WeibullDistribution.java
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* 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,
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* 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.gamma.LogGamma;
import org.apache.commons.rng.UniformRandomProvider;
import org.apache.commons.rng.sampling.distribution.ZigguratSampler;
/**
* Implementation of the Weibull distribution.
*
* <p>The probability density function of \( X \) is:
*
* <p>\[ f(x;k,\lambda) = \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}} \]
*
* <p>for \( k > 0 \) the shape,
* \( \lambda > 0 \) the scale, and
* \( x \in (0, \infty) \).
*
* <p>Note the special cases:
* <ul>
* <li>\( k = 1 \) is the exponential distribution
* <li>\( k = 2 \) is the Rayleigh distribution with scale \( \sigma = \frac {\lambda}{\sqrt{2}} \)
* </ul>
*
* @see <a href="https://en.wikipedia.org/wiki/Weibull_distribution">Weibull distribution (Wikipedia)</a>
* @see <a href="https://mathworld.wolfram.com/WeibullDistribution.html">Weibull distribution (MathWorld)</a>
*/
public final class WeibullDistribution extends AbstractContinuousDistribution {
/** Support lower bound. */
private static final double SUPPORT_LO = 0;
/** Support upper bound. */
private static final double SUPPORT_HI = Double.POSITIVE_INFINITY;
/** The shape parameter. */
private final double shape;
/** The scale parameter. */
private final double scale;
/** shape / scale. */
private final double shapeOverScale;
/** log(shape / scale). */
private final double logShapeOverScale;
/**
* @param shape Shape parameter.
* @param scale Scale parameter.
*/
private WeibullDistribution(double shape,
double scale) {
this.scale = scale;
this.shape = shape;
shapeOverScale = shape / scale;
logShapeOverScale = Math.log(shapeOverScale);
}
/**
* Creates a Weibull distribution.
*
* @param shape Shape parameter.
* @param scale Scale parameter.
* @return the distribution
* @throws IllegalArgumentException if {@code shape <= 0} or {@code scale <= 0}.
*/
public static WeibullDistribution of(double shape,
double scale) {
if (shape <= 0) {
throw new DistributionException(DistributionException.NOT_STRICTLY_POSITIVE,
shape);
}
if (scale <= 0) {
throw new DistributionException(DistributionException.NOT_STRICTLY_POSITIVE,
scale);
}
return new WeibullDistribution(shape, scale);
}
/**
* Gets the shape parameter of this distribution.
*
* @return the shape parameter.
*/
public double getShape() {
return shape;
}
/**
* Gets the scale parameter of this distribution.
*
* @return the scale parameter.
*/
public double getScale() {
return scale;
}
/** {@inheritDoc}
*
* <p>Returns the limit when {@code x = 0}:
* <ul>
* <li>{@code shape < 1}: Infinity
* <li>{@code shape == 1}: 1 / scale
* <li>{@code shape > 1}: 0
* </ul>
*/
@Override
public double density(double x) {
if (x <= SUPPORT_LO || x >= SUPPORT_HI) {
// Special case x=0
if (x == SUPPORT_LO && shape <= 1) {
return shape == 1 ?
// Exponential distribution
shapeOverScale :
Double.POSITIVE_INFINITY;
}
return 0;
}
final double xscale = x / scale;
final double xscalepow = Math.pow(xscale, shape - 1);
/*
* Math.pow(x / scale, shape) =
* Math.pow(xscale, shape) =
* Math.pow(xscale, shape - 1) * xscale
*/
final double xscalepowshape = xscalepow * xscale;
return shapeOverScale * xscalepow * Math.exp(-xscalepowshape);
}
/** {@inheritDoc}
*
* <p>Returns the limit when {@code x = 0}:
* <ul>
* <li>{@code shape < 1}: Infinity
* <li>{@code shape == 1}: log(1 / scale)
* <li>{@code shape > 1}: -Infinity
* </ul>
*/
@Override
public double logDensity(double x) {
if (x <= SUPPORT_LO || x >= SUPPORT_HI) {
// Special case x=0
if (x == SUPPORT_LO && shape <= 1) {
return shape == 1 ?
// Exponential distribution
logShapeOverScale :
Double.POSITIVE_INFINITY;
}
return Double.NEGATIVE_INFINITY;
}
final double xscale = x / scale;
final double logxscalepow = Math.log(xscale) * (shape - 1);
/*
* Math.pow(x / scale, shape) =
* Math.pow(xscale, shape) =
* Math.pow(xscale, shape - 1) * xscale
* Math.exp(log(xscale) * (shape - 1)) * xscale
*/
final double xscalepowshape = Math.exp(logxscalepow) * xscale;
return logShapeOverScale + logxscalepow - xscalepowshape;
}
/** {@inheritDoc} */
@Override
public double cumulativeProbability(double x) {
if (x <= SUPPORT_LO) {
return 0;
}
return -Math.expm1(-Math.pow(x / scale, shape));
}
/** {@inheritDoc} */
@Override
public double survivalProbability(double x) {
if (x <= SUPPORT_LO) {
return 1;
}
return Math.exp(-Math.pow(x / scale, shape));
}
/**
* {@inheritDoc}
*
* <p>Returns {@code 0} when {@code p == 0} and
* {@link Double#POSITIVE_INFINITY} when {@code p == 1}.
*/
@Override
public double inverseCumulativeProbability(double p) {
ArgumentUtils.checkProbability(p);
if (p == 0) {
return 0.0;
} else if (p == 1) {
return Double.POSITIVE_INFINITY;
}
return scale * Math.pow(-Math.log1p(-p), 1.0 / shape);
}
/**
* {@inheritDoc}
*
* <p>Returns {@code 0} when {@code p == 1} and
* {@link Double#POSITIVE_INFINITY} when {@code p == 0}.
*/
@Override
public double inverseSurvivalProbability(double p) {
ArgumentUtils.checkProbability(p);
if (p == 1) {
return 0.0;
} else if (p == 0) {
return Double.POSITIVE_INFINITY;
}
return scale * Math.pow(-Math.log(p), 1.0 / shape);
}
/**
* {@inheritDoc}
*
* <p>For shape parameter \( k \) and scale parameter \( \lambda \), the mean is:
*
* <p>\[ \lambda \, \Gamma(1+\frac{1}{k}) \]
*
* <p>where \( \Gamma \) is the Gamma-function.
*/
@Override
public double getMean() {
final double sh = getShape();
final double sc = getScale();
// Special case of exponential when shape is 1
return sh == 1 ? sc : sc * Math.exp(LogGamma.value(1 + (1 / sh)));
}
/**
* {@inheritDoc}
*
* <p>For shape parameter \( k \) and scale parameter \( \lambda \), the variance is:
*
* <p>\[ \lambda^2 \left[ \Gamma\left(1+\frac{2}{k}\right) -
* \left(\Gamma\left(1+\frac{1}{k}\right)\right)^2 \right] \]
*
* <p>where \( \Gamma \) is the Gamma-function.
*/
@Override
public double getVariance() {
final double sh = getShape();
final double sc = getScale();
final double mn = getMean();
// Special case of exponential when shape is 1
return sh == 1 ?
sc * sc :
(sc * sc) * Math.exp(LogGamma.value(1 + (2 / sh))) -
(mn * mn);
}
/**
* {@inheritDoc}
*
* <p>The lower bound of the support is always 0.
*
* @return 0.
*/
@Override
public double getSupportLowerBound() {
return SUPPORT_LO;
}
/**
* {@inheritDoc}
*
* <p>The upper bound of the support is always positive infinity.
*
* @return {@link Double#POSITIVE_INFINITY positive infinity}.
*/
@Override
public double getSupportUpperBound() {
return SUPPORT_HI;
}
/** {@inheritDoc} */
@Override
public ContinuousDistribution.Sampler createSampler(final UniformRandomProvider rng) {
// Special case: shape=1 is the exponential distribution
if (shape == 1) {
// Exponential distribution sampler.
return ZigguratSampler.Exponential.of(rng, scale)::sample;
}
return super.createSampler(rng);
}
}