ParetoDistribution.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.util.FastMath;

  27. /**
  28.  * Implementation of the Pareto distribution.
  29.  *
  30.  * <p>
  31.  * <strong>Parameters:</strong>
  32.  * The probability distribution function of {@code X} is given by (for {@code x >= k}):
  33.  * <pre>
  34.  *  α * k^α / x^(α + 1)
  35.  * </pre>
  36.  * <p>
  37.  * <ul>
  38.  * <li>{@code k} is the <em>scale</em> parameter: this is the minimum possible value of {@code X},</li>
  39.  * <li>{@code α} is the <em>shape</em> parameter: this is the Pareto index</li>
  40.  * </ul>
  41.  *
  42.  * @see <a href="http://en.wikipedia.org/wiki/Pareto_distribution">
  43.  * Pareto distribution (Wikipedia)</a>
  44.  * @see <a href="http://mathworld.wolfram.com/ParetoDistribution.html">
  45.  * Pareto distribution (MathWorld)</a>
  46.  *
  47.  * @since 3.3
  48.  */
  49. public class ParetoDistribution extends AbstractRealDistribution {

  51.     /** Default inverse cumulative probability accuracy. */
  52.     public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY = 1e-9;

  54.     /** Serializable version identifier. */
  55.     private static final long serialVersionUID = 20130424;

  57.     /** The scale parameter of this distribution. */
  58.     private final double scale;

  60.     /** The shape parameter of this distribution. */
  61.     private final double shape;

  63.     /** Inverse cumulative probability accuracy. */
  64.     private final double solverAbsoluteAccuracy;

  66.     /**
  67.      * Create a Pareto distribution with a scale of {@code 1} and a shape of {@code 1}.
  68.      */
  69.     public ParetoDistribution() {
  70.         this(1, 1);
  71.     }

  73.     /**
  74.      * Create a Pareto distribution using the specified scale and shape.
  75.      * <p>
  76.      * <b>Note:</b> this constructor will implicitly create an instance of
  77.      * {@link Well19937c} as random generator to be used for sampling only (see
  78.      * {@link #sample()} and {@link #sample(int)}). In case no sampling is
  79.      * needed for the created distribution, it is advised to pass {@code null}
  80.      * as random generator via the appropriate constructors to avoid the
  81.      * additional initialisation overhead.
  82.      *
  83.      * @param scale the scale parameter of this distribution
  84.      * @param shape the shape parameter of this distribution
  85.      * @throws NotStrictlyPositiveException if {@code scale <= 0} or {@code shape <= 0}.
  86.      */
  87.     public ParetoDistribution(double scale, double shape)
  88.         throws NotStrictlyPositiveException {
  89.         this(scale, shape, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
  90.     }

  92.     /**
  93.      * Create a Pareto distribution using the specified scale, shape and
  94.      * inverse cumulative distribution accuracy.
  95.      * <p>
  96.      * <b>Note:</b> this constructor will implicitly create an instance of
  97.      * {@link Well19937c} as random generator to be used for sampling only (see
  98.      * {@link #sample()} and {@link #sample(int)}). In case no sampling is
  99.      * needed for the created distribution, it is advised to pass {@code null}
  100.      * as random generator via the appropriate constructors to avoid the
  101.      * additional initialisation overhead.
  102.      *
  103.      * @param scale the scale parameter of this distribution
  104.      * @param shape the shape parameter of this distribution
  105.      * @param inverseCumAccuracy Inverse cumulative probability accuracy.
  106.      * @throws NotStrictlyPositiveException if {@code scale <= 0} or {@code shape <= 0}.
  107.      */
  108.     public ParetoDistribution(double scale, double shape, double inverseCumAccuracy)
  109.         throws NotStrictlyPositiveException {
  110.         this(new Well19937c(), scale, shape, inverseCumAccuracy);
  111.     }

  113.     /**
  114.      * Creates a Pareto distribution.
  115.      *
  116.      * @param rng Random number generator.
  117.      * @param scale Scale parameter of this distribution.
  118.      * @param shape Shape parameter of this distribution.
  119.      * @throws NotStrictlyPositiveException if {@code scale <= 0} or {@code shape <= 0}.
  120.      */
  121.     public ParetoDistribution(RandomGenerator rng, double scale, double shape)
  122.         throws NotStrictlyPositiveException {
  123.         this(rng, scale, shape, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
  124.     }

  126.     /**
  127.      * Creates a Pareto distribution.
  128.      *
  129.      * @param rng Random number generator.
  130.      * @param scale Scale parameter of this distribution.
  131.      * @param shape Shape parameter of this distribution.
  132.      * @param inverseCumAccuracy Inverse cumulative probability accuracy.
  133.      * @throws NotStrictlyPositiveException if {@code scale <= 0} or {@code shape <= 0}.
  134.      */
  135.     public ParetoDistribution(RandomGenerator rng,
  136.                               double scale,
  137.                               double shape,
  138.                               double inverseCumAccuracy)
  139.         throws NotStrictlyPositiveException {
  140.         super(rng);

  142.         if (scale <= 0) {
  143.             throw new NotStrictlyPositiveException(LocalizedFormats.SCALE, scale);
  144.         }

  146.         if (shape <= 0) {
  147.             throw new NotStrictlyPositiveException(LocalizedFormats.SHAPE, shape);
  148.         }

  150.         this.scale = scale;
  151.         this.shape = shape;
  152.         this.solverAbsoluteAccuracy = inverseCumAccuracy;
  153.     }

  155.     /**
  156.      * Returns the scale parameter of this distribution.
  157.      *
  158.      * @return the scale parameter
  159.      */
  160.     public double getScale() {
  161.         return scale;
  162.     }

  164.     /**
  165.      * Returns the shape parameter of this distribution.
  166.      *
  167.      * @return the shape parameter
  168.      */
  169.     public double getShape() {
  170.         return shape;
  171.     }

  173.     /**
  174.      * {@inheritDoc}
  175.      * <p>
  176.      * For scale {@code k}, and shape {@code α} of this distribution, the PDF
  177.      * is given by
  178.      * <ul>
  179.      * <li>{@code 0} if {@code x < k},</li>
  180.      * <li>{@code α * k^α / x^(α + 1)} otherwise.</li>
  181.      * </ul>
  182.      */
  183.     public double density(double x) {
  184.         if (x < scale) {
  185.             return 0;
  186.         }
  187.         return FastMath.pow(scale, shape) / FastMath.pow(x, shape + 1) * shape;
  188.     }

  190.     /** {@inheritDoc}
  191.      *
  192.      * See documentation of {@link #density(double)} for computation details.
  193.      */
  194.     @Override
  195.     public double logDensity(double x) {
  196.         if (x < scale) {
  197.             return Double.NEGATIVE_INFINITY;
  198.         }
  199.         return FastMath.log(scale) * shape - FastMath.log(x) * (shape + 1) + FastMath.log(shape);
  200.     }

  202.     /**
  203.      * {@inheritDoc}
  204.      * <p>
  205.      * For scale {@code k}, and shape {@code α} of this distribution, the CDF is given by
  206.      * <ul>
  207.      * <li>{@code 0} if {@code x < k},</li>
  208.      * <li>{@code 1 - (k / x)^α} otherwise.</li>
  209.      * </ul>
  210.      */
  211.     public double cumulativeProbability(double x)  {
  212.         if (x <= scale) {
  213.             return 0;
  214.         }
  215.         return 1 - FastMath.pow(scale / x, shape);
  216.     }

  218.     /**
  219.      * {@inheritDoc}
  220.      *
  221.      * @deprecated See {@link RealDistribution#cumulativeProbability(double,double)}
  222.      */
  223.     @Override
  224.     @Deprecated
  225.     public double cumulativeProbability(double x0, double x1)
  226.         throws NumberIsTooLargeException {
  227.         return probability(x0, x1);
  228.     }

  230.     /** {@inheritDoc} */
  231.     @Override
  232.     protected double getSolverAbsoluteAccuracy() {
  233.         return solverAbsoluteAccuracy;
  234.     }

  236.     /**
  237.      * {@inheritDoc}
  238.      * <p>
  239.      * For scale {@code k} and shape {@code α}, the mean is given by
  240.      * <ul>
  241.      * <li>{@code ∞} if {@code α <= 1},</li>
  242.      * <li>{@code α * k / (α - 1)} otherwise.</li>
  243.      * </ul>
  244.      */
  245.     public double getNumericalMean() {
  246.         if (shape <= 1) {
  247.             return Double.POSITIVE_INFINITY;
  248.         }
  249.         return shape * scale / (shape - 1);
  250.     }

  252.     /**
  253.      * {@inheritDoc}
  254.      * <p>
  255.      * For scale {@code k} and shape {@code α}, the variance is given by
  256.      * <ul>
  257.      * <li>{@code ∞} if {@code 1 < α <= 2},</li>
  258.      * <li>{@code k^2 * α / ((α - 1)^2 * (α - 2))} otherwise.</li>
  259.      * </ul>
  260.      */
  261.     public double getNumericalVariance() {
  262.         if (shape <= 2) {
  263.             return Double.POSITIVE_INFINITY;
  264.         }
  265.         double s = shape - 1;
  266.         return scale * scale * shape / (s * s) / (shape - 2);
  267.     }

  269.     /**
  270.      * {@inheritDoc}
  271.      * <p>
  272.      * The lower bound of the support is equal to the scale parameter {@code k}.
  273.      *
  274.      * @return lower bound of the support
  275.      */
  276.     public double getSupportLowerBound() {
  277.         return scale;
  278.     }

  280.     /**
  281.      * {@inheritDoc}
  282.      * <p>
  283.      * The upper bound of the support is always positive infinity no matter the parameters.
  284.      *
  285.      * @return upper bound of the support (always {@code Double.POSITIVE_INFINITY})
  286.      */
  287.     public double getSupportUpperBound() {
  288.         return Double.POSITIVE_INFINITY;
  289.     }

  291.     /** {@inheritDoc} */
  292.     public boolean isSupportLowerBoundInclusive() {
  293.         return true;
  294.     }

  296.     /** {@inheritDoc} */
  297.     public boolean isSupportUpperBoundInclusive() {
  298.         return false;
  299.     }

  301.     /**
  302.      * {@inheritDoc}
  303.      * <p>
  304.      * The support of this distribution is connected.
  305.      *
  306.      * @return {@code true}
  307.      */
  308.     public boolean isSupportConnected() {
  309.         return true;
  310.     }

  312.     /** {@inheritDoc} */
  313.     @Override
  314.     public double sample()  {
  315.         final double n = random.nextDouble();
  316.         return scale / FastMath.pow(n, 1 / shape);
  317.     }
  318. }
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