/*
* 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.math3.distribution;
import org.apache.commons.math3.exception.NotStrictlyPositiveException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.random.RandomGenerator;
import org.apache.commons.math3.random.Well19937c;
import org.apache.commons.math3.special.Gamma;
/**
* 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 {
/**
* Default inverse cumulative probability accuracy.
* @since 2.1
*/
public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY = 1e-9;
/** 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;
/** Inverse cumulative probability accuracy. */
private final double solverAbsoluteAccuracy;
/**
* Creates a new gamma distribution with specified values of the shape and
* scale parameters.
* <p>
* <b>Note:</b> this constructor will implicitly create an instance of
* {@link Well19937c} as random generator to be used for sampling only (see
* {@link #sample()} and {@link #sample(int)}). In case no sampling is
* needed for the created distribution, it is advised to pass {@code null}
* as random generator via the appropriate constructors to avoid the
* additional initialisation overhead.
*
* @param shape the shape parameter
* @param scale the scale parameter
* @throws NotStrictlyPositiveException if {@code shape <= 0} or
* {@code scale <= 0}.
*/
public GammaDistribution(double shape, double scale) throws NotStrictlyPositiveException {
this(shape, scale, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
}
/**
* Creates a new gamma distribution with specified values of the shape and
* scale parameters.
* <p>
* <b>Note:</b> this constructor will implicitly create an instance of
* {@link Well19937c} as random generator to be used for sampling only (see
* {@link #sample()} and {@link #sample(int)}). In case no sampling is
* needed for the created distribution, it is advised to pass {@code null}
* as random generator via the appropriate constructors to avoid the
* additional initialisation overhead.
*
* @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_INVERSE_ABSOLUTE_ACCURACY}).
* @throws NotStrictlyPositiveException if {@code shape <= 0} or
* {@code scale <= 0}.
* @since 2.1
*/
public GammaDistribution(double shape, double scale, double inverseCumAccuracy)
throws NotStrictlyPositiveException {
this(new Well19937c(), shape, scale, inverseCumAccuracy);
}
/**
* Creates a Gamma distribution.
*
* @param rng Random number generator.
* @param shape the shape parameter
* @param scale the scale parameter
* @throws NotStrictlyPositiveException if {@code shape <= 0} or
* {@code scale <= 0}.
* @since 3.3
*/
public GammaDistribution(RandomGenerator rng, double shape, double scale)
throws NotStrictlyPositiveException {
this(rng, shape, scale, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
}
/**
* Creates a Gamma distribution.
*
* @param rng Random number generator.
* @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_INVERSE_ABSOLUTE_ACCURACY}).
* @throws NotStrictlyPositiveException if {@code shape <= 0} or
* {@code scale <= 0}.
* @since 3.1
*/
public GammaDistribution(RandomGenerator rng,
double shape,
double scale,
double inverseCumAccuracy)
throws NotStrictlyPositiveException {
super(rng);
if (shape <= 0) {
throw new NotStrictlyPositiveException(LocalizedFormats.SHAPE, shape);
}
if (scale <= 0) {
throw new NotStrictlyPositiveException(LocalizedFormats.SCALE, scale);
}
this.shape = shape;
this.scale = scale;
this.solverAbsoluteAccuracy = inverseCumAccuracy;
this.shiftedShape = shape + Gamma.LANCZOS_G + 0.5;
final double aux = Math.E / (2.0 * Math.PI * shiftedShape);
this.densityPrefactor2 = shape * Math.sqrt(aux) / Gamma.lanczos(shape);
this.logDensityPrefactor2 = Math.log(shape) + 0.5 * Math.log(aux) -
Math.log(Gamma.lanczos(shape));
this.densityPrefactor1 = this.densityPrefactor2 / scale *
Math.pow(shiftedShape, -shape) *
Math.exp(shape + Gamma.LANCZOS_G);
this.logDensityPrefactor1 = this.logDensityPrefactor2 - Math.log(scale) -
Math.log(shiftedShape) * shape +
shape + Gamma.LANCZOS_G;
this.minY = shape + Gamma.LANCZOS_G - Math.log(Double.MAX_VALUE);
this.maxLogY = Math.log(Double.MAX_VALUE) / (shape - 1.0);
}
/**
* Returns the shape parameter of {@code this} distribution.
*
* @return the shape parameter
* @deprecated as of version 3.1, {@link #getShape()} should be preferred.
* This method will be removed in version 4.0.
*/
@Deprecated
public double getAlpha() {
return shape;
}
/**
* Returns the shape parameter of {@code this} distribution.
*
* @return the shape parameter
* @since 3.1
*/
public double getShape() {
return shape;
}
/**
* Returns the scale parameter of {@code this} distribution.
*
* @return the scale parameter
* @deprecated as of version 3.1, {@link #getScale()} should be preferred.
* This method will be removed in version 4.0.
*/
@Deprecated
public double getBeta() {
return scale;
}
/**
* Returns the scale parameter of {@code this} distribution.
*
* @return the scale parameter
* @since 3.1
*/
public double getScale() {
return scale;
}
/** {@inheritDoc} */
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) || (Math.log(y) >= maxLogY)) {
/*
* Overflow.
*/
final double aux1 = (y - shiftedShape) / shiftedShape;
final double aux2 = shape * (Math.log1p(aux1) - aux1);
final double aux3 = -y * (Gamma.LANCZOS_G + 0.5) / shiftedShape +
Gamma.LANCZOS_G + aux2;
return densityPrefactor2 / x * Math.exp(aux3);
}
/*
* Natural calculation.
*/
return densityPrefactor1 * Math.exp(-y) * Math.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) || (Math.log(y) >= maxLogY)) {
/*
* Overflow.
*/
final double aux1 = (y - shiftedShape) / shiftedShape;
final double aux2 = shape * (Math.log1p(aux1) - aux1);
final double aux3 = -y * (Gamma.LANCZOS_G + 0.5) / shiftedShape +
Gamma.LANCZOS_G + aux2;
return logDensityPrefactor2 - Math.log(x) + aux3;
}
/*
* Natural calculation.
*/
return logDensityPrefactor1 - y + Math.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>
*/
public double cumulativeProbability(double x) {
double ret;
if (x <= 0) {
ret = 0;
} else {
ret = Gamma.regularizedGammaP(shape, x / scale);
}
return ret;
}
/** {@inheritDoc} */
@Override
protected double getSolverAbsoluteAccuracy() {
return solverAbsoluteAccuracy;
}
/**
* {@inheritDoc}
*
* For shape parameter {@code alpha} and scale parameter {@code beta}, the
* mean is {@code alpha * beta}.
*/
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}
*/
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)
*/
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)
*/
public double getSupportUpperBound() {
return Double.POSITIVE_INFINITY;
}
/** {@inheritDoc} */
public boolean isSupportLowerBoundInclusive() {
return true;
}
/** {@inheritDoc} */
public boolean isSupportUpperBoundInclusive() {
return false;
}
/**
* {@inheritDoc}
*
* The support of this distribution is connected.
*
* @return {@code true}
*/
public boolean isSupportConnected() {
return true;
}
/**
* <p>This implementation uses the following algorithms: </p>
*
* <p>For 0 < shape < 1: <br/>
* Ahrens, J. H. and Dieter, U., <i>Computer methods for
* sampling from gamma, beta, Poisson and binomial distributions.</i>
* Computing, 12, 223-246, 1974.</p>
*
* <p>For shape >= 1: <br/>
* Marsaglia and Tsang, <i>A Simple Method for Generating
* Gamma Variables.</i> ACM Transactions on Mathematical Software,
* Volume 26 Issue 3, September, 2000.</p>
*
* @return random value sampled from the Gamma(shape, scale) distribution
*/
@Override
public double sample() {
if (shape < 1) {
// [1]: p. 228, Algorithm GS
while (true) {
// Step 1:
final double u = random.nextDouble();
final double bGS = 1 + shape / Math.E;
final double p = bGS * u;
if (p <= 1) {
// Step 2:
final double x = Math.pow(p, 1 / shape);
final double u2 = random.nextDouble();
if (u2 > Math.exp(-x)) {
// Reject
continue;
} else {
return scale * x;
}
} else {
// Step 3:
final double x = -1 * Math.log((bGS - p) / shape);
final double u2 = random.nextDouble();
if (u2 > Math.pow(x, shape - 1)) {
// Reject
continue;
} else {
return scale * x;
}
}
}
}
// Now shape >= 1
final double d = shape - 0.333333333333333333;
final double c = 1 / (3 * Math.sqrt(d));
while (true) {
final double x = random.nextGaussian();
final double v = (1 + c * x) * (1 + c * x) * (1 + c * x);
if (v <= 0) {
continue;
}
final double x2 = x * x;
final double u = random.nextDouble();
// Squeeze
if (u < 1 - 0.0331 * x2 * x2) {
return scale * d * v;
}
if (Math.log(u) < 0.5 * x2 + d * (1 - v + Math.log(v))) {
return scale * d * v;
}
}
}
}