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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.

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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.special.Gamma;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathUtils;

/**
 * 

* 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. *

*

* References: *

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

* * @since 2.1 */ final class SaddlePointExpansion { /** 1/2 * log(2 π). */ private static final double HALF_LOG_2_PI = 0.5 * FastMath.log(MathUtils.TWO_PI); /** 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 */ }; /** * Default constructor. */ private SaddlePointExpansion() { super(); } /** * 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. * http://mathworld.wolfram.com/StirlingsSeries.html
  2. *
*

* * @param z the value. * @return the Striling's series error. */ static double getStirlingError(double z) { double ret; if (z < 15.0) { double z2 = 2.0 * z; if (FastMath.floor(z2) == z2) { ret = EXACT_STIRLING_ERRORS[(int) z2]; } else { ret = Gamma.logGamma(z + 1.0) - (z + 0.5) * FastMath.log(z) + z - HALF_LOG_2_PI; } } else { double z2 = z * z; ret = (0.083333333333333333333 - (0.00277777777777777777778 - (0.00079365079365079365079365 - (0.000595238095238095238095238 - 0.0008417508417508417508417508 / z2) / z2) / z2) / z2) / z; } return ret; } /** * 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. *
*

* * @param x the x value. * @param mu the average. * @return a part of the deviance. */ static double getDeviancePart(double x, double mu) { double ret; if (FastMath.abs(x - mu) < 0.1 * (x + mu)) { 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; } ret = s1; } else { ret = x * FastMath.log(x / mu) + mu - x; } return ret; } /** * Compute the logarithm of the PMF for a binomial distribution * using the saddle point expansion. * * @param x the value at which the probability is evaluated. * @param n the number of trials. * @param p the probability of success. * @param q the probability of failure (1 - p). * @return log(p(x)). */ static double logBinomialProbability(int x, int n, double p, double q) { double ret; if (x == 0) { if (p < 0.1) { ret = -getDeviancePart(n, n * q) - n * p; } else { ret = n * FastMath.log(q); } } else if (x == n) { if (q < 0.1) { ret = -getDeviancePart(n, n * p) - n * q; } else { ret = n * FastMath.log(p); } } else { ret = getStirlingError(n) - getStirlingError(x) - getStirlingError(n - x) - getDeviancePart(x, n * p) - getDeviancePart(n - x, n * q); double f = (MathUtils.TWO_PI * x * (n - x)) / n; ret = -0.5 * FastMath.log(f) + ret; } return ret; } }




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