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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.util.FastMath;

Implementation of the Pareto distribution.

Parameters: The probability distribution function of X is given by (for x >= k): 
 α * k^α / x^(α + 1)



k is the scale parameter: this is the minimum possible value of X,
α is the shape parameter: this is the Pareto index

See Also:

 * Pareto distribution (Wikipedia)

 * Pareto distribution (MathWorld)
Since:
3.3/**
 * Implementation of the Pareto distribution.
 *
 * <p>
 * <strong>Parameters:</strong>
 * The probability distribution function of {@code X} is given by (for {@code x >= k}):
 * <pre>
 *  α * k^α / x^(α + 1)
 * </pre>
 * <p>
 * <ul>
 * <li>{@code k} is the <em>scale</em> parameter: this is the minimum possible value of {@code X},</li>
 * <li>{@code α} is the <em>shape</em> parameter: this is the Pareto index</li>
 * </ul>
 *
 * @see <a href="http://en.wikipedia.org/wiki/Pareto_distribution">
 * Pareto distribution (Wikipedia)</a>
 * @see <a href="http://mathworld.wolfram.com/ParetoDistribution.html">
 * Pareto distribution (MathWorld)</a>
 *
 * @since 3.3
 */
public class ParetoDistribution extends AbstractRealDistribution {

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

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

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

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

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

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

    
Create a Pareto distribution using the specified scale and shape.

Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see 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:
scale – the scale parameter of this distribution
shape – the shape parameter of this distribution
Throws:
NotStrictlyPositiveException – if scale <= 0 or shape <= 0./**
     * Create a Pareto 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 scale <= 0} or {@code shape <= 0}.
     */
    public ParetoDistribution(double scale, double shape)
        throws NotStrictlyPositiveException {
        this(scale, shape, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
    }

    
Create a Pareto distribution using the specified scale, shape and
inverse cumulative distribution accuracy.

Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see 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:
scale – the scale parameter of this distribution
shape – the shape parameter of this distribution
inverseCumAccuracy – Inverse cumulative probability accuracy.
Throws:
NotStrictlyPositiveException – if scale <= 0 or shape <= 0./**
     * Create a Pareto 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 scale <= 0} or {@code shape <= 0}.
     */
    public ParetoDistribution(double scale, double shape, double inverseCumAccuracy)
        throws NotStrictlyPositiveException {
        this(new Well19937c(), scale, shape, inverseCumAccuracy);
    }

    
Creates a Pareto distribution.
Params:
rng – Random number generator.
scale – Scale parameter of this distribution.
shape – Shape parameter of this distribution.
Throws:
NotStrictlyPositiveException – if scale <= 0 or shape <= 0./**
     * Creates a Pareto distribution.
     *
     * @param rng Random number generator.
     * @param scale Scale parameter of this distribution.
     * @param shape Shape parameter of this distribution.
     * @throws NotStrictlyPositiveException if {@code scale <= 0} or {@code shape <= 0}.
     */
    public ParetoDistribution(RandomGenerator rng, double scale, double shape)
        throws NotStrictlyPositiveException {
        this(rng, scale, shape, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
    }

    
Creates a Pareto distribution.
Params:
rng – Random number generator.
scale – Scale parameter of this distribution.
shape – Shape parameter of this distribution.
inverseCumAccuracy – Inverse cumulative probability accuracy.
Throws:
NotStrictlyPositiveException – if scale <= 0 or shape <= 0./**
     * Creates a Pareto 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 scale <= 0} or {@code shape <= 0}.
     */
    public ParetoDistribution(RandomGenerator rng,
                              double scale,
                              double shape,
                              double inverseCumAccuracy)
        throws NotStrictlyPositiveException {
        super(rng);

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

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

        this.scale = scale;
        this.shape = shape;
        this.solverAbsoluteAccuracy = inverseCumAccuracy;
    }

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

    
Returns the shape parameter of this distribution.
Returns:
the shape parameter/**
     * Returns the shape parameter of this distribution.
     *
     * @return the shape parameter
     */
    public double getShape() {
        return shape;
    }

    
{@inheritDoc}
 For scale k, and shape α of this distribution, the PDF is given by 

0 if x < k,
α * k^α / x^(α + 1) otherwise.

/**
     * {@inheritDoc}
     * <p>
     * For scale {@code k}, and shape {@code α} of this distribution, the PDF
     * is given by
     * <ul>
     * <li>{@code 0} if {@code x < k},</li>
     * <li>{@code α * k^α / x^(α + 1)} otherwise.</li>
     * </ul>
     */
    public double density(double x) {
        if (x < scale) {
            return 0;
        }
        return FastMath.pow(scale, shape) / FastMath.pow(x, shape + 1) * shape;
    }

    
{@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 < scale) {
            return Double.NEGATIVE_INFINITY;
        }
        return FastMath.log(scale) * shape - FastMath.log(x) * (shape + 1) + FastMath.log(shape);
    }

    
{@inheritDoc}
 For scale k, and shape α of this distribution, the CDF is given by 

0 if x < k,
1 - (k / x)^α otherwise.

/**
     * {@inheritDoc}
     * <p>
     * For scale {@code k}, and shape {@code α} of this distribution, the CDF is given by
     * <ul>
     * <li>{@code 0} if {@code x < k},</li>
     * <li>{@code 1 - (k / x)^α} otherwise.</li>
     * </ul>
     */
    public double cumulativeProbability(double x)  {
        if (x <= scale) {
            return 0;
        }
        return 1 - FastMath.pow(scale / x, shape);
    }

    
{@inheritDoc}
Deprecated:
See RealDistribution.cumulativeProbability(double, double)/**
     * {@inheritDoc}
     *
     * @deprecated See {@link RealDistribution#cumulativeProbability(double,double)}
     */
    @Override
    @Deprecated
    public double cumulativeProbability(double x0, double x1)
        throws NumberIsTooLargeException {
        return probability(x0, x1);
    }

    
{@inheritDoc} /** {@inheritDoc} */
    @Override
    protected double getSolverAbsoluteAccuracy() {
        return solverAbsoluteAccuracy;
    }

    
{@inheritDoc}
 For scale k and shape α, the mean is given by 

∞ if α <= 1,
α * k / (α - 1) otherwise.

/**
     * {@inheritDoc}
     * <p>
     * For scale {@code k} and shape {@code α}, the mean is given by
     * <ul>
     * <li>{@code ∞} if {@code α <= 1},</li>
     * <li>{@code α * k / (α - 1)} otherwise.</li>
     * </ul>
     */
    public double getNumericalMean() {
        if (shape <= 1) {
            return Double.POSITIVE_INFINITY;
        }
        return shape * scale / (shape - 1);
    }

    
{@inheritDoc}
 For scale k and shape α, the variance is given by 

∞ if 1 < α <= 2,
k^2 * α / ((α - 1)^2 * (α - 2)) otherwise.

/**
     * {@inheritDoc}
     * <p>
     * For scale {@code k} and shape {@code α}, the variance is given by
     * <ul>
     * <li>{@code ∞} if {@code 1 < α <= 2},</li>
     * <li>{@code k^2 * α / ((α - 1)^2 * (α - 2))} otherwise.</li>
     * </ul>
     */
    public double getNumericalVariance() {
        if (shape <= 2) {
            return Double.POSITIVE_INFINITY;
        }
        double s = shape - 1;
        return scale * scale * shape / (s * s) / (shape - 2);
    }

    
{@inheritDoc}
 The lower bound of the support is equal to the scale parameter k. 
Returns:
lower bound of the support/**
     * {@inheritDoc}
     * <p>
     * The lower bound of the support is equal to the scale parameter {@code k}.
     *
     * @return lower bound of the support
     */
    public double getSupportLowerBound() {
        return scale;
    }

    
{@inheritDoc}

The upper bound of the support is always positive infinity no matter the parameters.
Returns:
upper bound of the support (always Double.POSITIVE_INFINITY)/**
     * {@inheritDoc}
     * <p>
     * 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} /** {@inheritDoc} */
    public boolean isSupportLowerBoundInclusive() {
        return true;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public boolean isSupportUpperBoundInclusive() {
        return false;
    }

    
{@inheritDoc}

The support of this distribution is connected.
Returns:
true/**
     * {@inheritDoc}
     * <p>
     * The support of this distribution is connected.
     *
     * @return {@code true}
     */
    public boolean isSupportConnected() {
        return true;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    @Override
    public double sample()  {
        final double n = random.nextDouble();
        return scale / FastMath.pow(n, 1 / shape);
    }
}