org.apache.commons.statistics.distribution.PascalDistribution Maven / Gradle / Ivy
/*
* 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.statistics.distribution;
import org.apache.commons.numbers.combinatorics.BinomialCoefficientDouble;
import org.apache.commons.numbers.combinatorics.LogBinomialCoefficient;
import org.apache.commons.numbers.gamma.RegularizedBeta;
/**
* Implementation of the Pascal distribution.
*
* The Pascal distribution is a special case of the negative binomial distribution
* where the number of successes parameter is an integer.
*
*
There are various ways to express the probability mass and distribution
* functions for the Pascal distribution. The present implementation represents
* the distribution of the number of failures before \( r \) successes occur.
* This is the convention adopted in e.g.
* MathWorld,
* but not in
* Wikipedia.
*
*
The probability mass function of \( X \) is:
*
*
\[ f(k; r, p) = \binom{k+r-1}{r-1} p^r \, (1-p)^k \]
*
*
for \( r \in \{1, 2, \dots\} \) the number of successes,
* \( p \in (0, 1] \) the probability of success,
* \( k \in \{0, 1, 2, \dots\} \) the total number of failures, and
*
*
\[ \binom{k+r-1}{r-1} = \frac{(k+r-1)!}{(r-1)! \, k!} \]
*
*
is the binomial coefficient.
*
*
The cumulative distribution function of \( X \) is:
*
*
\[ P(X \leq k) = I(p, r, k + 1) \]
*
*
where \( I \) is the regularized incomplete beta function.
*
* @see Negative binomial distribution (Wikipedia)
* @see Negative binomial distribution (MathWorld)
*/
public final class PascalDistribution extends AbstractDiscreteDistribution {
/** The number of successes. */
private final int numberOfSuccesses;
/** The probability of success. */
private final double probabilityOfSuccess;
/** The value of {@code log(p) * n}, where {@code p} is the probability of success
* and {@code n} is the number of successes, stored for faster computation. */
private final double logProbabilityOfSuccessByNumOfSuccesses;
/** The value of {@code log(1-p)}, where {@code p} is the probability of success,
* stored for faster computation. */
private final double log1mProbabilityOfSuccess;
/** The value of {@code p^n}, where {@code p} is the probability of success
* and {@code n} is the number of successes, stored for faster computation. */
private final double probabilityOfSuccessPowNumOfSuccesses;
/**
* @param r Number of successes.
* @param p Probability of success.
*/
private PascalDistribution(int r,
double p) {
numberOfSuccesses = r;
probabilityOfSuccess = p;
logProbabilityOfSuccessByNumOfSuccesses = Math.log(p) * numberOfSuccesses;
log1mProbabilityOfSuccess = Math.log1p(-p);
probabilityOfSuccessPowNumOfSuccesses = Math.pow(probabilityOfSuccess, numberOfSuccesses);
}
/**
* Create a Pascal distribution.
*
* @param r Number of successes.
* @param p Probability of success.
* @return the distribution
* @throws IllegalArgumentException if {@code r <= 0} or {@code p <= 0} or
* {@code p > 1}.
*/
public static PascalDistribution of(int r,
double p) {
if (r <= 0) {
throw new DistributionException(DistributionException.NOT_STRICTLY_POSITIVE, r);
}
if (p <= 0 ||
p > 1) {
throw new DistributionException(DistributionException.INVALID_NON_ZERO_PROBABILITY, p);
}
return new PascalDistribution(r, p);
}
/**
* Gets the number of successes parameter of this distribution.
*
* @return the number of successes.
*/
public int getNumberOfSuccesses() {
return numberOfSuccesses;
}
/**
* Gets the probability of success parameter of this distribution.
*
* @return the probability of success.
*/
public double getProbabilityOfSuccess() {
return probabilityOfSuccess;
}
/** {@inheritDoc} */
@Override
public double probability(int x) {
if (x <= 0) {
// Special case of x=0 exploiting cancellation.
return x == 0 ? probabilityOfSuccessPowNumOfSuccesses : 0.0;
}
final int n = x + numberOfSuccesses - 1;
if (n < 0) {
// overflow
return 0.0;
}
return BinomialCoefficientDouble.value(n, numberOfSuccesses - 1) *
probabilityOfSuccessPowNumOfSuccesses *
Math.pow(1.0 - probabilityOfSuccess, x);
}
/** {@inheritDoc} */
@Override
public double logProbability(int x) {
if (x <= 0) {
// Special case of x=0 exploiting cancellation.
return x == 0 ? logProbabilityOfSuccessByNumOfSuccesses : Double.NEGATIVE_INFINITY;
}
final int n = x + numberOfSuccesses - 1;
if (n < 0) {
// overflow
return Double.NEGATIVE_INFINITY;
}
return LogBinomialCoefficient.value(n, numberOfSuccesses - 1) +
logProbabilityOfSuccessByNumOfSuccesses +
log1mProbabilityOfSuccess * x;
}
/** {@inheritDoc} */
@Override
public double cumulativeProbability(int x) {
if (x < 0) {
return 0.0;
}
return RegularizedBeta.value(probabilityOfSuccess,
numberOfSuccesses, x + 1.0);
}
/** {@inheritDoc} */
@Override
public double survivalProbability(int x) {
if (x < 0) {
return 1.0;
}
return RegularizedBeta.complement(probabilityOfSuccess,
numberOfSuccesses, x + 1.0);
}
/**
* {@inheritDoc}
*
*
For number of successes \( r \) and probability of success \( p \),
* the mean is:
*
*
\[ \frac{r (1 - p)}{p} \]
*/
@Override
public double getMean() {
final double p = getProbabilityOfSuccess();
final double r = getNumberOfSuccesses();
return (r * (1 - p)) / p;
}
/**
* {@inheritDoc}
*
*
For number of successes \( r \) and probability of success \( p \),
* the variance is:
*
*
\[ \frac{r (1 - p)}{p^2} \]
*/
@Override
public double getVariance() {
final double p = getProbabilityOfSuccess();
final double r = getNumberOfSuccesses();
return r * (1 - p) / (p * p);
}
/**
* {@inheritDoc}
*
*
The lower bound of the support is always 0.
*
* @return 0.
*/
@Override
public int getSupportLowerBound() {
return 0;
}
/**
* {@inheritDoc}
*
*
The upper bound of the support is positive infinity except for the
* probability parameter {@code p = 1.0}.
*
* @return {@link Integer#MAX_VALUE} or 0.
*/
@Override
public int getSupportUpperBound() {
return probabilityOfSuccess < 1 ? Integer.MAX_VALUE : 0;
}
}