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* 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.exception.NotStrictlyPositiveException;
import org.apache.commons.math3.exception.OutOfRangeException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.special.Beta;
import org.apache.commons.math3.util.CombinatoricsUtils;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.random.RandomGenerator;
import org.apache.commons.math3.random.Well19937c;
/**
* <p>
* 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.
* </p>
* <p>
* 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 {@code r} successes occur.
* This is the convention adopted in e.g.
* <a href="http://mathworld.wolfram.com/NegativeBinomialDistribution.html">MathWorld</a>,
* but <em>not</em> in
* <a href="http://en.wikipedia.org/wiki/Negative_binomial_distribution">Wikipedia</a>.
* </p>
* <p>
* For a random variable {@code X} whose values are distributed according to this
* distribution, the probability mass function is given by<br/>
* {@code P(X = k) = C(k + r - 1, r - 1) * p^r * (1 - p)^k,}<br/>
* where {@code r} is the number of successes, {@code p} is the probability of
* success, and {@code X} is the total number of failures. {@code C(n, k)} is
* the binomial coefficient ({@code n} choose {@code k}). The mean and variance
* of {@code X} are<br/>
* {@code E(X) = (1 - p) * r / p, var(X) = (1 - p) * r / p^2.}<br/>
* Finally, the cumulative distribution function is given by<br/>
* {@code P(X <= k) = I(p, r, k + 1)},
* where I is the regularized incomplete Beta function.
* </p>
*
* @see <a href="http://en.wikipedia.org/wiki/Negative_binomial_distribution">
* Negative binomial distribution (Wikipedia)</a>
* @see <a href="http://mathworld.wolfram.com/NegativeBinomialDistribution.html">
* Negative binomial distribution (MathWorld)</a>
* @since 1.2 (changed to concrete class in 3.0)
*/
public class PascalDistribution extends AbstractIntegerDistribution {
/** Serializable version identifier. */
private static final long serialVersionUID = 6751309484392813623L;
/** The number of successes. */
private final int numberOfSuccesses;
/** The probability of success. */
private final double probabilityOfSuccess;
/** The value of {@code log(p)}, where {@code p} is the probability of success,
* stored for faster computation. */
private final double logProbabilityOfSuccess;
/** The value of {@code log(1-p)}, where {@code p} is the probability of success,
* stored for faster computation. */
private final double log1mProbabilityOfSuccess;
/**
* Create a Pascal distribution with the given number of successes and
* probability of success.
*
* @param r Number of successes.
* @param p Probability of success.
* @throws NotStrictlyPositiveException if the number of successes is not positive
* @throws OutOfRangeException if the probability of success is not in the
* range {@code [0, 1]}.
*/
public PascalDistribution(int r, double p)
throws NotStrictlyPositiveException, OutOfRangeException {
this(new Well19937c(), r, p);
}
/**
* Create a Pascal distribution with the given number of successes and
* probability of success.
*
* @param rng Random number generator.
* @param r Number of successes.
* @param p Probability of success.
* @throws NotStrictlyPositiveException if the number of successes is not positive
* @throws OutOfRangeException if the probability of success is not in the
* range {@code [0, 1]}.
* @since 3.1
*/
public PascalDistribution(RandomGenerator rng,
int r,
double p)
throws NotStrictlyPositiveException, OutOfRangeException {
super(rng);
if (r <= 0) {
throw new NotStrictlyPositiveException(LocalizedFormats.NUMBER_OF_SUCCESSES,
r);
}
if (p < 0 || p > 1) {
throw new OutOfRangeException(p, 0, 1);
}
numberOfSuccesses = r;
probabilityOfSuccess = p;
logProbabilityOfSuccess = FastMath.log(p);
log1mProbabilityOfSuccess = FastMath.log1p(-p);
}
/**
* Access the number of successes for this distribution.
*
* @return the number of successes.
*/
public int getNumberOfSuccesses() {
return numberOfSuccesses;
}
/**
* Access the probability of success for this distribution.
*
* @return the probability of success.
*/
public double getProbabilityOfSuccess() {
return probabilityOfSuccess;
}
/** {@inheritDoc} */
public double probability(int x) {
double ret;
if (x < 0) {
ret = 0.0;
} else {
ret = CombinatoricsUtils.binomialCoefficientDouble(x +
numberOfSuccesses - 1, numberOfSuccesses - 1) *
FastMath.pow(probabilityOfSuccess, numberOfSuccesses) *
FastMath.pow(1.0 - probabilityOfSuccess, x);
}
return ret;
}
/** {@inheritDoc} */
@Override
public double logProbability(int x) {
double ret;
if (x < 0) {
ret = Double.NEGATIVE_INFINITY;
} else {
ret = CombinatoricsUtils.binomialCoefficientLog(x +
numberOfSuccesses - 1, numberOfSuccesses - 1) +
logProbabilityOfSuccess * numberOfSuccesses +
log1mProbabilityOfSuccess * x;
}
return ret;
}
/** {@inheritDoc} */
public double cumulativeProbability(int x) {
double ret;
if (x < 0) {
ret = 0.0;
} else {
ret = Beta.regularizedBeta(probabilityOfSuccess,
numberOfSuccesses, x + 1.0);
}
return ret;
}
/**
* {@inheritDoc}
*
* For number of successes {@code r} and probability of success {@code p},
* the mean is {@code r * (1 - p) / p}.
*/
public double getNumericalMean() {
final double p = getProbabilityOfSuccess();
final double r = getNumberOfSuccesses();
return (r * (1 - p)) / p;
}
/**
* {@inheritDoc}
*
* For number of successes {@code r} and probability of success {@code p},
* the variance is {@code r * (1 - p) / p^2}.
*/
public double getNumericalVariance() {
final double p = getProbabilityOfSuccess();
final double r = getNumberOfSuccesses();
return r * (1 - p) / (p * p);
}
/**
* {@inheritDoc}
*
* The lower bound of the support is always 0 no matter the parameters.
*
* @return lower bound of the support (always 0)
*/
public int getSupportLowerBound() {
return 0;
}
/**
* {@inheritDoc}
*
* The upper bound of the support is always positive infinity no matter the
* parameters. Positive infinity is symbolized by {@code Integer.MAX_VALUE}.
*
* @return upper bound of the support (always {@code Integer.MAX_VALUE}
* for positive infinity)
*/
public int getSupportUpperBound() {
return Integer.MAX_VALUE;
}
/**
* {@inheritDoc}
*
* The support of this distribution is connected.
*
* @return {@code true}
*/
public boolean isSupportConnected() {
return true;
}
}