/*
* 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.random.RandomGenerator;
import org.apache.commons.math3.random.Well19937c;
import org.apache.commons.math3.special.Erf;
import org.apache.commons.math3.util.FastMath;
/**
* Implementation of the log-normal (gaussian) distribution.
*
* <p>
* <strong>Parameters:</strong>
* {@code X} is log-normally distributed if its natural logarithm {@code log(X)}
* is normally distributed. The probability distribution function of {@code X}
* is given by (for {@code x > 0})
* </p>
* <p>
* {@code exp(-0.5 * ((ln(x) - m) / s)^2) / (s * sqrt(2 * pi) * x)}
* </p>
* <ul>
* <li>{@code m} is the <em>scale</em> parameter: this is the mean of the
* normally distributed natural logarithm of this distribution,</li>
* <li>{@code s} is the <em>shape</em> parameter: this is the standard
* deviation of the normally distributed natural logarithm of this
* distribution.
* </ul>
*
* @see <a href="http://en.wikipedia.org/wiki/Log-normal_distribution">
* Log-normal distribution (Wikipedia)</a>
* @see <a href="http://mathworld.wolfram.com/LogNormalDistribution.html">
* Log Normal distribution (MathWorld)</a>
*
* @since 3.0
*/
public class LogNormalDistribution 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 = 20120112;
/** √(2 π) */
private static final double SQRT2PI = FastMath.sqrt(2 * FastMath.PI);
/** √(2) */
private static final double SQRT2 = FastMath.sqrt(2.0);
/** The scale parameter of this distribution. */
private final double scale;
/** The shape parameter of this distribution. */
private final double shape;
/** The value of {@code log(shape) + 0.5 * log(2*PI)} stored for faster computation. */
private final double logShapePlusHalfLog2Pi;
/** Inverse cumulative probability accuracy. */
private final double solverAbsoluteAccuracy;
/**
* Create a log-normal distribution, where the mean and standard deviation
* of the {@link NormalDistribution normally distributed} natural
* logarithm of the log-normal distribution are equal to zero and one
* respectively. In other words, the scale of the returned distribution is
* {@code 0}, while its shape is {@code 1}.
* <p>
* <b>Note:</b> this constructor will implicitly create an instance of
* {@link Well19937c} as random generator to be used for sampling only (see
* {@link #sample()} and {@link #sample(int)}). In case no sampling is
* needed for the created distribution, it is advised to pass {@code null}
* as random generator via the appropriate constructors to avoid the
* additional initialisation overhead.
*/
public LogNormalDistribution() {
this(0, 1);
}
/**
* Create a log-normal distribution using the specified scale and shape.
* <p>
* <b>Note:</b> this constructor will implicitly create an instance of
* {@link Well19937c} as random generator to be used for sampling only (see
* {@link #sample()} and {@link #sample(int)}). In case no sampling is
* needed for the created distribution, it is advised to pass {@code null}
* as random generator via the appropriate constructors to avoid the
* additional initialisation overhead.
*
* @param scale the scale parameter of this distribution
* @param shape the shape parameter of this distribution
* @throws NotStrictlyPositiveException if {@code shape <= 0}.
*/
public LogNormalDistribution(double scale, double shape)
throws NotStrictlyPositiveException {
this(scale, shape, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
}
/**
* Create a log-normal distribution using the specified scale, shape and
* inverse cumulative distribution accuracy.
* <p>
* <b>Note:</b> this constructor will implicitly create an instance of
* {@link Well19937c} as random generator to be used for sampling only (see
* {@link #sample()} and {@link #sample(int)}). In case no sampling is
* needed for the created distribution, it is advised to pass {@code null}
* as random generator via the appropriate constructors to avoid the
* additional initialisation overhead.
*
* @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 shape <= 0}.
*/
public LogNormalDistribution(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 shape <= 0}.
* @since 3.3
*/
public LogNormalDistribution(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 shape <= 0}.
* @since 3.1
*/
public LogNormalDistribution(RandomGenerator rng,
double scale,
double shape,
double inverseCumAccuracy)
throws NotStrictlyPositiveException {
super(rng);
if (shape <= 0) {
throw new NotStrictlyPositiveException(LocalizedFormats.SHAPE, shape);
}
this.scale = scale;
this.shape = shape;
this.logShapePlusHalfLog2Pi = FastMath.log(shape) + 0.5 * FastMath.log(2 * FastMath.PI);
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}
*
* For scale {@code m}, and shape {@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) - m) / s)^2) / (s * sqrt(2 * pi) * x)}
* otherwise.</li>
* </ul>
*/
public double density(double x) {
if (x <= 0) {
return 0;
}
final double x0 = FastMath.log(x) - scale;
final double x1 = x0 / shape;
return FastMath.exp(-0.5 * x1 * x1) / (shape * SQRT2PI * x);
}
/** {@inheritDoc}
*
* 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 = FastMath.log(x);
final double x0 = logX - scale;
final double x1 = x0 / shape;
return -0.5 * x1 * x1 - (logShapePlusHalfLog2Pi + logX);
}
/**
* {@inheritDoc}
*
* For scale {@code m}, and shape {@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) - m < 0} and {@code m - ln(x) > 40 * s}, as
* in these cases the actual value is within {@code Double.MIN_VALUE} of 0,
* <li>{@code 1} if {@code ln(x) - m >= 0} and {@code ln(x) - m > 40 * s},
* as in these cases the actual value is within {@code Double.MIN_VALUE} of
* 1,</li>
* <li>{@code 0.5 + 0.5 * erf((ln(x) - m) / (s * sqrt(2))} otherwise.</li>
* </ul>
*/
public double cumulativeProbability(double x) {
if (x <= 0) {
return 0;
}
final double dev = FastMath.log(x) - scale;
if (FastMath.abs(dev) > 40 * shape) {
return dev < 0 ? 0.0d : 1.0d;
}
return 0.5 + 0.5 * Erf.erf(dev / (shape * SQRT2));
}
/**
* {@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
public double probability(double x0,
double x1)
throws NumberIsTooLargeException {
if (x0 > x1) {
throw new NumberIsTooLargeException(LocalizedFormats.LOWER_ENDPOINT_ABOVE_UPPER_ENDPOINT,
x0, x1, true);
}
if (x0 <= 0 || x1 <= 0) {
return super.probability(x0, x1);
}
final double denom = shape * SQRT2;
final double v0 = (FastMath.log(x0) - scale) / denom;
final double v1 = (FastMath.log(x1) - scale) / denom;
return 0.5 * Erf.erf(v0, v1);
}
/** {@inheritDoc} */
@Override
protected double getSolverAbsoluteAccuracy() {
return solverAbsoluteAccuracy;
}
/**
* {@inheritDoc}
*
* For scale {@code m} and shape {@code s}, the mean is
* {@code exp(m + s^2 / 2)}.
*/
public double getNumericalMean() {
double s = shape;
return FastMath.exp(scale + (s * s / 2));
}
/**
* {@inheritDoc}
*
* For scale {@code m} and shape {@code s}, the variance is
* {@code (exp(s^2) - 1) * exp(2 * m + s^2)}.
*/
public double getNumericalVariance() {
final double s = shape;
final double ss = s * s;
return (FastMath.expm1(ss)) * FastMath.exp(2 * scale + ss);
}
/**
* {@inheritDoc}
*
* The lower bound of the support is always 0 no matter the parameters.
*
* @return lower bound of the support (always 0)
*/
public double getSupportLowerBound() {
return 0;
}
/**
* {@inheritDoc}
*
* 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}
*
* The support of this distribution is connected.
*
* @return {@code true}
*/
public boolean isSupportConnected() {
return true;
}
/** {@inheritDoc} */
@Override
public double sample() {
final double n = random.nextGaussian();
return FastMath.exp(scale + shape * n);
}
}