Class is obsolete.
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Gilles
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/*
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* Licensed to the Apache Software Foundation (ASF) under one or more
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* contributor license agreements. See the NOTICE file distributed with
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* this work for additional information regarding copyright ownership.
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* The ASF licenses this file to You under the Apache License, Version 2.0
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* (the "License"); you may not use this file except in compliance with
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* the License. You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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package org.apache.commons.math4.distribution;
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import org.apache.commons.numbers.gamma.LogGamma;
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import org.apache.commons.math4.util.FastMath;
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import org.apache.commons.math4.util.MathUtils;
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/**
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* <p>
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* Utility class used by various distributions to accurately compute their
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* respective probability mass functions. The implementation for this class is
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* based on the Catherine Loader's <a target="_blank"
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* href="http://www.herine.net/stat/software/dbinom.html">dbinom</a> routines.
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* </p>
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* <p>
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* This class is not intended to be called directly.
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* </p>
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* <p>
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* References:
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* <ol>
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* <li>Catherine Loader (2000). "Fast and Accurate Computation of Binomial
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* Probabilities.". <a target="_blank"
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* href="http://www.herine.net/stat/papers/dbinom.pdf">
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* http://www.herine.net/stat/papers/dbinom.pdf</a></li>
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* </ol>
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* </p>
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*
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* @since 2.1
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*/
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final class SaddlePointExpansion {
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/** 1/2 * log(2 π). */
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private static final double HALF_LOG_2_PI = 0.5 * FastMath.log(MathUtils.TWO_PI);
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/** exact Stirling expansion error for certain values. */
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private static final double[] EXACT_STIRLING_ERRORS = { 0.0, /* 0.0 */
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0.1534264097200273452913848, /* 0.5 */
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0.0810614667953272582196702, /* 1.0 */
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0.0548141210519176538961390, /* 1.5 */
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0.0413406959554092940938221, /* 2.0 */
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0.03316287351993628748511048, /* 2.5 */
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0.02767792568499833914878929, /* 3.0 */
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0.02374616365629749597132920, /* 3.5 */
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0.02079067210376509311152277, /* 4.0 */
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0.01848845053267318523077934, /* 4.5 */
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0.01664469118982119216319487, /* 5.0 */
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0.01513497322191737887351255, /* 5.5 */
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0.01387612882307074799874573, /* 6.0 */
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0.01281046524292022692424986, /* 6.5 */
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0.01189670994589177009505572, /* 7.0 */
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0.01110455975820691732662991, /* 7.5 */
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0.010411265261972096497478567, /* 8.0 */
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0.009799416126158803298389475, /* 8.5 */
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0.009255462182712732917728637, /* 9.0 */
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0.008768700134139385462952823, /* 9.5 */
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0.008330563433362871256469318, /* 10.0 */
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0.007934114564314020547248100, /* 10.5 */
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0.007573675487951840794972024, /* 11.0 */
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0.007244554301320383179543912, /* 11.5 */
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0.006942840107209529865664152, /* 12.0 */
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0.006665247032707682442354394, /* 12.5 */
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0.006408994188004207068439631, /* 13.0 */
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0.006171712263039457647532867, /* 13.5 */
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0.005951370112758847735624416, /* 14.0 */
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0.005746216513010115682023589, /* 14.5 */
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0.005554733551962801371038690 /* 15.0 */
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};
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/**
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* Default constructor.
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*/
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private SaddlePointExpansion() {
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super();
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}
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/**
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* Compute the error of Stirling's series at the given value.
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* <p>
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* References:
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* <ol>
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* <li>Eric W. Weisstein. "Stirling's Series." From MathWorld--A Wolfram Web
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* Resource. <a target="_blank"
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* href="http://mathworld.wolfram.com/StirlingsSeries.html">
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* http://mathworld.wolfram.com/StirlingsSeries.html</a></li>
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* </ol>
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* </p>
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*
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* @param z the value.
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* @return the Striling's series error.
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*/
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static double getStirlingError(double z) {
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double ret;
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if (z < 15.0) {
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double z2 = 2.0 * z;
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if (FastMath.floor(z2) == z2) {
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ret = EXACT_STIRLING_ERRORS[(int) z2];
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} else {
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ret = LogGamma.value(z + 1.0) - (z + 0.5) * FastMath.log(z) +
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z - HALF_LOG_2_PI;
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}
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} else {
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double z2 = z * z;
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ret = (0.083333333333333333333 -
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(0.00277777777777777777778 -
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(0.00079365079365079365079365 -
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(0.000595238095238095238095238 -
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0.0008417508417508417508417508 /
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z2) / z2) / z2) / z2) / z;
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}
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return ret;
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}
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/**
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* A part of the deviance portion of the saddle point approximation.
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* <p>
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* References:
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* <ol>
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* <li>Catherine Loader (2000). "Fast and Accurate Computation of Binomial
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* Probabilities.". <a target="_blank"
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* href="http://www.herine.net/stat/papers/dbinom.pdf">
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* http://www.herine.net/stat/papers/dbinom.pdf</a></li>
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* </ol>
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* </p>
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*
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* @param x the x value.
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* @param mu the average.
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* @return a part of the deviance.
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*/
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static double getDeviancePart(double x, double mu) {
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double ret;
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if (FastMath.abs(x - mu) < 0.1 * (x + mu)) {
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double d = x - mu;
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double v = d / (x + mu);
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double s1 = v * d;
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double s = Double.NaN;
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double ej = 2.0 * x * v;
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v *= v;
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int j = 1;
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while (s1 != s) {
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s = s1;
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ej *= v;
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s1 = s + ej / ((j * 2) + 1);
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++j;
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}
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ret = s1;
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} else {
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if (x == 0) {
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return mu;
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}
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ret = x * FastMath.log(x / mu) + mu - x;
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}
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return ret;
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}
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/**
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* Compute the logarithm of the PMF for a binomial distribution
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* using the saddle point expansion.
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*
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* @param x the value at which the probability is evaluated.
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* @param n the number of trials.
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* @param p the probability of success.
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* @param q the probability of failure (1 - p).
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* @return log(p(x)).
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*/
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static double logBinomialProbability(int x, int n, double p, double q) {
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double ret;
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if (x == 0) {
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if (p < 0.1) {
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ret = -getDeviancePart(n, n * q) - n * p;
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} else {
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if (n == 0) {
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return 0;
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}
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ret = n * FastMath.log(q);
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}
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} else if (x == n) {
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if (q < 0.1) {
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ret = -getDeviancePart(n, n * p) - n * q;
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} else {
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ret = n * FastMath.log(p);
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}
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} else {
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ret = getStirlingError(n) - getStirlingError(x) -
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getStirlingError(n - x) - getDeviancePart(x, n * p) -
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getDeviancePart(n - x, n * q);
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double f = (MathUtils.TWO_PI * x * (n - x)) / n;
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ret = -0.5 * FastMath.log(f) + ret;
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}
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return ret;
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}
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}
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