/*
 * 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.OutOfRangeException;
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;

This class implements the 
Lévy distribution.
Since:
3.2/**
 * This class implements the <a href="http://en.wikipedia.org/wiki/L%C3%A9vy_distribution">
 * L&eacute;vy distribution</a>.
 *
 * @since 3.2
 */
public class LevyDistribution extends AbstractRealDistribution {

    
Serializable UID. /** Serializable UID. */
    private static final long serialVersionUID = 20130314L;

    
Location parameter. /** Location parameter. */
    private final double mu;

    
Scale parameter. /** Scale parameter. */
    private final double c;  // Setting this to 1 returns a cumProb of 1.0

    
Half of c (for calculations). /** Half of c (for calculations). */
    private final double halfC;

    
Build a new instance.

Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see AbstractRealDistribution.sample() and AbstractRealDistribution.sample(int)). In case no sampling is needed for the created distribution, it is advised to pass null as random generator via the appropriate constructors to avoid the additional initialisation overhead. 
Params:
mu – location parameter
c – scale parameter
Since:
3.4/**
     * Build a new instance.
     * <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 mu location parameter
     * @param c scale parameter
     * @since 3.4
     */
    public LevyDistribution(final double mu, final double c) {
        this(new Well19937c(), mu, c);
    }

    
Creates a LevyDistribution.
Params:
rng – random generator to be used for sampling
mu – location
c – scale parameter/**
     * Creates a LevyDistribution.
     * @param rng random generator to be used for sampling
     * @param mu location
     * @param c scale parameter
     */
    public LevyDistribution(final RandomGenerator rng, final double mu, final double c) {
        super(rng);
        this.mu    = mu;
        this.c     = c;
        this.halfC = 0.5 * c;
    }

    
{@inheritDoc}

From Wikipedia: The probability density function of the Lévy distribution
over the domain is

f(x; μ, c) = √(c / 2π) * e-c / 2 (x - μ) / (x - μ)3/2

 For this distribution, X, this method returns P(X < x). If x is less than location parameter μ, Double.NaN is returned, as in these cases the distribution is not defined. 
/** {@inheritDoc}
    * <p>
    * From Wikipedia: The probability density function of the L&eacute;vy distribution
    * over the domain is
    * </p>
    * <pre>
    * f(x; &mu;, c) = &radic;(c / 2&pi;) * e<sup>-c / 2 (x - &mu;)</sup> / (x - &mu;)<sup>3/2</sup>
    * </pre>
    * <p>
    * For this distribution, {@code X}, this method returns {@code P(X < x)}.
    * If {@code x} is less than location parameter &mu;, {@code Double.NaN} is
    * returned, as in these cases the distribution is not defined.
    * </p>
    */
    public double density(final double x) {
        if (x < mu) {
            return Double.NaN;
        }

        final double delta = x - mu;
        final double f     = halfC / delta;
        return FastMath.sqrt(f / FastMath.PI) * FastMath.exp(-f) /delta;
    }

    
{@inheritDoc} See documentation of density(double) for computation details. /** {@inheritDoc}
     *
     * See documentation of {@link #density(double)} for computation details.
     */
    @Override
    public double logDensity(double x) {
        if (x < mu) {
            return Double.NaN;
        }

        final double delta = x - mu;
        final double f     = halfC / delta;
        return 0.5 * FastMath.log(f / FastMath.PI) - f - FastMath.log(delta);
    }

    
{@inheritDoc}

From Wikipedia: the cumulative distribution function is

f(x; u, c) = erfc (√ (c / 2 (x - u )))

/** {@inheritDoc}
     * <p>
     * From Wikipedia: the cumulative distribution function is
     * </p>
     * <pre>
     * f(x; u, c) = erfc (&radic; (c / 2 (x - u )))
     * </pre>
     */
    public double cumulativeProbability(final double x) {
        if (x < mu) {
            return Double.NaN;
        }
        return Erf.erfc(FastMath.sqrt(halfC / (x - mu)));
    }

    
{@inheritDoc} /** {@inheritDoc} */
    @Override
    public double inverseCumulativeProbability(final double p) throws OutOfRangeException {
        if (p < 0.0 || p > 1.0) {
            throw new OutOfRangeException(p, 0, 1);
        }
        final double t = Erf.erfcInv(p);
        return mu + halfC / (t * t);
    }

    
Get the scale parameter of the distribution.
Returns:
scale parameter of the distribution/** Get the scale parameter of the distribution.
     * @return scale parameter of the distribution
     */
    public double getScale() {
        return c;
    }

    
Get the location parameter of the distribution.
Returns:
location parameter of the distribution/** Get the location parameter of the distribution.
     * @return location parameter of the distribution
     */
    public double getLocation() {
        return mu;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public double getNumericalMean() {
        return Double.POSITIVE_INFINITY;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public double getNumericalVariance() {
        return Double.POSITIVE_INFINITY;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public double getSupportLowerBound() {
        return mu;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public double getSupportUpperBound() {
        return Double.POSITIVE_INFINITY;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public boolean isSupportLowerBoundInclusive() {
        // there is a division by x-mu in the computation, so density
        // is not finite at lower bound, bound must be excluded
        return false;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public boolean isSupportUpperBoundInclusive() {
        // upper bound is infinite, so it must be excluded
        return false;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public boolean isSupportConnected() {
        return true;
    }

}