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
*
* https://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.
*/
/*
* This is not the original file distributed by the Apache Software Foundation
* It has been modified by the Hipparchus project
*/
package org.hipparchus.distribution.continuous;
import org.hipparchus.exception.LocalizedCoreFormats;
import org.hipparchus.exception.MathIllegalArgumentException;
import org.hipparchus.special.Gamma;
import org.hipparchus.util.FastMath;
/**
* Implementation of the Gamma distribution.
*
* @see <a href="http://en.wikipedia.org/wiki/Gamma_distribution">Gamma distribution (Wikipedia)</a>
* @see <a href="http://mathworld.wolfram.com/GammaDistribution.html">Gamma distribution (MathWorld)</a>
*/
public class GammaDistribution extends AbstractRealDistribution {
/** Serializable version identifier. */
private static final long serialVersionUID = 20120524L;
/** The shape parameter. */
private final double shape;
/** The scale parameter. */
private final double scale;
/**
* The constant value of {@code shape + g + 0.5}, where {@code g} is the
* Lanczos constant {@link Gamma#LANCZOS_G}.
*/
private final double shiftedShape;
/**
* The constant value of
* {@code shape / scale * sqrt(e / (2 * pi * (shape + g + 0.5))) / L(shape)},
* where {@code L(shape)} is the Lanczos approximation returned by
* {@link Gamma#lanczos(double)}. This prefactor is used in
* {@link #density(double)}, when no overflow occurs with the natural
* calculation.
*/
private final double densityPrefactor1;
/**
* The constant value of
* {@code log(shape / scale * sqrt(e / (2 * pi * (shape + g + 0.5))) / L(shape))},
* where {@code L(shape)} is the Lanczos approximation returned by
* {@link Gamma#lanczos(double)}. This prefactor is used in
* {@link #logDensity(double)}, when no overflow occurs with the natural
* calculation.
*/
private final double logDensityPrefactor1;
/**
* The constant value of
* {@code shape * sqrt(e / (2 * pi * (shape + g + 0.5))) / L(shape)},
* where {@code L(shape)} is the Lanczos approximation returned by
* {@link Gamma#lanczos(double)}. This prefactor is used in
* {@link #density(double)}, when overflow occurs with the natural
* calculation.
*/
private final double densityPrefactor2;
/**
* The constant value of
* {@code log(shape * sqrt(e / (2 * pi * (shape + g + 0.5))) / L(shape))},
* where {@code L(shape)} is the Lanczos approximation returned by
* {@link Gamma#lanczos(double)}. This prefactor is used in
* {@link #logDensity(double)}, when overflow occurs with the natural
* calculation.
*/
private final double logDensityPrefactor2;
/**
* Lower bound on {@code y = x / scale} for the selection of the computation
* method in {@link #density(double)}. For {@code y <= minY}, the natural
* calculation overflows.
*/
private final double minY;
/**
* Upper bound on {@code log(y)} ({@code y = x / scale}) for the selection
* of the computation method in {@link #density(double)}. For
* {@code log(y) >= maxLogY}, the natural calculation overflows.
*/
private final double maxLogY;
/**
* Creates a new gamma distribution with specified values of the shape and
* scale parameters.
*
* @param shape the shape parameter
* @param scale the scale parameter
* @throws MathIllegalArgumentException if {@code shape <= 0} or
* {@code scale <= 0}.
*/
public GammaDistribution(double shape, double scale) throws MathIllegalArgumentException {
this(shape, scale, DEFAULT_SOLVER_ABSOLUTE_ACCURACY);
}
/**
* Creates a Gamma distribution.
*
* @param shape the shape parameter
* @param scale the scale parameter
* @param inverseCumAccuracy the maximum absolute error in inverse
* cumulative probability estimates (defaults to
* {@link #DEFAULT_SOLVER_ABSOLUTE_ACCURACY}).
* @throws MathIllegalArgumentException if {@code shape <= 0} or
* {@code scale <= 0}.
*/
public GammaDistribution(final double shape,
final double scale,
final double inverseCumAccuracy)
throws MathIllegalArgumentException {
super(inverseCumAccuracy);
if (shape <= 0) {
throw new MathIllegalArgumentException(LocalizedCoreFormats.SHAPE, shape);
}
if (scale <= 0) {
throw new MathIllegalArgumentException(LocalizedCoreFormats.SCALE, scale);
}
this.shape = shape;
this.scale = scale;
this.shiftedShape = shape + Gamma.LANCZOS_G + 0.5;
final double aux = FastMath.E / (2.0 * FastMath.PI * shiftedShape);
this.densityPrefactor2 = shape * FastMath.sqrt(aux) / Gamma.lanczos(shape);
this.logDensityPrefactor2 = FastMath.log(shape) + 0.5 * FastMath.log(aux) -
FastMath.log(Gamma.lanczos(shape));
this.densityPrefactor1 = this.densityPrefactor2 / scale *
FastMath.pow(shiftedShape, -shape) *
FastMath.exp(shape + Gamma.LANCZOS_G);
this.logDensityPrefactor1 = this.logDensityPrefactor2 - FastMath.log(scale) -
FastMath.log(shiftedShape) * shape +
shape + Gamma.LANCZOS_G;
this.minY = shape + Gamma.LANCZOS_G - FastMath.log(Double.MAX_VALUE);
this.maxLogY = FastMath.log(Double.MAX_VALUE) / (shape - 1.0);
}
/**
* Returns the shape parameter of {@code this} distribution.
*
* @return the shape parameter
*/
public double getShape() {
return shape;
}
/**
* Returns the scale parameter of {@code this} distribution.
*
* @return the scale parameter
*/
public double getScale() {
return scale;
}
/** {@inheritDoc} */
@Override
public double density(double x) {
/* The present method must return the value of
*
* 1 x a - x
* ---------- (-) exp(---)
* x Gamma(a) b b
*
* where a is the shape parameter, and b the scale parameter.
* Substituting the Lanczos approximation of Gamma(a) leads to the
* following expression of the density
*
* a e 1 y a
* - sqrt(------------------) ---- (-----------) exp(a - y + g),
* x 2 pi (a + g + 0.5) L(a) a + g + 0.5
*
* where y = x / b. The above formula is the "natural" computation, which
* is implemented when no overflow is likely to occur. If overflow occurs
* with the natural computation, the following identity is used. It is
* based on the BOOST library
* http://www.boost.org/doc/libs/1_35_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/sf_gamma/igamma.html
* Formula (15) needs adaptations, which are detailed below.
*
* y a
* (-----------) exp(a - y + g)
* a + g + 0.5
* y - a - g - 0.5 y (g + 0.5)
* = exp(a log1pm(---------------) - ----------- + g),
* a + g + 0.5 a + g + 0.5
*
* where log1pm(z) = log(1 + z) - z. Therefore, the value to be
* returned is
*
* a e 1
* - sqrt(------------------) ----
* x 2 pi (a + g + 0.5) L(a)
* y - a - g - 0.5 y (g + 0.5)
* * exp(a log1pm(---------------) - ----------- + g).
* a + g + 0.5 a + g + 0.5
*/
if (x < 0) {
return 0;
}
final double y = x / scale;
if ((y <= minY) || (FastMath.log(y) >= maxLogY)) {
/*
* Overflow.
*/
final double aux1 = (y - shiftedShape) / shiftedShape;
final double aux2 = shape * (FastMath.log1p(aux1) - aux1);
final double aux3 = -y * (Gamma.LANCZOS_G + 0.5) / shiftedShape +
Gamma.LANCZOS_G + aux2;
return densityPrefactor2 / x * FastMath.exp(aux3);
}
/*
* Natural calculation.
*/
return densityPrefactor1 * FastMath.exp(-y) * FastMath.pow(y, shape - 1);
}
/** {@inheritDoc} **/
@Override
public double logDensity(double x) {
/*
* see the comment in {@link #density(double)} for computation details
*/
if (x < 0) {
return Double.NEGATIVE_INFINITY;
}
final double y = x / scale;
if ((y <= minY) || (FastMath.log(y) >= maxLogY)) {
/*
* Overflow.
*/
final double aux1 = (y - shiftedShape) / shiftedShape;
final double aux2 = shape * (FastMath.log1p(aux1) - aux1);
final double aux3 = -y * (Gamma.LANCZOS_G + 0.5) / shiftedShape +
Gamma.LANCZOS_G + aux2;
return logDensityPrefactor2 - FastMath.log(x) + aux3;
}
/*
* Natural calculation.
*/
return logDensityPrefactor1 - y + FastMath.log(y) * (shape - 1);
}
/**
* {@inheritDoc}
*
* The implementation of this method is based on:
* <ul>
* <li>
* <a href="http://mathworld.wolfram.com/Chi-SquaredDistribution.html">
* Chi-Squared Distribution</a>, equation (9).
* </li>
* <li>Casella, G., & Berger, R. (1990). <i>Statistical Inference</i>.
* Belmont, CA: Duxbury Press.
* </li>
* </ul>
*/
@Override
public double cumulativeProbability(double x) {
double ret;
if (x <= 0) {
ret = 0;
} else {
ret = Gamma.regularizedGammaP(shape, x / scale);
}
return ret;
}
/**
* {@inheritDoc}
*
* For shape parameter {@code alpha} and scale parameter {@code beta}, the
* mean is {@code alpha * beta}.
*/
@Override
public double getNumericalMean() {
return shape * scale;
}
/**
* {@inheritDoc}
*
* For shape parameter {@code alpha} and scale parameter {@code beta}, the
* variance is {@code alpha * beta^2}.
*
* @return {@inheritDoc}
*/
@Override
public double getNumericalVariance() {
return shape * scale * scale;
}
/**
* {@inheritDoc}
*
* The lower bound of the support is always 0 no matter the parameters.
*
* @return lower bound of the support (always 0)
*/
@Override
public double getSupportLowerBound() {
return 0;
}
/**
* {@inheritDoc}
*
* The upper bound of the support is always positive infinity
* no matter the parameters.
*
* @return upper bound of the support (always Double.POSITIVE_INFINITY)
*/
@Override
public double getSupportUpperBound() {
return Double.POSITIVE_INFINITY;
}
/**
* {@inheritDoc}
*
* The support of this distribution is connected.
*
* @return {@code true}
*/
@Override
public boolean isSupportConnected() {
return true;
}
}