GammaDistribution.java
/*
* 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.LogGamma;
import org.apache.commons.numbers.gamma.RegularizedGamma;
import org.apache.commons.rng.UniformRandomProvider;
import org.apache.commons.rng.sampling.distribution.AhrensDieterMarsagliaTsangGammaSampler;
/**
* Implementation of the gamma distribution.
*
* <p>The probability density function of \( X \) is:
*
* <p>\[ f(x;k,\theta) = \frac{x^{k-1}e^{-x/\theta}}{\theta^k\Gamma(k)} \]
*
* <p>for \( k > 0 \) the shape, \( \theta > 0 \) the scale, \( \Gamma(k) \) is the gamma function
* and \( x \in (0, \infty) \).
*
* @see <a href="https://en.wikipedia.org/wiki/Gamma_distribution">Gamma distribution (Wikipedia)</a>
* @see <a href="https://mathworld.wolfram.com/GammaDistribution.html">Gamma distribution (MathWorld)</a>
*/
public final class GammaDistribution extends AbstractContinuousDistribution {
/** Support lower bound. */
private static final double SUPPORT_LO = 0;
/** Support upper bound. */
private static final double SUPPORT_HI = Double.POSITIVE_INFINITY;
/** The shape parameter. */
private final double shape;
/** The scale parameter. */
private final double scale;
/** Precomputed term for the log density: {@code -log(gamma(shape)) - log(scale)}. */
private final double minusLogGammaShapeMinusLogScale;
/** Cached value for inverse probability function. */
private final double mean;
/** Cached value for inverse probability function. */
private final double variance;
/**
* @param shape Shape parameter.
* @param scale Scale parameter.
*/
private GammaDistribution(double shape,
double scale) {
this.shape = shape;
this.scale = scale;
this.minusLogGammaShapeMinusLogScale = -LogGamma.value(shape) - Math.log(scale);
mean = shape * scale;
variance = shape * scale * scale;
}
/**
* Creates a gamma distribution.
*
* @param shape Shape parameter.
* @param scale Scale parameter.
* @return the distribution
* @throws IllegalArgumentException if {@code shape <= 0} or {@code scale <= 0}.
*/
public static GammaDistribution of(double shape,
double scale) {
if (shape <= 0) {
throw new DistributionException(DistributionException.NOT_STRICTLY_POSITIVE, shape);
}
if (scale <= 0) {
throw new DistributionException(DistributionException.NOT_STRICTLY_POSITIVE, scale);
}
return new GammaDistribution(shape, scale);
}
/**
* Gets the shape parameter of this distribution.
*
* @return the shape parameter.
*/
public double getShape() {
return shape;
}
/**
* Gets the scale parameter of this distribution.
*
* @return the scale parameter.
*/
public double getScale() {
return scale;
}
/** {@inheritDoc}
*
* <p>Returns the limit when {@code x = 0}:
* <ul>
* <li>{@code shape < 1}: Infinity
* <li>{@code shape == 1}: 1 / scale
* <li>{@code shape > 1}: 0
* </ul>
*/
@Override
public double density(double x) {
if (x <= SUPPORT_LO ||
x >= SUPPORT_HI) {
// Special case x=0
if (x == SUPPORT_LO && shape <= 1) {
return shape == 1 ?
1 / scale :
Double.POSITIVE_INFINITY;
}
return 0;
}
return RegularizedGamma.P.derivative(shape, x / scale) / scale;
}
/** {@inheritDoc}
*
* <p>Returns the limit when {@code x = 0}:
* <ul>
* <li>{@code shape < 1}: Infinity
* <li>{@code shape == 1}: -log(scale)
* <li>{@code shape > 1}: -Infinity
* </ul>
*/
@Override
public double logDensity(double x) {
if (x <= SUPPORT_LO ||
x >= SUPPORT_HI) {
// Special case x=0
if (x == SUPPORT_LO && shape <= 1) {
return shape == 1 ?
-Math.log(scale) :
Double.POSITIVE_INFINITY;
}
return Double.NEGATIVE_INFINITY;
}
final double y = x / scale;
// More accurate to log the density when it is finite.
// See NUMBERS-174: 'Log of the Gamma P Derivative'
final double p = RegularizedGamma.P.derivative(shape, y) / scale;
if (p <= Double.MAX_VALUE && p >= Double.MIN_NORMAL) {
return Math.log(p);
}
// Use the log computation
return minusLogGammaShapeMinusLogScale - y + Math.log(y) * (shape - 1);
}
/** {@inheritDoc} */
@Override
public double cumulativeProbability(double x) {
if (x <= SUPPORT_LO) {
return 0;
} else if (x >= SUPPORT_HI) {
return 1;
}
return RegularizedGamma.P.value(shape, x / scale);
}
/** {@inheritDoc} */
@Override
public double survivalProbability(double x) {
if (x <= SUPPORT_LO) {
return 1;
} else if (x >= SUPPORT_HI) {
return 0;
}
return RegularizedGamma.Q.value(shape, x / scale);
}
/**
* {@inheritDoc}
*
* <p>For shape parameter \( k \) and scale parameter \( \theta \), the
* mean is \( k \theta \).
*/
@Override
public double getMean() {
return mean;
}
/**
* {@inheritDoc}
*
* <p>For shape parameter \( k \) and scale parameter \( \theta \), the
* variance is \( k \theta^2 \).
*/
@Override
public double getVariance() {
return variance;
}
/**
* {@inheritDoc}
*
* <p>The lower bound of the support is always 0.
*
* @return 0.
*/
@Override
public double getSupportLowerBound() {
return SUPPORT_LO;
}
/**
* {@inheritDoc}
*
* <p>The upper bound of the support is always positive infinity.
*
* @return {@link Double#POSITIVE_INFINITY positive infinity}.
*/
@Override
public double getSupportUpperBound() {
return SUPPORT_HI;
}
/** {@inheritDoc} */
@Override
public ContinuousDistribution.Sampler createSampler(final UniformRandomProvider rng) {
// Gamma distribution sampler.
return AhrensDieterMarsagliaTsangGammaSampler.of(rng, shape, scale)::sample;
}
}