/*
 * 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;
import org.apache.commons.math3.util.CombinatoricsUtils;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathUtils;

/**
 * Implementation of the Poisson distribution.
 *
 * @see <a href="http://en.wikipedia.org/wiki/Poisson_distribution">Poisson distribution (Wikipedia)</a>
 * @see <a href="http://mathworld.wolfram.com/PoissonDistribution.html">Poisson distribution (MathWorld)</a>
 */
public class PoissonDistribution extends AbstractIntegerDistribution {
    /**
     * Default maximum number of iterations for cumulative probability calculations.
     * @since 2.1
     */
    public static final int DEFAULT_MAX_ITERATIONS = 10000000;
    /**
     * Default convergence criterion.
     * @since 2.1
     */
    public static final double DEFAULT_EPSILON = 1e-12;
    /** Serializable version identifier. */
    private static final long serialVersionUID = -3349935121172596109L;
    /** Distribution used to compute normal approximation. */
    private final NormalDistribution normal;
    /** Distribution needed for the {@link #sample()} method. */
    private final ExponentialDistribution exponential;
    /** Mean of the distribution. */
    private final double mean;

    /**
     * Maximum number of iterations for cumulative probability. Cumulative
     * probabilities are estimated using either Lanczos series approximation
     * of {@link Gamma#regularizedGammaP(double, double, double, int)}
     * or continued fraction approximation of
     * {@link Gamma#regularizedGammaQ(double, double, double, int)}.
     */
    private final int maxIterations;

    /** Convergence criterion for cumulative probability. */
    private final double epsilon;

    /**
     * Creates a new Poisson distribution with specified mean.
     * <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 p the Poisson mean
     * @throws NotStrictlyPositiveException if {@code p <= 0}.
     */
    public PoissonDistribution(double p) throws NotStrictlyPositiveException {
        this(p, DEFAULT_EPSILON, DEFAULT_MAX_ITERATIONS);
    }

    /**
     * Creates a new Poisson distribution with specified mean, convergence
     * criterion and maximum number of iterations.
     * <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 p Poisson mean.
     * @param epsilon Convergence criterion for cumulative probabilities.
     * @param maxIterations the maximum number of iterations for cumulative
     * probabilities.
     * @throws NotStrictlyPositiveException if {@code p <= 0}.
     * @since 2.1
     */
    public PoissonDistribution(double p, double epsilon, int maxIterations)
    throws NotStrictlyPositiveException {
        this(new Well19937c(), p, epsilon, maxIterations);
    }

    /**
     * Creates a new Poisson distribution with specified mean, convergence
     * criterion and maximum number of iterations.
     *
     * @param rng Random number generator.
     * @param p Poisson mean.
     * @param epsilon Convergence criterion for cumulative probabilities.
     * @param maxIterations the maximum number of iterations for cumulative
     * probabilities.
     * @throws NotStrictlyPositiveException if {@code p <= 0}.
     * @since 3.1
     */
    public PoissonDistribution(RandomGenerator rng,
                               double p,
                               double epsilon,
                               int maxIterations)
    throws NotStrictlyPositiveException {
        super(rng);

        if (p <= 0) {
            throw new NotStrictlyPositiveException(LocalizedFormats.MEAN, p);
        }
        mean = p;
        this.epsilon = epsilon;
        this.maxIterations = maxIterations;

        // Use the same RNG instance as the parent class.
        normal = new NormalDistribution(rng, p, FastMath.sqrt(p),
                                        NormalDistribution.DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
        exponential = new ExponentialDistribution(rng, 1,
                                                  ExponentialDistribution.DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
    }

    /**
     * Creates a new Poisson distribution with the specified mean and
     * convergence criterion.
     *
     * @param p Poisson mean.
     * @param epsilon Convergence criterion for cumulative probabilities.
     * @throws NotStrictlyPositiveException if {@code p <= 0}.
     * @since 2.1
     */
    public PoissonDistribution(double p, double epsilon)
    throws NotStrictlyPositiveException {
        this(p, epsilon, DEFAULT_MAX_ITERATIONS);
    }

    /**
     * Creates a new Poisson distribution with the specified mean and maximum
     * number of iterations.
     *
     * @param p Poisson mean.
     * @param maxIterations Maximum number of iterations for cumulative
     * probabilities.
     * @since 2.1
     */
    public PoissonDistribution(double p, int maxIterations) {
        this(p, DEFAULT_EPSILON, maxIterations);
    }

    /**
     * Get the mean for the distribution.
     *
     * @return the mean for the distribution.
     */
    public double getMean() {
        return mean;
    }

    /** {@inheritDoc} */
    public double probability(int x) {
        final double logProbability = logProbability(x);
        return logProbability == Double.NEGATIVE_INFINITY ? 0 : FastMath.exp(logProbability);
    }

    /** {@inheritDoc} */
    @Override
    public double logProbability(int x) {
        double ret;
        if (x < 0 || x == Integer.MAX_VALUE) {
            ret = Double.NEGATIVE_INFINITY;
        } else if (x == 0) {
            ret = -mean;
        } else {
            ret = -SaddlePointExpansion.getStirlingError(x) -
                  SaddlePointExpansion.getDeviancePart(x, mean) -
                  0.5 * FastMath.log(MathUtils.TWO_PI) - 0.5 * FastMath.log(x);
        }
        return ret;
    }

    /** {@inheritDoc} */
    public double cumulativeProbability(int x) {
        if (x < 0) {
            return 0;
        }
        if (x == Integer.MAX_VALUE) {
            return 1;
        }
        return Gamma.regularizedGammaQ((double) x + 1, mean, epsilon,
                                       maxIterations);
    }

    /**
     * Calculates the Poisson distribution function using a normal
     * approximation. The {@code N(mean, sqrt(mean))} distribution is used
     * to approximate the Poisson distribution. The computation uses
     * "half-correction" (evaluating the normal distribution function at
     * {@code x + 0.5}).
     *
     * @param x Upper bound, inclusive.
     * @return the distribution function value calculated using a normal
     * approximation.
     */
    public double normalApproximateProbability(int x)  {
        // calculate the probability using half-correction
        return normal.cumulativeProbability(x + 0.5);
    }

    /**
     * {@inheritDoc}
     *
     * For mean parameter {@code p}, the mean is {@code p}.
     */
    public double getNumericalMean() {
        return getMean();
    }

    /**
     * {@inheritDoc}
     *
     * For mean parameter {@code p}, the variance is {@code p}.
     */
    public double getNumericalVariance() {
        return getMean();
    }

    /**
     * {@inheritDoc}
     *
     * The lower bound of the support is always 0 no matter the mean parameter.
     *
     * @return lower bound of the support (always 0)
     */
    public int getSupportLowerBound() {
        return 0;
    }

    /**
     * {@inheritDoc}
     *
     * The upper bound of the support is positive infinity,
     * regardless of the parameter values. There is no integer infinity,
     * so this method returns {@code Integer.MAX_VALUE}.
     *
     * @return upper bound of the support (always {@code Integer.MAX_VALUE} for
     * positive infinity)
     */
    public int getSupportUpperBound() {
        return Integer.MAX_VALUE;
    }

    /**
     * {@inheritDoc}
     *
     * The support of this distribution is connected.
     *
     * @return {@code true}
     */
    public boolean isSupportConnected() {
        return true;
    }

    /**
     * {@inheritDoc}
     * <p>
     * <strong>Algorithm Description</strong>:
     * <ul>
     *  <li>For small means, uses simulation of a Poisson process
     *   using Uniform deviates, as described
     *   <a href="http://mathaa.epfl.ch/cours/PMMI2001/interactive/rng7.htm"> here</a>.
     *   The Poisson process (and hence value returned) is bounded by 1000 * mean.
     *  </li>
     *  <li>For large means, uses the rejection algorithm described in
     *   <blockquote>
     *    Devroye, Luc. (1981).<i>The Computer Generation of Poisson Random Variables</i><br>
     *    <strong>Computing</strong> vol. 26 pp. 197-207.<br>
     *   </blockquote>
     *  </li>
     * </ul>
     * </p>
     *
     * @return a random value.
     * @since 2.2
     */
    @Override
    public int sample() {
        return (int) FastMath.min(nextPoisson(mean), Integer.MAX_VALUE);
    }

    /**
     * @param meanPoisson Mean of the Poisson distribution.
     * @return the next sample.
     */
    private long nextPoisson(double meanPoisson) {
        final double pivot = 40.0d;
        if (meanPoisson < pivot) {
            double p = FastMath.exp(-meanPoisson);
            long n = 0;
            double r = 1.0d;
            double rnd = 1.0d;

            while (n < 1000 * meanPoisson) {
                rnd = random.nextDouble();
                r *= rnd;
                if (r >= p) {
                    n++;
                } else {
                    return n;
                }
            }
            return n;
        } else {
            final double lambda = FastMath.floor(meanPoisson);
            final double lambdaFractional = meanPoisson - lambda;
            final double logLambda = FastMath.log(lambda);
            final double logLambdaFactorial = CombinatoricsUtils.factorialLog((int) lambda);
            final long y2 = lambdaFractional < Double.MIN_VALUE ? 0 : nextPoisson(lambdaFractional);
            final double delta = FastMath.sqrt(lambda * FastMath.log(32 * lambda / FastMath.PI + 1));
            final double halfDelta = delta / 2;
            final double twolpd = 2 * lambda + delta;
            final double a1 = FastMath.sqrt(FastMath.PI * twolpd) * FastMath.exp(1 / (8 * lambda));
            final double a2 = (twolpd / delta) * FastMath.exp(-delta * (1 + delta) / twolpd);
            final double aSum = a1 + a2 + 1;
            final double p1 = a1 / aSum;
            final double p2 = a2 / aSum;
            final double c1 = 1 / (8 * lambda);

            double x = 0;
            double y = 0;
            double v = 0;
            int a = 0;
            double t = 0;
            double qr = 0;
            double qa = 0;
            for (;;) {
                final double u = random.nextDouble();
                if (u <= p1) {
                    final double n = random.nextGaussian();
                    x = n * FastMath.sqrt(lambda + halfDelta) - 0.5d;
                    if (x > delta || x < -lambda) {
                        continue;
                    }
                    y = x < 0 ? FastMath.floor(x) : FastMath.ceil(x);
                    final double e = exponential.sample();
                    v = -e - (n * n / 2) + c1;
                } else {
                    if (u > p1 + p2) {
                        y = lambda;
                        break;
                    } else {
                        x = delta + (twolpd / delta) * exponential.sample();
                        y = FastMath.ceil(x);
                        v = -exponential.sample() - delta * (x + 1) / twolpd;
                    }
                }
                a = x < 0 ? 1 : 0;
                t = y * (y + 1) / (2 * lambda);
                if (v < -t && a == 0) {
                    y = lambda + y;
                    break;
                }
                qr = t * ((2 * y + 1) / (6 * lambda) - 1);
                qa = qr - (t * t) / (3 * (lambda + a * (y + 1)));
                if (v < qa) {
                    y = lambda + y;
                    break;
                }
                if (v > qr) {
                    continue;
                }
                if (v < y * logLambda - CombinatoricsUtils.factorialLog((int) (y + lambda)) + logLambdaFactorial) {
                    y = lambda + y;
                    break;
                }
            }
            return y2 + (long) y;
        }
    }
}