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Diffstat (limited to 'src/main/java/org/apache/commons/math3/distribution/SaddlePointExpansion.java')
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diff --git a/src/main/java/org/apache/commons/math3/distribution/SaddlePointExpansion.java b/src/main/java/org/apache/commons/math3/distribution/SaddlePointExpansion.java new file mode 100644 index 0000000..9dbceec --- /dev/null +++ b/src/main/java/org/apache/commons/math3/distribution/SaddlePointExpansion.java @@ -0,0 +1,199 @@ +/* + * 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 <a + * target="_blank" href="http://www.herine.net/stat/software/dbinom.html">dbinom</a> routines. + * + * <p>This class is not intended to be called directly. + * + * <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> + * </ol> + * + * @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> + * </ol> + * + * @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> + * </ol> + * + * @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; + } +} |