TDistribution.java

/*
 * 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
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 *
 *      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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package org.apache.commons.statistics.distribution;

import org.apache.commons.numbers.gamma.Beta;
import org.apache.commons.numbers.gamma.LogBeta;
import org.apache.commons.numbers.gamma.RegularizedBeta;
import org.apache.commons.rng.UniformRandomProvider;
import org.apache.commons.rng.sampling.distribution.TSampler;

/**
 * Implementation of Student's t-distribution.
 *
 * <p>The probability density function of \( X \) is:
 *
 * <p>\[ f(x; v) = \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{t^2}{\nu} \right)^{\!-\frac{\nu+1}{2}} \]
 *
 * <p>for \( v &gt; 0 \) the degrees of freedom,
 * \( \Gamma \) is the gamma function, and
 * \( x \in (-\infty, \infty) \).
 *
 * @see <a href="https://en.wikipedia.org/wiki/Student%27s_t-distribution">Student&#39;s t-distribution (Wikipedia)</a>
 * @see <a href="https://mathworld.wolfram.com/Studentst-Distribution.html">Student&#39;s t-distribution (MathWorld)</a>
 */
public abstract class TDistribution extends AbstractContinuousDistribution {
    /** The degrees of freedom. */
    private final double degreesOfFreedom;

    /**
     * Specialisation of the T-distribution used when there are infinite degrees of freedom.
     * In this case the distribution matches a normal distribution. This is used when the
     * variance is not different from 1.0.
     *
     * <p>This delegates all methods to the standard normal distribution. Instances are
     * allowed to provide access to the degrees of freedom used during construction.
     */
    private static class NormalTDistribution extends TDistribution {
        /** A standard normal distribution used for calculations.
         * This is immutable and thread-safe and can be used across instances. */
        private static final NormalDistribution STANDARD_NORMAL = NormalDistribution.of(0, 1);

        /**
         * @param degreesOfFreedom Degrees of freedom.
         */
        NormalTDistribution(double degreesOfFreedom) {
            super(degreesOfFreedom);
        }

        @Override
        public double density(double x) {
            return STANDARD_NORMAL.density(x);
        }

        @Override
        public double probability(double x0, double x1) {
            return STANDARD_NORMAL.probability(x0, x1);
        }

        @Override
        public double logDensity(double x) {
            return STANDARD_NORMAL.logDensity(x);
        }

        @Override
        public double cumulativeProbability(double x) {
            return STANDARD_NORMAL.cumulativeProbability(x);
        }

        @Override
        public double inverseCumulativeProbability(double p) {
            return STANDARD_NORMAL.inverseCumulativeProbability(p);
        }

        // Survival probability functions inherit the symmetry operations from the TDistribution

        @Override
        public double getMean() {
            return 0;
        }

        @Override
        public double getVariance() {
            return 1.0;
        }

        @Override
        public Sampler createSampler(UniformRandomProvider rng) {
            return STANDARD_NORMAL.createSampler(rng);
        }
    }

    /**
     * Implementation of Student's T-distribution.
     */
    private static class StudentsTDistribution extends TDistribution {
        /** 2. */
        private static final double TWO = 2;
        /** The threshold for the density function where the
         * power function base minus 1 is close to zero. */
        private static final double CLOSE_TO_ZERO = 0.25;

        /** -(v + 1) / 2, where v = degrees of freedom. */
        private final double mvp1Over2;
        /** Density normalisation factor, sqrt(v) * beta(1/2, v/2), where v = degrees of freedom. */
        private final double densityNormalisation;
        /** Log density normalisation term, 0.5 * log(v) + log(beta(1/2, v/2)), where v = degrees of freedom. */
        private final double logDensityNormalisation;
        /** Cached value for inverse probability function. */
        private final double mean;
        /** Cached value for inverse probability function. */
        private final double variance;

        /**
         * @param degreesOfFreedom Degrees of freedom.
         * @param variance Precomputed variance
         */
        StudentsTDistribution(double degreesOfFreedom, double variance) {
            super(degreesOfFreedom);

            mvp1Over2 = -0.5 * (degreesOfFreedom + 1);
            densityNormalisation = Math.sqrt(degreesOfFreedom) * Beta.value(0.5, degreesOfFreedom / 2);
            logDensityNormalisation = 0.5 * Math.log(degreesOfFreedom) + LogBeta.value(0.5, degreesOfFreedom / 2);
            mean = degreesOfFreedom > 1 ? 0 : Double.NaN;
            this.variance = variance;
        }

        /**
         * @param degreesOfFreedom Degrees of freedom.
         * @return the variance
         */
        static double computeVariance(double degreesOfFreedom) {
            if (degreesOfFreedom == Double.POSITIVE_INFINITY) {
                return 1;
            } else if (degreesOfFreedom > TWO) {
                return degreesOfFreedom / (degreesOfFreedom - 2);
            } else if (degreesOfFreedom > 1) {
                return Double.POSITIVE_INFINITY;
            } else {
                return Double.NaN;
            }
        }

        @Override
        public double density(double x) {
            //          1                       -(v+1)/2
            // ------------------- * (1 + t^2/v)
            // sqrt(v) B(1/2, v/2)

            final double t2OverV = x * x / getDegreesOfFreedom();
            if (t2OverV < CLOSE_TO_ZERO) {
                // Avoid power function when the base is close to 1
                return Math.exp(Math.log1p(t2OverV) * mvp1Over2) / densityNormalisation;
            }
            return Math.pow(1 + t2OverV, mvp1Over2) / densityNormalisation;
        }

        @Override
        public double logDensity(double x) {
            return Math.log1p(x * x / getDegreesOfFreedom()) * mvp1Over2 - logDensityNormalisation;
        }

        @Override
        public double cumulativeProbability(double x) {
            if (x == 0) {
                return 0.5;
            }
            final double v = getDegreesOfFreedom();

            // cdf(t) = 1 - 0.5 * I_x(t)(v/2, 1/2)
            // where x(t) = v / (v + t^2)
            //
            // When t^2 << v using the regularized beta results
            // in loss of precision. Use the complement instead:
            // I[x](a,b) = 1 - I[1-x](b,a)
            // x   = v   / (v + t^2)
            // 1-x = t^2 / (v + t^2)
            // Use the threshold documented in the Boost t-distribution:
            // https://www.boost.org/doc/libs/1_78_0/libs/math/doc/html/math_toolkit/dist_ref/dists/students_t_dist.html

            final double t2 = x * x;
            final double z;
            if (v < 2 * t2) {
                z = RegularizedBeta.value(v / (v + t2), v / 2, 0.5) / 2;
            } else {
                z = RegularizedBeta.complement(t2 / (v + t2), 0.5, v / 2) / 2;
            }
            return x > 0 ? 1 - z : z;
        }

        @Override
        public double getMean() {
            return mean;
        }

        @Override
        public double getVariance() {
            return variance;
        }

        @Override
        double getMedian() {
            // Overridden for the probability(double, double) method.
            // This is intentionally not a public method.
            return 0;
        }

        @Override
        public Sampler createSampler(UniformRandomProvider rng) {
            // T distribution sampler.
            return TSampler.of(rng, getDegreesOfFreedom())::sample;
        }
    }

    /**
     * @param degreesOfFreedom Degrees of freedom.
     */
    TDistribution(double degreesOfFreedom) {
        this.degreesOfFreedom = degreesOfFreedom;
    }

    /**
     * Creates a Student's t-distribution.
     *
     * @param degreesOfFreedom Degrees of freedom.
     * @return the distribution
     * @throws IllegalArgumentException if {@code degreesOfFreedom <= 0}
     */
    public static TDistribution of(double degreesOfFreedom) {
        if (degreesOfFreedom <= 0) {
            throw new DistributionException(DistributionException.NOT_STRICTLY_POSITIVE,
                                            degreesOfFreedom);
        }
        // If the variance converges to 1 use a NormalDistribution.
        // Occurs at 2^55 = 3.60e16
        final double variance = StudentsTDistribution.computeVariance(degreesOfFreedom);
        if (variance == 1) {
            return new NormalTDistribution(degreesOfFreedom);
        }
        return new StudentsTDistribution(degreesOfFreedom, variance);
    }

    /**
     * Gets the degrees of freedom parameter of this distribution.
     *
     * @return the degrees of freedom.
     */
    public double getDegreesOfFreedom() {
        return degreesOfFreedom;
    }

    /** {@inheritDoc} */
    @Override
    public double survivalProbability(double x) {
        // Exploit symmetry
        return cumulativeProbability(-x);
    }

    /** {@inheritDoc} */
    @Override
    public double inverseSurvivalProbability(double p) {
        // Exploit symmetry
        // Subtract from 0 avoids returning -0.0 for p=0.5
        return 0.0 - inverseCumulativeProbability(p);
    }

    /**
     * {@inheritDoc}
     *
     * <p>For degrees of freedom parameter \( v \), the mean is:
     *
     * <p>\[ \mathbb{E}[X] = \begin{cases}
     *       0                &amp; \text{for } v \gt 1 \\
     *       \text{undefined} &amp; \text{otherwise}
     *       \end{cases} \]
     *
     * @return the mean, or {@link Double#NaN NaN} if it is not defined.
     */
    @Override
    public abstract double getMean();

    /**
     * {@inheritDoc}
     *
     * <p>For degrees of freedom parameter \( v \), the variance is:
     *
     * <p>\[ \operatorname{var}[X] = \begin{cases}
     *       \frac{v}{v - 2}  &amp; \text{for } v \gt 2 \\
     *       \infty           &amp; \text{for } 1 \lt v \le 2 \\
     *       \text{undefined} &amp; \text{otherwise}
     *       \end{cases} \]
     *
     * @return the variance, or {@link Double#NaN NaN} if it is not defined.
     */
    @Override
    public abstract double getVariance();

    /**
     * {@inheritDoc}
     *
     * <p>The lower bound of the support is always negative infinity.
     *
     * @return {@linkplain Double#NEGATIVE_INFINITY negative infinity}.
     */
    @Override
    public double getSupportLowerBound() {
        return Double.NEGATIVE_INFINITY;
    }

    /**
     * {@inheritDoc}
     *
     * <p>The upper bound of the support is always positive infinity.
     *
     * @return {@linkplain Double#POSITIVE_INFINITY positive infinity}.
     */
    @Override
    public double getSupportUpperBound() {
        return Double.POSITIVE_INFINITY;
    }
}