/*
* 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;
/**
* <p>
* 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 <a target="_blank"
* href="http://www.herine.net/stat/software/dbinom.html">dbinom</a> routines.
* </p>
* <p>
* This class is not intended to be called directly.
* </p>
* <p>
* References:
* <ol>
* <li>Catherine Loader (2000). "Fast and Accurate Computation of Binomial
* Probabilities.". <a target="_blank"
* href="http://www.herine.net/stat/papers/dbinom.pdf">
* http://www.herine.net/stat/papers/dbinom.pdf</a></li>
* </ol>
* </p>
*
* @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.
* <p>
* References:
* <ol>
* <li>Eric W. Weisstein. "Stirling's Series." From MathWorld--A Wolfram Web
* Resource. <a target="_blank"
* href="http://mathworld.wolfram.com/StirlingsSeries.html">
* http://mathworld.wolfram.com/StirlingsSeries.html</a></li>
* </ol>
* </p>
*
* @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.
* <p>
* References:
* <ol>
* <li>Catherine Loader (2000). "Fast and Accurate Computation of Binomial
* Probabilities.". <a target="_blank"
* href="http://www.herine.net/stat/papers/dbinom.pdf">
* http://www.herine.net/stat/papers/dbinom.pdf</a></li>
* </ol>
* </p>
*
* @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;
}
}