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* 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
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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:
- Catherine Loader (2000). "Fast and Accurate Computation of Binomial
Probabilities.".
http://www.herine.net/stat/papers/dbinom.pdf
Since: 2.1
/**
* <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). /** 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. /** 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.
/**
* Default constructor.
*/
private SaddlePointExpansion() {
super();
}
Compute the error of Stirling's series at the given value.
References:
- Eric W. Weisstein. "Stirling's Series." From MathWorld--A Wolfram Web
Resource.
http://mathworld.wolfram.com/StirlingsSeries.html
Params: - z – the value.
Returns: the Striling's series error.
/**
* 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.
References:
- Catherine Loader (2000). "Fast and Accurate Computation of Binomial
Probabilities.".
http://www.herine.net/stat/papers/dbinom.pdf
Params: - x – the x value.
- mu – the average.
Returns: a part of the deviance.
/**
* 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.
Params: - x – the value at which the probability is evaluated.
- n – the number of trials.
- p – the probability of success.
- q – the probability of failure (1 - p).
Returns: log(p(x)).
/**
* 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;
}
}