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/*
* 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.NumberIsTooSmallException;
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.Beta;
import org.apache.commons.math3.special.Gamma;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.Precision;
/**
* Implements the Beta distribution.
*
* @see Beta distribution
* @since 2.0 (changed to concrete class in 3.0)
*/
public class BetaDistribution 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 = -1221965979403477668L;
/** First shape parameter. */
private final double alpha;
/** Second shape parameter. */
private final double beta;
/** Normalizing factor used in density computations.
* updated whenever alpha or beta are changed.
*/
private double z;
/** Inverse cumulative probability accuracy. */
private final double solverAbsoluteAccuracy;
/**
* Build a new instance.
*
* Note: 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 alpha First shape parameter (must be positive).
* @param beta Second shape parameter (must be positive).
*/
public BetaDistribution(double alpha, double beta) {
this(alpha, beta, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
}
/**
* Build a new instance.
*
* Note: 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 alpha First shape parameter (must be positive).
* @param beta Second shape parameter (must be positive).
* @param inverseCumAccuracy Maximum absolute error in inverse
* cumulative probability estimates (defaults to
* {@link #DEFAULT_INVERSE_ABSOLUTE_ACCURACY}).
* @since 2.1
*/
public BetaDistribution(double alpha, double beta, double inverseCumAccuracy) {
this(new Well19937c(), alpha, beta, inverseCumAccuracy);
}
/**
* Creates a β distribution.
*
* @param rng Random number generator.
* @param alpha First shape parameter (must be positive).
* @param beta Second shape parameter (must be positive).
* @since 3.3
*/
public BetaDistribution(RandomGenerator rng, double alpha, double beta) {
this(rng, alpha, beta, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
}
/**
* Creates a β distribution.
*
* @param rng Random number generator.
* @param alpha First shape parameter (must be positive).
* @param beta Second shape parameter (must be positive).
* @param inverseCumAccuracy Maximum absolute error in inverse
* cumulative probability estimates (defaults to
* {@link #DEFAULT_INVERSE_ABSOLUTE_ACCURACY}).
* @since 3.1
*/
public BetaDistribution(RandomGenerator rng,
double alpha,
double beta,
double inverseCumAccuracy) {
super(rng);
this.alpha = alpha;
this.beta = beta;
z = Double.NaN;
solverAbsoluteAccuracy = inverseCumAccuracy;
}
/**
* Access the first shape parameter, {@code alpha}.
*
* @return the first shape parameter.
*/
public double getAlpha() {
return alpha;
}
/**
* Access the second shape parameter, {@code beta}.
*
* @return the second shape parameter.
*/
public double getBeta() {
return beta;
}
/** Recompute the normalization factor. */
private void recomputeZ() {
if (Double.isNaN(z)) {
z = Gamma.logGamma(alpha) + Gamma.logGamma(beta) - Gamma.logGamma(alpha + beta);
}
}
/** {@inheritDoc} */
public double density(double x) {
final double logDensity = logDensity(x);
return logDensity == Double.NEGATIVE_INFINITY ? 0 : FastMath.exp(logDensity);
}
/** {@inheritDoc} **/
@Override
public double logDensity(double x) {
recomputeZ();
if (x < 0 || x > 1) {
return Double.NEGATIVE_INFINITY;
} else if (x == 0) {
if (alpha < 1) {
throw new NumberIsTooSmallException(LocalizedFormats.CANNOT_COMPUTE_BETA_DENSITY_AT_0_FOR_SOME_ALPHA, alpha, 1, false);
}
return Double.NEGATIVE_INFINITY;
} else if (x == 1) {
if (beta < 1) {
throw new NumberIsTooSmallException(LocalizedFormats.CANNOT_COMPUTE_BETA_DENSITY_AT_1_FOR_SOME_BETA, beta, 1, false);
}
return Double.NEGATIVE_INFINITY;
} else {
double logX = FastMath.log(x);
double log1mX = FastMath.log1p(-x);
return (alpha - 1) * logX + (beta - 1) * log1mX - z;
}
}
/** {@inheritDoc} */
public double cumulativeProbability(double x) {
if (x <= 0) {
return 0;
} else if (x >= 1) {
return 1;
} else {
return Beta.regularizedBeta(x, alpha, beta);
}
}
/**
* Return the absolute accuracy setting of the solver used to estimate
* inverse cumulative probabilities.
*
* @return the solver absolute accuracy.
* @since 2.1
*/
@Override
protected double getSolverAbsoluteAccuracy() {
return solverAbsoluteAccuracy;
}
/**
* {@inheritDoc}
*
* For first shape parameter {@code alpha} and second shape parameter
* {@code beta}, the mean is {@code alpha / (alpha + beta)}.
*/
public double getNumericalMean() {
final double a = getAlpha();
return a / (a + getBeta());
}
/**
* {@inheritDoc}
*
* For first shape parameter {@code alpha} and second shape parameter
* {@code beta}, the variance is
* {@code (alpha * beta) / [(alpha + beta)^2 * (alpha + beta + 1)]}.
*/
public double getNumericalVariance() {
final double a = getAlpha();
final double b = getBeta();
final double alphabetasum = a + b;
return (a * b) / ((alphabetasum * alphabetasum) * (alphabetasum + 1));
}
/**
* {@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 1 no matter the parameters.
*
* @return upper bound of the support (always 1)
*/
public double getSupportUpperBound() {
return 1;
}
/** {@inheritDoc} */
public boolean isSupportLowerBoundInclusive() {
return false;
}
/** {@inheritDoc} */
public boolean isSupportUpperBoundInclusive() {
return false;
}
/**
* {@inheritDoc}
*
* The support of this distribution is connected.
*
* @return {@code true}
*/
public boolean isSupportConnected() {
return true;
}
/** {@inheritDoc}
*
* Sampling is performed using Cheng algorithms:
*
*
* R. C. H. Cheng, "Generating beta variates with nonintegral shape parameters.".
* Communications of the ACM, 21, 317–322, 1978.
*
*/
@Override
public double sample() {
return ChengBetaSampler.sample(random, alpha, beta);
}
/** Utility class implementing Cheng's algorithms for beta distribution sampling.
*
* R. C. H. Cheng, "Generating beta variates with nonintegral shape parameters.".
* Communications of the ACM, 21, 317–322, 1978.
*
* @since 3.6
*/
private static final class ChengBetaSampler {
/**
* Returns one sample using Cheng's sampling algorithm.
* @param random random generator to use
* @param alpha distribution first shape parameter
* @param beta distribution second shape parameter
* @return sampled value
*/
static double sample(RandomGenerator random, final double alpha, final double beta) {
final double a = FastMath.min(alpha, beta);
final double b = FastMath.max(alpha, beta);
if (a > 1) {
return algorithmBB(random, alpha, a, b);
} else {
return algorithmBC(random, alpha, b, a);
}
}
/**
* Returns one sample using Cheng's BB algorithm, when both α and β are greater than 1.
* @param random random generator to use
* @param a0 distribution first shape parameter (α)
* @param a min(α, β) where α, β are the two distribution shape parameters
* @param b max(α, β) where α, β are the two distribution shape parameters
* @return sampled value
*/
private static double algorithmBB(RandomGenerator random,
final double a0,
final double a,
final double b) {
final double alpha = a + b;
final double beta = FastMath.sqrt((alpha - 2.) / (2. * a * b - alpha));
final double gamma = a + 1. / beta;
double r;
double w;
double t;
do {
final double u1 = random.nextDouble();
final double u2 = random.nextDouble();
final double v = beta * (FastMath.log(u1) - FastMath.log1p(-u1));
w = a * FastMath.exp(v);
final double z = u1 * u1 * u2;
r = gamma * v - 1.3862944;
final double s = a + r - w;
if (s + 2.609438 >= 5 * z) {
break;
}
t = FastMath.log(z);
if (s >= t) {
break;
}
} while (r + alpha * (FastMath.log(alpha) - FastMath.log(b + w)) < t);
w = FastMath.min(w, Double.MAX_VALUE);
return Precision.equals(a, a0) ? w / (b + w) : b / (b + w);
}
/**
* Returns one sample using Cheng's BC algorithm, when at least one of α and β is smaller than 1.
* @param random random generator to use
* @param a0 distribution first shape parameter (α)
* @param a max(α, β) where α, β are the two distribution shape parameters
* @param b min(α, β) where α, β are the two distribution shape parameters
* @return sampled value
*/
private static double algorithmBC(RandomGenerator random,
final double a0,
final double a,
final double b) {
final double alpha = a + b;
final double beta = 1. / b;
final double delta = 1. + a - b;
final double k1 = delta * (0.0138889 + 0.0416667 * b) / (a * beta - 0.777778);
final double k2 = 0.25 + (0.5 + 0.25 / delta) * b;
double w;
for (;;) {
final double u1 = random.nextDouble();
final double u2 = random.nextDouble();
final double y = u1 * u2;
final double z = u1 * y;
if (u1 < 0.5) {
if (0.25 * u2 + z - y >= k1) {
continue;
}
} else {
if (z <= 0.25) {
final double v = beta * (FastMath.log(u1) - FastMath.log1p(-u1));
w = a * FastMath.exp(v);
break;
}
if (z >= k2) {
continue;
}
}
final double v = beta * (FastMath.log(u1) - FastMath.log1p(-u1));
w = a * FastMath.exp(v);
if (alpha * (FastMath.log(alpha) - FastMath.log(b + w) + v) - 1.3862944 >= FastMath.log(z)) {
break;
}
}
w = FastMath.min(w, Double.MAX_VALUE);
return Precision.equals(a, a0) ? w / (b + w) : b / (b + w);
}
}
}