LogNormalDistribution.java

  1. /*
  2.  * Licensed to the Apache Software Foundation (ASF) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * The ASF licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *      http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */

  18. package org.apache.commons.math3.distribution;

  20. import org.apache.commons.math3.exception.NotStrictlyPositiveException;
  21. import org.apache.commons.math3.exception.NumberIsTooLargeException;
  22. import org.apache.commons.math3.exception.util.LocalizedFormats;
  23. import org.apache.commons.math3.random.RandomGenerator;
  24. import org.apache.commons.math3.random.Well19937c;
  25. import org.apache.commons.math3.special.Erf;
  26. import org.apache.commons.math3.util.FastMath;

  28. /**
  29.  * Implementation of the log-normal (gaussian) distribution.
  30.  *
  31.  * <p>
  32.  * <strong>Parameters:</strong>
  33.  * {@code X} is log-normally distributed if its natural logarithm {@code log(X)}
  34.  * is normally distributed. The probability distribution function of {@code X}
  35.  * is given by (for {@code x > 0})
  36.  * </p>
  37.  * <p>
  38.  * {@code exp(-0.5 * ((ln(x) - m) / s)^2) / (s * sqrt(2 * pi) * x)}
  39.  * </p>
  40.  * <ul>
  41.  * <li>{@code m} is the <em>scale</em> parameter: this is the mean of the
  42.  * normally distributed natural logarithm of this distribution,</li>
  43.  * <li>{@code s} is the <em>shape</em> parameter: this is the standard
  44.  * deviation of the normally distributed natural logarithm of this
  45.  * distribution.
  46.  * </ul>
  47.  *
  48.  * @see <a href="http://en.wikipedia.org/wiki/Log-normal_distribution">
  49.  * Log-normal distribution (Wikipedia)</a>
  50.  * @see <a href="http://mathworld.wolfram.com/LogNormalDistribution.html">
  51.  * Log Normal distribution (MathWorld)</a>
  52.  *
  53.  * @since 3.0
  54.  */
  55. public class LogNormalDistribution extends AbstractRealDistribution {
  56.     /** Default inverse cumulative probability accuracy. */
  57.     public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY = 1e-9;

  59.     /** Serializable version identifier. */
  60.     private static final long serialVersionUID = 20120112;

  62.     /** &radic;(2 &pi;) */
  63.     private static final double SQRT2PI = FastMath.sqrt(2 * FastMath.PI);

  65.     /** &radic;(2) */
  66.     private static final double SQRT2 = FastMath.sqrt(2.0);

  68.     /** The scale parameter of this distribution. */
  69.     private final double scale;

  71.     /** The shape parameter of this distribution. */
  72.     private final double shape;
  73.     /** The value of {@code log(shape) + 0.5 * log(2*PI)} stored for faster computation. */
  74.     private final double logShapePlusHalfLog2Pi;

  76.     /** Inverse cumulative probability accuracy. */
  77.     private final double solverAbsoluteAccuracy;

  79.     /**
  80.      * Create a log-normal distribution, where the mean and standard deviation
  81.      * of the {@link NormalDistribution normally distributed} natural
  82.      * logarithm of the log-normal distribution are equal to zero and one
  83.      * respectively. In other words, the scale of the returned distribution is
  84.      * {@code 0}, while its shape is {@code 1}.
  85.      * <p>
  86.      * <b>Note:</b> this constructor will implicitly create an instance of
  87.      * {@link Well19937c} as random generator to be used for sampling only (see
  88.      * {@link #sample()} and {@link #sample(int)}). In case no sampling is
  89.      * needed for the created distribution, it is advised to pass {@code null}
  90.      * as random generator via the appropriate constructors to avoid the
  91.      * additional initialisation overhead.
  92.      */
  93.     public LogNormalDistribution() {
  94.         this(0, 1);
  95.     }

  97.     /**
  98.      * Create a log-normal distribution using the specified scale and shape.
  99.      * <p>
  100.      * <b>Note:</b> this constructor will implicitly create an instance of
  101.      * {@link Well19937c} as random generator to be used for sampling only (see
  102.      * {@link #sample()} and {@link #sample(int)}). In case no sampling is
  103.      * needed for the created distribution, it is advised to pass {@code null}
  104.      * as random generator via the appropriate constructors to avoid the
  105.      * additional initialisation overhead.
  106.      *
  107.      * @param scale the scale parameter of this distribution
  108.      * @param shape the shape parameter of this distribution
  109.      * @throws NotStrictlyPositiveException if {@code shape <= 0}.
  110.      */
  111.     public LogNormalDistribution(double scale, double shape)
  112.         throws NotStrictlyPositiveException {
  113.         this(scale, shape, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
  114.     }

  116.     /**
  117.      * Create a log-normal distribution using the specified scale, shape and
  118.      * inverse cumulative distribution accuracy.
  119.      * <p>
  120.      * <b>Note:</b> this constructor will implicitly create an instance of
  121.      * {@link Well19937c} as random generator to be used for sampling only (see
  122.      * {@link #sample()} and {@link #sample(int)}). In case no sampling is
  123.      * needed for the created distribution, it is advised to pass {@code null}
  124.      * as random generator via the appropriate constructors to avoid the
  125.      * additional initialisation overhead.
  126.      *
  127.      * @param scale the scale parameter of this distribution
  128.      * @param shape the shape parameter of this distribution
  129.      * @param inverseCumAccuracy Inverse cumulative probability accuracy.
  130.      * @throws NotStrictlyPositiveException if {@code shape <= 0}.
  131.      */
  132.     public LogNormalDistribution(double scale, double shape, double inverseCumAccuracy)
  133.         throws NotStrictlyPositiveException {
  134.         this(new Well19937c(), scale, shape, inverseCumAccuracy);
  135.     }

  137.     /**
  138.      * Creates a log-normal distribution.
  139.      *
  140.      * @param rng Random number generator.
  141.      * @param scale Scale parameter of this distribution.
  142.      * @param shape Shape parameter of this distribution.
  143.      * @throws NotStrictlyPositiveException if {@code shape <= 0}.
  144.      * @since 3.3
  145.      */
  146.     public LogNormalDistribution(RandomGenerator rng, double scale, double shape)
  147.         throws NotStrictlyPositiveException {
  148.         this(rng, scale, shape, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
  149.     }

  151.     /**
  152.      * Creates a log-normal distribution.
  153.      *
  154.      * @param rng Random number generator.
  155.      * @param scale Scale parameter of this distribution.
  156.      * @param shape Shape parameter of this distribution.
  157.      * @param inverseCumAccuracy Inverse cumulative probability accuracy.
  158.      * @throws NotStrictlyPositiveException if {@code shape <= 0}.
  159.      * @since 3.1
  160.      */
  161.     public LogNormalDistribution(RandomGenerator rng,
  162.                                  double scale,
  163.                                  double shape,
  164.                                  double inverseCumAccuracy)
  165.         throws NotStrictlyPositiveException {
  166.         super(rng);

  168.         if (shape <= 0) {
  169.             throw new NotStrictlyPositiveException(LocalizedFormats.SHAPE, shape);
  170.         }

  172.         this.scale = scale;
  173.         this.shape = shape;
  174.         this.logShapePlusHalfLog2Pi = FastMath.log(shape) + 0.5 * FastMath.log(2 * FastMath.PI);
  175.         this.solverAbsoluteAccuracy = inverseCumAccuracy;
  176.     }

  178.     /**
  179.      * Returns the scale parameter of this distribution.
  180.      *
  181.      * @return the scale parameter
  182.      */
  183.     public double getScale() {
  184.         return scale;
  185.     }

  187.     /**
  188.      * Returns the shape parameter of this distribution.
  189.      *
  190.      * @return the shape parameter
  191.      */
  192.     public double getShape() {
  193.         return shape;
  194.     }

  196.     /**
  197.      * {@inheritDoc}
  198.      *
  199.      * For scale {@code m}, and shape {@code s} of this distribution, the PDF
  200.      * is given by
  201.      * <ul>
  202.      * <li>{@code 0} if {@code x <= 0},</li>
  203.      * <li>{@code exp(-0.5 * ((ln(x) - m) / s)^2) / (s * sqrt(2 * pi) * x)}
  204.      * otherwise.</li>
  205.      * </ul>
  206.      */
  207.     public double density(double x) {
  208.         if (x <= 0) {
  209.             return 0;
  210.         }
  211.         final double x0 = FastMath.log(x) - scale;
  212.         final double x1 = x0 / shape;
  213.         return FastMath.exp(-0.5 * x1 * x1) / (shape * SQRT2PI * x);
  214.     }

  216.     /** {@inheritDoc}
  217.      *
  218.      * See documentation of {@link #density(double)} for computation details.
  219.      */
  220.     @Override
  221.     public double logDensity(double x) {
  222.         if (x <= 0) {
  223.             return Double.NEGATIVE_INFINITY;
  224.         }
  225.         final double logX = FastMath.log(x);
  226.         final double x0 = logX - scale;
  227.         final double x1 = x0 / shape;
  228.         return -0.5 * x1 * x1 - (logShapePlusHalfLog2Pi + logX);
  229.     }

  231.     /**
  232.      * {@inheritDoc}
  233.      *
  234.      * For scale {@code m}, and shape {@code s} of this distribution, the CDF
  235.      * is given by
  236.      * <ul>
  237.      * <li>{@code 0} if {@code x <= 0},</li>
  238.      * <li>{@code 0} if {@code ln(x) - m < 0} and {@code m - ln(x) > 40 * s}, as
  239.      * in these cases the actual value is within {@code Double.MIN_VALUE} of 0,
  240.      * <li>{@code 1} if {@code ln(x) - m >= 0} and {@code ln(x) - m > 40 * s},
  241.      * as in these cases the actual value is within {@code Double.MIN_VALUE} of
  242.      * 1,</li>
  243.      * <li>{@code 0.5 + 0.5 * erf((ln(x) - m) / (s * sqrt(2))} otherwise.</li>
  244.      * </ul>
  245.      */
  246.     public double cumulativeProbability(double x)  {
  247.         if (x <= 0) {
  248.             return 0;
  249.         }
  250.         final double dev = FastMath.log(x) - scale;
  251.         if (FastMath.abs(dev) > 40 * shape) {
  252.             return dev < 0 ? 0.0d : 1.0d;
  253.         }
  254.         return 0.5 + 0.5 * Erf.erf(dev / (shape * SQRT2));
  255.     }

  257.     /**
  258.      * {@inheritDoc}
  259.      *
  260.      * @deprecated See {@link RealDistribution#cumulativeProbability(double,double)}
  261.      */
  262.     @Override@Deprecated
  263.     public double cumulativeProbability(double x0, double x1)
  264.         throws NumberIsTooLargeException {
  265.         return probability(x0, x1);
  266.     }

  268.     /** {@inheritDoc} */
  269.     @Override
  270.     public double probability(double x0,
  271.                               double x1)
  272.         throws NumberIsTooLargeException {
  273.         if (x0 > x1) {
  274.             throw new NumberIsTooLargeException(LocalizedFormats.LOWER_ENDPOINT_ABOVE_UPPER_ENDPOINT,
  275.                                                 x0, x1, true);
  276.         }
  277.         if (x0 <= 0 || x1 <= 0) {
  278.             return super.probability(x0, x1);
  279.         }
  280.         final double denom = shape * SQRT2;
  281.         final double v0 = (FastMath.log(x0) - scale) / denom;
  282.         final double v1 = (FastMath.log(x1) - scale) / denom;
  283.         return 0.5 * Erf.erf(v0, v1);
  284.     }

  286.     /** {@inheritDoc} */
  287.     @Override
  288.     protected double getSolverAbsoluteAccuracy() {
  289.         return solverAbsoluteAccuracy;
  290.     }

  292.     /**
  293.      * {@inheritDoc}
  294.      *
  295.      * For scale {@code m} and shape {@code s}, the mean is
  296.      * {@code exp(m + s^2 / 2)}.
  297.      */
  298.     public double getNumericalMean() {
  299.         double s = shape;
  300.         return FastMath.exp(scale + (s * s / 2));
  301.     }

  303.     /**
  304.      * {@inheritDoc}
  305.      *
  306.      * For scale {@code m} and shape {@code s}, the variance is
  307.      * {@code (exp(s^2) - 1) * exp(2 * m + s^2)}.
  308.      */
  309.     public double getNumericalVariance() {
  310.         final double s = shape;
  311.         final double ss = s * s;
  312.         return (FastMath.expm1(ss)) * FastMath.exp(2 * scale + ss);
  313.     }

  315.     /**
  316.      * {@inheritDoc}
  317.      *
  318.      * The lower bound of the support is always 0 no matter the parameters.
  319.      *
  320.      * @return lower bound of the support (always 0)
  321.      */
  322.     public double getSupportLowerBound() {
  323.         return 0;
  324.     }

  326.     /**
  327.      * {@inheritDoc}
  328.      *
  329.      * The upper bound of the support is always positive infinity
  330.      * no matter the parameters.
  331.      *
  332.      * @return upper bound of the support (always
  333.      * {@code Double.POSITIVE_INFINITY})
  334.      */
  335.     public double getSupportUpperBound() {
  336.         return Double.POSITIVE_INFINITY;
  337.     }

  339.     /** {@inheritDoc} */
  340.     public boolean isSupportLowerBoundInclusive() {
  341.         return true;
  342.     }

  344.     /** {@inheritDoc} */
  345.     public boolean isSupportUpperBoundInclusive() {
  346.         return false;
  347.     }

  349.     /**
  350.      * {@inheritDoc}
  351.      *
  352.      * The support of this distribution is connected.
  353.      *
  354.      * @return {@code true}
  355.      */
  356.     public boolean isSupportConnected() {
  357.         return true;
  358.     }

  360.     /** {@inheritDoc} */
  361.     @Override
  362.     public double sample()  {
  363.         final double n = random.nextGaussian();
  364.         return FastMath.exp(scale + shape * n);
  365.     }
  366. }
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