jidt/java/source/infodynamics/utils/commonsmath3/distribution/BetaDistribution.java

/*
 *  Java Information Dynamics Toolkit (JIDT)
 *  Copyright (C) 2017, Joseph T. Lizier
 *  
 *  This program is free software: you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation, either version 3 of the License, or
 *  (at your option) any later version.
 *  
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *  
 *  You should have received a copy of the GNU General Public License
 *  along with this program.  If not, see <http://www.gnu.org/licenses/>.
*/

/*
 * This class was originally distributed as part of the Apache Commons
 *  Math3 library (3.6.1), under the Apache License Version 2.0, which is 
 *  copied below. This Apache 2 software is now included as a derivative
 *  work in the GPLv3 licensed JIDT project, as per:
 *  http://www.apache.org/licenses/GPL-compatibility.html
 *  
 * The original Apache source code has been modified as follows:
 * -- We have modified package names to sit inside the JIDT structure.
 */

/*
 * 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 infodynamics.utils.commonsmath3.distribution;

import infodynamics.utils.commonsmath3.exception.NumberIsTooSmallException;
import infodynamics.utils.commonsmath3.exception.util.LocalizedFormats;
import infodynamics.utils.commonsmath3.random.RandomGenerator;
import infodynamics.utils.commonsmath3.random.Well19937c;
import infodynamics.utils.commonsmath3.special.Beta;
import infodynamics.utils.commonsmath3.special.Gamma;
import infodynamics.utils.commonsmath3.util.FastMath;
import infodynamics.utils.commonsmath3.util.Precision;

/**
 * Implements the Beta distribution.
 *
 * @see <a href="http://en.wikipedia.org/wiki/Beta_distribution">Beta distribution</a>
 * @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.
     * <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 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.
     * <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 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 &beta; 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 &beta; 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}
    * <p>
    * Sampling is performed using Cheng algorithms:
    * </p>
    * <p>
    * R. C. H. Cheng, "Generating beta variates with nonintegral shape parameters.".
    *                 Communications of the ACM, 21, 317–322, 1978.
    * </p>
    */
    @Override
    public double sample() {
        return ChengBetaSampler.sample(random, alpha, beta);
    }

    /** Utility class implementing Cheng's algorithms for beta distribution sampling.
     * <p>
     * R. C. H. Cheng, "Generating beta variates with nonintegral shape parameters.".
     *                 Communications of the ACM, 21, 317–322, 1978.
     * </p>
     * @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 &alpha; and &beta; are greater than 1.
         * @param random random generator to use
         * @param a0 distribution first shape parameter (&alpha;)
         * @param a min(&alpha;, &beta;) where &alpha;, &beta; are the two distribution shape parameters
         * @param b max(&alpha;, &beta;) where &alpha;, &beta; 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 &alpha; and &beta; is smaller than 1.
         * @param random random generator to use
         * @param a0 distribution first shape parameter (&alpha;)
         * @param a max(&alpha;, &beta;) where &alpha;, &beta; are the two distribution shape parameters
         * @param b min(&alpha;, &beta;) where &alpha;, &beta; 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);
        }

    }
}