
     /** {@inheritDoc} */
     @Override
     public double logProbability(int x) {
         if (x <= 0 || x > numberOfElements) {
             return Double.NEGATIVE_INFINITY;
         }
 
         return -Math.log(x) * exponent - logNthHarmonic;
     }
 
     /** {@inheritDoc} */
     @Override
     public double cumulativeProbability(final int x) {
         if (x <= 0) {
             return 0;
         } else if (x >= numberOfElements) {
             return 1;
         }
 
         return generalizedHarmonic(x, exponent) / nthHarmonic;
     }
 
     /** {@inheritDoc} */
     @Override
     public double survivalProbability(int x) {
         if (x <= 0) {
             return 1;
         } else if (x >= numberOfElements) {
             return 0;
         }
 
         // See http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Zipf.pdf
         // S(x) = P(X > x) = ((x+1)^a Hn,a - (x+1)^a Hx+1,a + 1) / ((x+1)^a Hn,a)
         // where a = exponent and Hx,a is the generalized harmonic for x with exponent a.
         final double z = Math.pow(x + 1.0, exponent);
         // Compute generalizedHarmonic(x, exponent) and generalizedHarmonic(x+1, exponent)
         final double hx = generalizedHarmonic(x, exponent);
         final double hx1 = hx + Math.pow(x + 1.0, -exponent);
         // Compute the survival function
         final double p = (z * (nthHarmonic - hx1) + 1) / (z * nthHarmonic);
         // May overflow for large exponent so validate the probability.
         // If this occurs revert to 1 - CDF(x), reusing the generalized harmonic for x
         return p <= 1.0 ? p : 1.0 - hx / nthHarmonic;
     }
 
     /**
      * {@inheritDoc}
      *
      * <p>For number of elements \( N \) and exponent \( s \), the mean is:
      *
      * <p>\[ \frac{H_{N,s-1}}{H_{N,s}} \]
      *
      * <p>where \( H_{N,k} \) is the
      * <a href="https://en.wikipedia.org/wiki/Harmonic_number#Generalized_harmonic_numbers">
      * generalized harmonic number</a> of order \( N \) of \( k \).
      */
     @Override
     public double getMean() {
         final int N = getNumberOfElements();
         final double s = getExponent();
 
         final double Hs1 = generalizedHarmonicAscendingSum(N, s - 1);
 
         return Hs1 / nthHarmonic;
     }
 
     /**
      * {@inheritDoc}
      *
      * <p>For number of elements \( N \) and exponent \( s \), the variance is:
      *
      * <p>\[ \frac{H_{N,s-2}}{H_{N,s}} - \frac{H_{N,s-1}^2}{H_{N,s}^2} \]
      *
      * <p>where \( H_{N,k} \) is the
      * <a href="https://en.wikipedia.org/wiki/Harmonic_number#Generalized_harmonic_numbers">
      * generalized harmonic number</a> of order \( N \) of \( k \).
      */
     @Override
     public double getVariance() {
         final int N = getNumberOfElements();
         final double s = getExponent();
 
         final double Hs2 = generalizedHarmonicAscendingSum(N, s - 2);
         final double Hs1 = generalizedHarmonicAscendingSum(N, s - 1);
         final double Hs = nthHarmonic;
 
         return (Hs2 / Hs) - ((Hs1 * Hs1) / (Hs * Hs));
     }
 
     /**
      * Calculates the Nth generalized harmonic number. See
      * <a href="https://mathworld.wolfram.com/HarmonicSeries.html">Harmonic
      * Series</a>.
      *
      * <p>Assumes {@code exponent > 0} to arrange the terms to sum from small to large.
      *
      * @param n Term in the series to calculate (must be larger than 1)
      * @param m Exponent (special case {@code m = 1} is the harmonic series).
      * @return the n<sup>th</sup> generalized harmonic number.
      */
     private static double generalizedHarmonic(final int n, final double m) {
         double value = 0;
         // Sum small to large
         for (int k = n; k >= 1; k--) {
             value += Math.pow(k, -m);
         }
         return value;
     }
 
     /**
      * Calculates the Nth generalized harmonic number.
      *
      * <p>Checks the value of the {@code exponent} to arrange the terms to sum from from small to large.
      *
      * @param n Term in the series to calculate (must be larger than 1)
      * @param m Exponent (special case {@code m = 1} is the harmonic series).
      * @return the n<sup>th</sup> generalized harmonic number.
      */
     private static double generalizedHarmonicAscendingSum(final int n, final double m) {
         double value = 0;
         // Sum small to large
         // If m < 0 then sum ascending, otherwise descending
         if (m < 0) {
             for (int k = 1; k <= n; k++) {
                 value += Math.pow(k, -m);
             }
         } else {
             for (int k = n; k >= 1; k--) {
                 value += Math.pow(k, -m);
             }
         }
         return value;
     }
 
     /**
      * {@inheritDoc}
      *
      * <p>The lower bound of the support is always 1.
      *
      * @return 1.
      */
     @Override
     public int getSupportLowerBound() {
         return 1;
     }
 
     /**
      * {@inheritDoc}
      *
      * <p>The upper bound of the support is the number of elements.
      *
      * @return number of elements.
      */
     @Override
     public int getSupportUpperBound() {
         return getNumberOfElements();
     }
 
     /** {@inheritDoc} */
     @Override
     public DiscreteDistribution.Sampler createSampler(final UniformRandomProvider rng) {
         // Zipf distribution sampler.
         return RejectionInversionZipfSampler.of(rng, numberOfElements, exponent)::sample;
     }
 }