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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.statistics.distribution;

/**
 * Utility class used by various distributions to accurately compute their
 * respective probability mass functions. The implementation for this class is
 * based on the Catherine Loader's
 * dbinom routines.
 *
 * This class is not intended to be called directly.
 */
final class SaddlePointExpansionUtils {
    /** 2 π. */
    private static final double TWO_PI = 2 * Math.PI;
    /** 1/10. */
    private static final double ONE_TENTH = 0.1;
    /** The threshold value for switching the method to compute th Stirling error. */
    private static final int STIRLING_ERROR_THRESHOLD = 15;

    /** Exact Stirling expansion error for certain values. */
    private static final double[] EXACT_STIRLING_ERRORS = {
        0.0, /* 0.0 */
        0.1534264097200273452913848, /* 0.5 */
        0.0810614667953272582196702, /* 1.0 */
        0.0548141210519176538961390, /* 1.5 */
        0.0413406959554092940938221, /* 2.0 */
        0.03316287351993628748511048, /* 2.5 */
        0.02767792568499833914878929, /* 3.0 */
        0.02374616365629749597132920, /* 3.5 */
        0.02079067210376509311152277, /* 4.0 */
        0.01848845053267318523077934, /* 4.5 */
        0.01664469118982119216319487, /* 5.0 */
        0.01513497322191737887351255, /* 5.5 */
        0.01387612882307074799874573, /* 6.0 */
        0.01281046524292022692424986, /* 6.5 */
        0.01189670994589177009505572, /* 7.0 */
        0.01110455975820691732662991, /* 7.5 */
        0.010411265261972096497478567, /* 8.0 */
        0.009799416126158803298389475, /* 8.5 */
        0.009255462182712732917728637, /* 9.0 */
        0.008768700134139385462952823, /* 9.5 */
        0.008330563433362871256469318, /* 10.0 */
        0.007934114564314020547248100, /* 10.5 */
        0.007573675487951840794972024, /* 11.0 */
        0.007244554301320383179543912, /* 11.5 */
        0.006942840107209529865664152, /* 12.0 */
        0.006665247032707682442354394, /* 12.5 */
        0.006408994188004207068439631, /* 13.0 */
        0.006171712263039457647532867, /* 13.5 */
        0.005951370112758847735624416, /* 14.0 */
        0.005746216513010115682023589, /* 14.5 */
        0.005554733551962801371038690 /* 15.0 */
    };

    /**
     * Forbid construction.
     */
    private SaddlePointExpansionUtils() {}

    /**
     * Compute the error of Stirling's series at the given value.
     * 

* References: *

    *
  1. Eric W. Weisstein. "Stirling's Series." From MathWorld--A Wolfram Web * Resource. * https://mathworld.wolfram.com/StirlingsSeries.html
  2. *
*

* *

Note: This function has been modified for integer {@code z}.

* * @param z Value at which the function is evaluated. * @return the Stirling's series error. */ static double getStirlingError(int z) { if (z <= STIRLING_ERROR_THRESHOLD) { return EXACT_STIRLING_ERRORS[2 * z]; } final double z2 = (double) z * z; return (0.083333333333333333333 - (0.00277777777777777777778 - (0.00079365079365079365079365 - (0.000595238095238095238095238 - 0.0008417508417508417508417508 / z2) / z2) / z2) / z2) / z; } /** * A part of the deviance portion of the saddle point approximation. *

* References: *

    *
  1. Catherine Loader (2000). "Fast and Accurate Computation of Binomial * Probabilities.". * http://www.herine.net/stat/papers/dbinom.pdf
  2. *
*

* *

Note: This function has been modified for integer {@code x}.

* * @param x Value at which the function is evaluated. * @param mu Average. * @return a part of the deviance. */ static double getDeviancePart(int x, double mu) { if (Math.abs(x - mu) < 0.1 * (x + mu)) { final double d = x - mu; double v = d / (x + mu); double s1 = v * d; double s = Double.NaN; double ej = 2.0 * x * v; v *= v; int j = 1; while (s1 != s) { s = s1; ej *= v; s1 = s + ej / ((j * 2) + 1); ++j; } return s1; } else if (x == 0) { return mu; } return x * Math.log(x / mu) + mu - x; } /** * Compute the logarithm of the PMF for a binomial distribution * using the saddle point expansion. * * @param x Value at which the probability is evaluated. * @param n Number of trials. * @param p Probability of success. * @param q Probability of failure (1 - p). * @return log(p(x)). */ static double logBinomialProbability(int x, int n, double p, double q) { if (x == 0) { if (p < ONE_TENTH) { // Subtract from 0 avoids returning -0.0 for p=0.0 return 0.0 - getDeviancePart(n, n * q) - n * p; } else if (n == 0) { return 0; } return n * Math.log(q); } else if (x == n) { if (q < ONE_TENTH) { // Subtract from 0 avoids returning -0.0 for p=1.0 return 0.0 - getDeviancePart(n, n * p) - n * q; } return n * Math.log(p); } final int nMx = n - x; final double ret = getStirlingError(n) - getStirlingError(x) - getStirlingError(nMx) - getDeviancePart(x, n * p) - getDeviancePart(nMx, n * q); final double f = (TWO_PI * x * nMx) / n; return -0.5 * Math.log(f) + ret; } }




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