RejectionInversionZipfSampler.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
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package org.apache.commons.rng.sampling.distribution;

import org.apache.commons.rng.UniformRandomProvider;

/**
 * Implementation of the <a href="https://en.wikipedia.org/wiki/Zipf's_law">Zipf distribution</a>.
 *
 * <p>Sampling uses {@link UniformRandomProvider#nextDouble()}.</p>
 *
 * @since 1.0
 */
public class RejectionInversionZipfSampler
    extends SamplerBase
    implements SharedStateDiscreteSampler {

    /** The implementation of the sample method. */
    private final SharedStateDiscreteSampler delegate;

    /**
     * Implements the rejection-inversion method for the Zipf distribution.
     */
    private static final class RejectionInversionZipfSamplerImpl implements SharedStateDiscreteSampler {
        /** Threshold below which Taylor series will be used. */
        private static final double TAYLOR_THRESHOLD = 1e-8;
        /** 1/2. */
        private static final double F_1_2 = 0.5;
        /** 1/3. */
        private static final double F_1_3 = 1d / 3;
        /** 1/4. */
        private static final double F_1_4 = 0.25;
        /** Number of elements. */
        private final int numberOfElements;
        /** Exponent parameter of the distribution. */
        private final double exponent;
        /** {@code hIntegral(1.5) - 1}. */
        private final double hIntegralX1;
        /** {@code hIntegral(numberOfElements + 0.5)}. */
        private final double hIntegralNumberOfElements;
        /** {@code hIntegralX1 - hIntegralNumberOfElements}. */
        private final double r;
        /** {@code 2 - hIntegralInverse(hIntegral(2.5) - h(2)}. */
        private final double s;
        /** Underlying source of randomness. */
        private final UniformRandomProvider rng;

        /**
         * @param rng Generator of uniformly distributed random numbers.
         * @param numberOfElements Number of elements (must be {@code > 0}).
         * @param exponent Exponent (must be {@code > 0}).
         */
        RejectionInversionZipfSamplerImpl(UniformRandomProvider rng,
                                          int numberOfElements,
                                          double exponent) {
            this.rng = rng;
            this.numberOfElements = numberOfElements;
            this.exponent = exponent;
            this.hIntegralX1 = hIntegral(1.5) - 1;
            this.hIntegralNumberOfElements = hIntegral(numberOfElements + F_1_2);
            this.r = hIntegralX1 - hIntegralNumberOfElements;
            this.s = 2 - hIntegralInverse(hIntegral(2.5) - h(2));
        }

        /**
         * @param rng Generator of uniformly distributed random numbers.
         * @param source Source to copy.
         */
        private RejectionInversionZipfSamplerImpl(UniformRandomProvider rng,
                                                  RejectionInversionZipfSamplerImpl source) {
            this.rng = rng;
            this.numberOfElements = source.numberOfElements;
            this.exponent = source.exponent;
            this.hIntegralX1 = source.hIntegralX1;
            this.hIntegralNumberOfElements = source.hIntegralNumberOfElements;
            this.r = source.r;
            this.s = source.s;
        }

        @Override
        public int sample() {
            // The paper describes an algorithm for exponents larger than 1
            // (Algorithm ZRI).
            // The original method uses
            //   H(x) = (v + x)^(1 - q) / (1 - q)
            // as the integral of the hat function.
            // This function is undefined for q = 1, which is the reason for
            // the limitation of the exponent.
            // If instead the integral function
            //   H(x) = ((v + x)^(1 - q) - 1) / (1 - q)
            // is used,
            // for which a meaningful limit exists for q = 1, the method works
            // for all positive exponents.
            // The following implementation uses v = 0 and generates integral
            // number in the range [1, numberOfElements].
            // This is different to the original method where v is defined to
            // be positive and numbers are taken from [0, i_max].
            // This explains why the implementation looks slightly different.

            while (true) {
                final double u = hIntegralNumberOfElements + rng.nextDouble() * r;
                // u is uniformly distributed in (hIntegralX1, hIntegralNumberOfElements]

                final double x = hIntegralInverse(u);
                int k = (int) (x + F_1_2);

                // Limit k to the range [1, numberOfElements] if it would be outside
                // due to numerical inaccuracies.
                if (k < 1) {
                    k = 1;
                } else if (k > numberOfElements) {
                    k = numberOfElements;
                }

                // Here, the distribution of k is given by:
                //
                //   P(k = 1) = C * (hIntegral(1.5) - hIntegralX1) = C
                //   P(k = m) = C * (hIntegral(m + 1/2) - hIntegral(m - 1/2)) for m >= 2
                //
                //   where C = 1 / (hIntegralNumberOfElements - hIntegralX1)

                if (k - x <= s || u >= hIntegral(k + F_1_2) - h(k)) {

                    // Case k = 1:
                    //
                    //   The right inequality is always true, because replacing k by 1 gives
                    //   u >= hIntegral(1.5) - h(1) = hIntegralX1 and u is taken from
                    //   (hIntegralX1, hIntegralNumberOfElements].
                    //
                    //   Therefore, the acceptance rate for k = 1 is P(accepted | k = 1) = 1
                    //   and the probability that 1 is returned as random value is
                    //   P(k = 1 and accepted) = P(accepted | k = 1) * P(k = 1) = C = C / 1^exponent
                    //
                    // Case k >= 2:
                    //
                    //   The left inequality (k - x <= s) is just a short cut
                    //   to avoid the more expensive evaluation of the right inequality
                    //   (u >= hIntegral(k + 0.5) - h(k)) in many cases.
                    //
                    //   If the left inequality is true, the right inequality is also true:
                    //     Theorem 2 in the paper is valid for all positive exponents, because
                    //     the requirements h'(x) = -exponent/x^(exponent + 1) < 0 and
                    //     (-1/hInverse'(x))'' = (1+1/exponent) * x^(1/exponent-1) >= 0
                    //     are both fulfilled.
                    //     Therefore, f(x) = x - hIntegralInverse(hIntegral(x + 0.5) - h(x))
                    //     is a non-decreasing function. If k - x <= s holds,
                    //     k - x <= s + f(k) - f(2) is obviously also true which is equivalent to
                    //     -x <= -hIntegralInverse(hIntegral(k + 0.5) - h(k)),
                    //     -hIntegralInverse(u) <= -hIntegralInverse(hIntegral(k + 0.5) - h(k)),
                    //     and finally u >= hIntegral(k + 0.5) - h(k).
                    //
                    //   Hence, the right inequality determines the acceptance rate:
                    //   P(accepted | k = m) = h(m) / (hIntegrated(m+1/2) - hIntegrated(m-1/2))
                    //   The probability that m is returned is given by
                    //   P(k = m and accepted) = P(accepted | k = m) * P(k = m) = C * h(m) = C / m^exponent.
                    //
                    // In both cases the probabilities are proportional to the probability mass function
                    // of the Zipf distribution.

                    return k;
                }
            }
        }

        /** {@inheritDoc} */
        @Override
        public String toString() {
            return "Rejection inversion Zipf deviate [" + rng.toString() + "]";
        }

        @Override
        public SharedStateDiscreteSampler withUniformRandomProvider(UniformRandomProvider rng) {
            return new RejectionInversionZipfSamplerImpl(rng, this);
        }

        /**
         * {@code H(x)} is defined as
         * <ul>
         *  <li>{@code (x^(1 - exponent) - 1) / (1 - exponent)}, if {@code exponent != 1}</li>
         *  <li>{@code log(x)}, if {@code exponent == 1}</li>
         * </ul>
         * H(x) is an integral function of h(x), the derivative of H(x) is h(x).
         *
         * @param x Free parameter.
         * @return {@code H(x)}.
         */
        private double hIntegral(final double x) {
            final double logX = Math.log(x);
            return helper2((1 - exponent) * logX) * logX;
        }

        /**
         * {@code h(x) = 1 / x^exponent}.
         *
         * @param x Free parameter.
         * @return {@code h(x)}.
         */
        private double h(final double x) {
            return Math.exp(-exponent * Math.log(x));
        }

        /**
         * The inverse function of {@code H(x)}.
         *
         * @param x Free parameter
         * @return y for which {@code H(y) = x}.
         */
        private double hIntegralInverse(final double x) {
            double t = x * (1 - exponent);
            if (t < -1) {
                // Limit value to the range [-1, +inf).
                // t could be smaller than -1 in some rare cases due to numerical errors.
                t = -1;
            }
            return Math.exp(helper1(t) * x);
        }

        /**
         * Helper function that calculates {@code log(1 + x) / x}.
         * <p>
         * A Taylor series expansion is used, if x is close to 0.
         * </p>
         *
         * @param x A value larger than or equal to -1.
         * @return {@code log(1 + x) / x}.
         */
        private static double helper1(final double x) {
            if (Math.abs(x) > TAYLOR_THRESHOLD) {
                return Math.log1p(x) / x;
            }
            return 1 - x * (F_1_2 - x * (F_1_3 - F_1_4 * x));
        }

        /**
         * Helper function to calculate {@code (exp(x) - 1) / x}.
         * <p>
         * A Taylor series expansion is used, if x is close to 0.
         * </p>
         *
         * @param x Free parameter.
         * @return {@code (exp(x) - 1) / x} if x is non-zero, or 1 if x = 0.
         */
        private static double helper2(final double x) {
            if (Math.abs(x) > TAYLOR_THRESHOLD) {
                return Math.expm1(x) / x;
            }
            return 1 + x * F_1_2 * (1 + x * F_1_3 * (1 + F_1_4 * x));
        }
    }

    /**
     * This instance delegates sampling. Use the factory method
     * {@link #of(UniformRandomProvider, int, double)} to create an optimal sampler.
     *
     * @param rng Generator of uniformly distributed random numbers.
     * @param numberOfElements Number of elements.
     * @param exponent Exponent.
     * @throws IllegalArgumentException if {@code numberOfElements <= 0}
     * or {@code exponent < 0}.
     */
    public RejectionInversionZipfSampler(UniformRandomProvider rng,
                                         int numberOfElements,
                                         double exponent) {
        this(of(rng, numberOfElements, exponent));
    }

    /**
     * Private constructor used by to prevent partially initialized object if the construction
     * of the delegate throws. In future versions the public constructor should be removed.
     *
     * @param delegate Delegate.
     */
    private RejectionInversionZipfSampler(SharedStateDiscreteSampler delegate) {
        super(null);
        this.delegate = delegate;
    }

    /**
     * Rejection inversion sampling method for a discrete, bounded Zipf
     * distribution that is based on the method described in
     * <blockquote>
     *   Wolfgang Hörmann and Gerhard Derflinger.
     *   <i>"Rejection-inversion to generate variates from monotone discrete
     *    distributions",</i><br>
     *   <strong>ACM Transactions on Modeling and Computer Simulation</strong> (TOMACS) 6.3 (1996): 169-184.
     * </blockquote>
     */
    @Override
    public int sample() {
        return delegate.sample();
    }

    /** {@inheritDoc} */
    @Override
    public String toString() {
        return delegate.toString();
    }

    /**
     * {@inheritDoc}
     *
     * @since 1.3
     */
    @Override
    public SharedStateDiscreteSampler withUniformRandomProvider(UniformRandomProvider rng) {
        return delegate.withUniformRandomProvider(rng);
    }

    /**
     * Creates a new Zipf distribution sampler.
     *
     * <p>Note when {@code exponent = 0} the Zipf distribution reduces to a
     * discrete uniform distribution over the interval {@code [1, n]} with {@code n}
     * the number of elements.
     *
     * @param rng Generator of uniformly distributed random numbers.
     * @param numberOfElements Number of elements.
     * @param exponent Exponent.
     * @return the sampler
     * @throws IllegalArgumentException if {@code numberOfElements <= 0} or
     * {@code exponent < 0}.
     * @since 1.3
     */
    public static SharedStateDiscreteSampler of(UniformRandomProvider rng,
                                                int numberOfElements,
                                                double exponent) {
        if (numberOfElements <= 0) {
            throw new IllegalArgumentException("number of elements is not strictly positive: " + numberOfElements);
        }
        InternalUtils.requirePositive(exponent, "exponent");

        // When the exponent is at the limit of 0 the distribution PMF reduces to 1 / n
        // and sampling can use a discrete uniform sampler.
        return exponent > 0 ?
            new RejectionInversionZipfSamplerImpl(rng, numberOfElements, exponent) :
            DiscreteUniformSampler.of(rng, 1, numberOfElements);
    }
}