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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.NumberIsTooLargeException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.random.RandomGenerator;
import org.apache.commons.math3.random.Well19937c;
/**
* Implementation of the Pareto distribution.
*
* <p>
* <strong>Parameters:</strong>
* The probability distribution function of {@code X} is given by (for {@code x >= k}):
* <pre>
* α * k^α / x^(α + 1)
* </pre>
* <p>
* <ul>
* <li>{@code k} is the <em>scale</em> parameter: this is the minimum possible value of {@code X},</li>
* <li>{@code α} is the <em>shape</em> parameter: this is the Pareto index</li>
* </ul>
*
* @see <a href="http://en.wikipedia.org/wiki/Pareto_distribution">
* Pareto distribution (Wikipedia)</a>
* @see <a href="http://mathworld.wolfram.com/ParetoDistribution.html">
* Pareto distribution (MathWorld)</a>
*
* @since 3.3
*/
public class ParetoDistribution extends AbstractRealDistribution {
/** Default inverse cumulative probability accuracy. */
public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY = 1e-9;
/** Serializable version identifier. */
private static final long serialVersionUID = 20130424;
/** The scale parameter of this distribution. */
private final double scale;
/** The shape parameter of this distribution. */
private final double shape;
/** Inverse cumulative probability accuracy. */
private final double solverAbsoluteAccuracy;
/**
* Create a Pareto distribution with a scale of {@code 1} and a shape of {@code 1}.
*/
public ParetoDistribution() {
this(1, 1);
}
/**
* Create a Pareto distribution using the specified scale and shape.
*
* @param scale the scale parameter of this distribution
* @param shape the shape parameter of this distribution
* @throws NotStrictlyPositiveException if {@code scale <= 0} or {@code shape <= 0}.
*/
public ParetoDistribution(double scale, double shape)
throws NotStrictlyPositiveException {
this(scale, shape, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
}
/**
* Create a Pareto distribution using the specified scale, shape and
* inverse cumulative distribution accuracy.
*
* @param scale the scale parameter of this distribution
* @param shape the shape parameter of this distribution
* @param inverseCumAccuracy Inverse cumulative probability accuracy.
* @throws NotStrictlyPositiveException if {@code scale <= 0} or {@code shape <= 0}.
*/
public ParetoDistribution(double scale, double shape, double inverseCumAccuracy)
throws NotStrictlyPositiveException {
this(new Well19937c(), scale, shape, inverseCumAccuracy);
}
/**
* Creates a log-normal distribution.
*
* @param rng Random number generator.
* @param scale Scale parameter of this distribution.
* @param shape Shape parameter of this distribution.
* @throws NotStrictlyPositiveException if {@code scale <= 0} or {@code shape <= 0}.
*/
public ParetoDistribution(RandomGenerator rng, double scale, double shape)
throws NotStrictlyPositiveException {
this(rng, scale, shape, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
}
/**
* Creates a log-normal distribution.
*
* @param rng Random number generator.
* @param scale Scale parameter of this distribution.
* @param shape Shape parameter of this distribution.
* @param inverseCumAccuracy Inverse cumulative probability accuracy.
* @throws NotStrictlyPositiveException if {@code scale <= 0} or {@code shape <= 0}.
*/
public ParetoDistribution(RandomGenerator rng,
double scale,
double shape,
double inverseCumAccuracy)
throws NotStrictlyPositiveException {
super(rng);
if (scale <= 0) {
throw new NotStrictlyPositiveException(LocalizedFormats.SCALE, scale);
}
if (shape <= 0) {
throw new NotStrictlyPositiveException(LocalizedFormats.SHAPE, shape);
}
this.scale = scale;
this.shape = shape;
this.solverAbsoluteAccuracy = inverseCumAccuracy;
}
/**
* Returns the scale parameter of this distribution.
*
* @return the scale parameter
*/
public double getScale() {
return scale;
}
/**
* Returns the shape parameter of this distribution.
*
* @return the shape parameter
*/
public double getShape() {
return shape;
}
/**
* {@inheritDoc}
* <p>
* For scale {@code k}, and shape {@code α} of this distribution, the PDF
* is given by
* <ul>
* <li>{@code 0} if {@code x < k},</li>
* <li>{@code α * k^α / x^(α + 1)} otherwise.</li>
* </ul>
*/
public double density(double x) {
if (x < scale) {
return 0;
}
return FastMath.pow(scale, shape) / FastMath.pow(x, shape + 1) * shape;
}
/** {@inheritDoc}
*
* See documentation of {@link #density(double)} for computation details.
*/
@Override
public double logDensity(double x) {
if (x < scale) {
return Double.NEGATIVE_INFINITY;
}
return FastMath.log(scale) * shape - FastMath.log(x) * (shape + 1) + FastMath.log(shape);
}
/**
* {@inheritDoc}
* <p>
* For scale {@code k}, and shape {@code α} of this distribution, the CDF is given by
* <ul>
* <li>{@code 0} if {@code x < k},</li>
* <li>{@code 1 - (k / x)^α} otherwise.</li>
* </ul>
*/
public double cumulativeProbability(double x) {
if (x <= scale) {
return 0;
}
return 1 - FastMath.pow(scale / x, shape);
}
/**
* {@inheritDoc}
*
* @deprecated See {@link RealDistribution#cumulativeProbability(double,double)}
*/
@Override
@Deprecated
public double cumulativeProbability(double x0, double x1)
throws NumberIsTooLargeException {
return probability(x0, x1);
}
/** {@inheritDoc} */
@Override
protected double getSolverAbsoluteAccuracy() {
return solverAbsoluteAccuracy;
}
/**
* {@inheritDoc}
* <p>
* For scale {@code k} and shape {@code α}, the mean is given by
* <ul>
* <li>{@code ∞} if {@code α <= 1},</li>
* <li>{@code α * k / (α - 1)} otherwise.</li>
* </ul>
*/
public double getNumericalMean() {
if (shape <= 1) {
return Double.POSITIVE_INFINITY;
}
return shape * scale / (shape - 1);
}
/**
* {@inheritDoc}
* <p>
* For scale {@code k} and shape {@code α}, the variance is given by
* <ul>
* <li>{@code ∞} if {@code 1 < α <= 2},</li>
* <li>{@code k^2 * α / ((α - 1)^2 * (α - 2))} otherwise.</li>
* </ul>
*/
public double getNumericalVariance() {
if (shape <= 2) {
return Double.POSITIVE_INFINITY;
}
double s = shape - 1;
return scale * scale * shape / (s * s) / (shape - 2);
}
/**
* {@inheritDoc}
* <p>
* The lower bound of the support is equal to the scale parameter {@code k}.
*
* @return lower bound of the support
*/
public double getSupportLowerBound() {
return scale;
}
/**
* {@inheritDoc}
* <p>
* The upper bound of the support is always positive infinity no matter the parameters.
*
* @return upper bound of the support (always {@code Double.POSITIVE_INFINITY})
*/
public double getSupportUpperBound() {
return Double.POSITIVE_INFINITY;
}
/** {@inheritDoc} */
public boolean isSupportLowerBoundInclusive() {
return true;
}
/** {@inheritDoc} */
public boolean isSupportUpperBoundInclusive() {
return false;
}
/**
* {@inheritDoc}
* <p>
* The support of this distribution is connected.
*
* @return {@code true}
*/
public boolean isSupportConnected() {
return true;
}
/** {@inheritDoc} */
@Override
public double sample() {
final double n = random.nextDouble();
return scale / FastMath.pow(n, 1 / shape);
}
}