LogNormalDistribution.java
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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.statistics.distribution;
import org.apache.commons.numbers.gamma.ErfDifference;
import org.apache.commons.numbers.gamma.Erfc;
import org.apache.commons.numbers.gamma.InverseErfc;
import org.apache.commons.rng.UniformRandomProvider;
import org.apache.commons.rng.sampling.distribution.LogNormalSampler;
import org.apache.commons.rng.sampling.distribution.ZigguratSampler;
/**
* Implementation of the log-normal distribution.
*
* <p>\( X \) is log-normally distributed if its natural logarithm \( \ln(x) \)
* is normally distributed. The probability density function of \( X \) is:
*
* <p>\[ f(x; \mu, \sigma) = \frac 1 {x\sigma\sqrt{2\pi\,}} e^{-{\frac 1 2}\left( \frac{\ln x-\mu}{\sigma} \right)^2 } \]
*
* <p>for \( \mu \) the mean of the normally distributed natural logarithm of this distribution,
* \( \sigma > 0 \) the standard deviation of the normally distributed natural logarithm of this
* distribution, and
* \( x \in (0, \infty) \).
*
* @see <a href="https://en.wikipedia.org/wiki/Log-normal_distribution">Log-normal distribution (Wikipedia)</a>
* @see <a href="https://mathworld.wolfram.com/LogNormalDistribution.html">Log-normal distribution (MathWorld)</a>
*/
public final class LogNormalDistribution extends AbstractContinuousDistribution {
/** 0.5 * ln(2 * pi). Computed to 25-digits precision. */
private static final double HALF_LOG_TWO_PI = 0.9189385332046727417803297;
/** √(2 π). */
private static final double SQRT2PI = Math.sqrt(2 * Math.PI);
/** The mu parameter of this distribution. */
private final double mu;
/** The sigma parameter of this distribution. */
private final double sigma;
/** The value of {@code log(sigma) + 0.5 * log(2*PI)} stored for faster computation. */
private final double logSigmaPlusHalfLog2Pi;
/** Sigma multiplied by sqrt(2). */
private final double sigmaSqrt2;
/** Sigma multiplied by sqrt(2 * pi). */
private final double sigmaSqrt2Pi;
/**
* @param mu Mean of the natural logarithm of the distribution values.
* @param sigma Standard deviation of the natural logarithm of the distribution values.
*/
private LogNormalDistribution(double mu,
double sigma) {
this.mu = mu;
this.sigma = sigma;
logSigmaPlusHalfLog2Pi = Math.log(sigma) + HALF_LOG_TWO_PI;
sigmaSqrt2 = ExtendedPrecision.sqrt2xx(sigma);
sigmaSqrt2Pi = sigma * SQRT2PI;
}
/**
* Creates a log-normal distribution.
*
* @param mu Mean of the natural logarithm of the distribution values.
* @param sigma Standard deviation of the natural logarithm of the distribution values.
* @return the distribution
* @throws IllegalArgumentException if {@code sigma <= 0}.
*/
public static LogNormalDistribution of(double mu,
double sigma) {
if (sigma <= 0) {
throw new DistributionException(DistributionException.NOT_STRICTLY_POSITIVE, sigma);
}
return new LogNormalDistribution(mu, sigma);
}
/**
* Gets the {@code mu} parameter of this distribution.
* This is the mean of the natural logarithm of the distribution values,
* not the mean of distribution.
*
* @return the mu parameter.
*/
public double getMu() {
return mu;
}
/**
* Gets the {@code sigma} parameter of this distribution.
* This is the standard deviation of the natural logarithm of the distribution values,
* not the standard deviation of distribution.
*
* @return the sigma parameter.
*/
public double getSigma() {
return sigma;
}
/**
* {@inheritDoc}
*
* <p>For {@code mu}, and sigma {@code s} of this distribution, the PDF
* is given by
* <ul>
* <li>{@code 0} if {@code x <= 0},</li>
* <li>{@code exp(-0.5 * ((ln(x) - mu) / s)^2) / (s * sqrt(2 * pi) * x)}
* otherwise.</li>
* </ul>
*/
@Override
public double density(double x) {
if (x <= 0) {
return 0;
}
final double x0 = Math.log(x) - mu;
final double x1 = x0 / sigma;
return Math.exp(-0.5 * x1 * x1) / (sigmaSqrt2Pi * x);
}
/** {@inheritDoc} */
@Override
public double probability(double x0,
double x1) {
if (x0 > x1) {
throw new DistributionException(DistributionException.INVALID_RANGE_LOW_GT_HIGH,
x0, x1);
}
if (x0 <= 0) {
return super.probability(x0, x1);
}
// Assumes x1 >= x0 && x0 > 0
final double v0 = (Math.log(x0) - mu) / sigmaSqrt2;
final double v1 = (Math.log(x1) - mu) / sigmaSqrt2;
return 0.5 * ErfDifference.value(v0, v1);
}
/** {@inheritDoc}
*
* <p>See documentation of {@link #density(double)} for computation details.
*/
@Override
public double logDensity(double x) {
if (x <= 0) {
return Double.NEGATIVE_INFINITY;
}
final double logX = Math.log(x);
final double x0 = logX - mu;
final double x1 = x0 / sigma;
return -0.5 * x1 * x1 - (logSigmaPlusHalfLog2Pi + logX);
}
/**
* {@inheritDoc}
*
* <p>For {@code mu}, and sigma {@code s} of this distribution, the CDF
* is given by
* <ul>
* <li>{@code 0} if {@code x <= 0},</li>
* <li>{@code 0} if {@code ln(x) - mu < 0} and {@code mu - ln(x) > 40 * s}, as
* in these cases the actual value is within {@link Double#MIN_VALUE} of 0,
* <li>{@code 1} if {@code ln(x) - mu >= 0} and {@code ln(x) - mu > 40 * s},
* as in these cases the actual value is within {@link Double#MIN_VALUE} of
* 1,</li>
* <li>{@code 0.5 + 0.5 * erf((ln(x) - mu) / (s * sqrt(2))} otherwise.</li>
* </ul>
*/
@Override
public double cumulativeProbability(double x) {
if (x <= 0) {
return 0;
}
final double dev = Math.log(x) - mu;
return 0.5 * Erfc.value(-dev / sigmaSqrt2);
}
/** {@inheritDoc} */
@Override
public double survivalProbability(double x) {
if (x <= 0) {
return 1;
}
final double dev = Math.log(x) - mu;
return 0.5 * Erfc.value(dev / sigmaSqrt2);
}
/** {@inheritDoc} */
@Override
public double inverseCumulativeProbability(double p) {
ArgumentUtils.checkProbability(p);
return Math.exp(mu - sigmaSqrt2 * InverseErfc.value(2 * p));
}
/** {@inheritDoc} */
@Override
public double inverseSurvivalProbability(double p) {
ArgumentUtils.checkProbability(p);
return Math.exp(mu + sigmaSqrt2 * InverseErfc.value(2 * p));
}
/**
* {@inheritDoc}
*
* <p>For \( \mu \) the mean of the normally distributed natural logarithm of
* this distribution, \( \sigma > 0 \) the standard deviation of the normally
* distributed natural logarithm of this distribution, the mean is:
*
* <p>\[ \exp(\mu + \frac{\sigma^2}{2}) \]
*/
@Override
public double getMean() {
final double s = sigma;
return Math.exp(mu + (s * s / 2));
}
/**
* {@inheritDoc}
*
* <p>For \( \mu \) the mean of the normally distributed natural logarithm of
* this distribution, \( \sigma > 0 \) the standard deviation of the normally
* distributed natural logarithm of this distribution, the variance is:
*
* <p>\[ [\exp(\sigma^2) - 1)] \exp(2 \mu + \sigma^2) \]
*/
@Override
public double getVariance() {
final double s = sigma;
final double ss = s * s;
return Math.expm1(ss) * Math.exp(2 * mu + ss);
}
/**
* {@inheritDoc}
*
* <p>The lower bound of the support is always 0.
*
* @return 0.
*/
@Override
public double getSupportLowerBound() {
return 0;
}
/**
* {@inheritDoc}
*
* <p>The upper bound of the support is always positive infinity.
*
* @return {@link Double#POSITIVE_INFINITY positive infinity}.
*/
@Override
public double getSupportUpperBound() {
return Double.POSITIVE_INFINITY;
}
/** {@inheritDoc} */
@Override
public ContinuousDistribution.Sampler createSampler(final UniformRandomProvider rng) {
// Log normal distribution sampler.
final ZigguratSampler.NormalizedGaussian gaussian = ZigguratSampler.NormalizedGaussian.of(rng);
return LogNormalSampler.of(gaussian, mu, sigma)::sample;
}
}