ZipfDistribution.java
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* 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
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*
* 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,
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* See the License for the specific language governing permissions and
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*/
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;
}
}