org.apache.commons.math3.distribution

Class CauchyDistribution

java.lang.Object

org.apache.commons.math3.distribution.AbstractRealDistribution

org.apache.commons.math3.distribution.CauchyDistribution

All Implemented Interfaces:

Serializable, RealDistribution

public class CauchyDistribution
extends AbstractRealDistribution

Implementation of the Cauchy distribution.

Since:

1.1 (changed to concrete class in 3.0)

Version:

$Id: CauchyDistribution.java 1244107 2012-02-14 16:17:55Z erans $

See Also:

Cauchy distribution (Wikipedia), Cauchy Distribution (MathWorld), Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
static double	DEFAULT_INVERSE_ABSOLUTE_ACCURACY Default inverse cumulative probability accuracy.

Fields inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution

randomData, SOLVER_DEFAULT_ABSOLUTE_ACCURACY

Constructor Summary

Constructors

Constructor and Description
CauchyDistribution() Creates a Cauchy distribution with the median equal to zero and scale equal to one.
CauchyDistribution(double median, double scale) Creates a Cauchy distribution using the given median and scale.
CauchyDistribution(double median, double scale, double inverseCumAccuracy) Creates a Cauchy distribution using the given median and scale.

Method Summary

Methods

Modifier and Type	Method and Description
double	cumulativeProbability(double x) For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x).
double	density(double x) Returns the probability density function (PDF) of this distribution evaluated at the specified point x.
double	getMedian() Access the median.
double	getNumericalMean() Use this method to get the numerical value of the mean of this distribution.
double	getNumericalVariance() Use this method to get the numerical value of the variance of this distribution.
double	getScale() Access the scale parameter.
protected double	getSolverAbsoluteAccuracy() Returns the solver absolute accuracy for inverse cumulative computation.
double	getSupportLowerBound() Access the lower bound of the support.
double	getSupportUpperBound() Access the upper bound of the support.
double	inverseCumulativeProbability(double p) Computes the quantile function of this distribution.
boolean	isSupportConnected() Use this method to get information about whether the support is connected, i.e.
boolean	isSupportLowerBoundInclusive() Use this method to get information about whether the lower bound of the support is inclusive or not.
boolean	isSupportUpperBoundInclusive() Use this method to get information about whether the upper bound of the support is inclusive or not.
double	probability(double x) For a random variable X whose values are distributed according to this distribution, this method returns P(X = x).

Methods inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution

cumulativeProbability, reseedRandomGenerator, sample, sample

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

DEFAULT_INVERSE_ABSOLUTE_ACCURACY

public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY

Default inverse cumulative probability accuracy.

Since:

2.1

See Also:

Constant Field Values

Constructor Detail

CauchyDistribution

public CauchyDistribution()

Creates a Cauchy distribution with the median equal to zero and scale equal to one.

CauchyDistribution

public CauchyDistribution(double median,
                  double scale)

Creates a Cauchy distribution using the given median and scale.

Parameters:

median - Median for this distribution.

scale - Scale parameter for this distribution.

CauchyDistribution

public CauchyDistribution(double median,
                  double scale,
                  double inverseCumAccuracy)

Creates a Cauchy distribution using the given median and scale.

Parameters:

median - Median for this distribution.

scale - Scale parameter for this distribution.

inverseCumAccuracy - Maximum absolute error in inverse cumulative probability estimates (defaults to DEFAULT_INVERSE_ABSOLUTE_ACCURACY).

Throws:

NotStrictlyPositiveException - if scale <= 0.

Since:

2.1

Method Detail

cumulativeProbability

public double cumulativeProbability(double x)

For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x). In other words, this method represents the (cumulative) distribution function (CDF) for this distribution.

Parameters:

x - the point at which the CDF is evaluated

Returns:

the probability that a random variable with this distribution takes a value less than or equal to x

getMedian

public double getMedian()

Access the median.

Returns:

the median for this distribution.

getScale

public double getScale()

Access the scale parameter.

Returns:

the scale parameter for this distribution.

probability

public double probability(double x)

For a random variable X whose values are distributed according to this distribution, this method returns P(X = x). In other words, this method represents the probability mass function (PMF) for the distribution. For this distribution P(X = x) always evaluates to 0.

Parameters:

x - the point at which the PMF is evaluated

Returns:

0

density

public double density(double x)

Returns the probability density function (PDF) of this distribution evaluated at the specified point x. In general, the PDF is the derivative of the CDF. If the derivative does not exist at x, then an appropriate replacement should be returned, e.g. Double.POSITIVE_INFINITY, Double.NaN, or the limit inferior or limit superior of the difference quotient.

Parameters:

x - the point at which the PDF is evaluated

Returns:

the value of the probability density function at point x

inverseCumulativeProbability

public double inverseCumulativeProbability(double p)
                                    throws OutOfRangeException

Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is

• inf{x in R | P(X<=x) >= p} for 0 < p <= 1,
• inf{x in R | P(X<=x) > 0} for p = 0.

The default implementation returns

• RealDistribution.getSupportLowerBound() for p = 0,
• RealDistribution.getSupportUpperBound() for p = 1.

Returns Double.NEGATIVE_INFINITY when p == 0 and Double.POSITIVE_INFINITY when p == 1.

Specified by:

inverseCumulativeProbability in interface RealDistribution

Overrides:

inverseCumulativeProbability in class AbstractRealDistribution

Parameters:

p - the cumulative probability

Returns:

the smallest p-quantile of this distribution (largest 0-quantile for p = 0)

Throws:

OutOfRangeException - if p < 0 or p > 1

getSolverAbsoluteAccuracy

protected double getSolverAbsoluteAccuracy()

Returns the solver absolute accuracy for inverse cumulative computation. You can override this method in order to use a Brent solver with an absolute accuracy different from the default.

Overrides:

getSolverAbsoluteAccuracy in class AbstractRealDistribution

Returns:

the maximum absolute error in inverse cumulative probability estimates

getNumericalMean

public double getNumericalMean()

Use this method to get the numerical value of the mean of this distribution. The mean is always undefined no matter the parameters.

Returns:

mean (always Double.NaN)

getNumericalVariance

public double getNumericalVariance()

Use this method to get the numerical value of the variance of this distribution. The variance is always undefined no matter the parameters.

Returns:

variance (always Double.NaN)

getSupportLowerBound

public double getSupportLowerBound()

Access the lower bound of the support. This method must return the same value as inverseCumulativeProbability(0). In other words, this method must return

inf {x in R | P(X <= x) > 0}.

The lower bound of the support is always negative infinity no matter the parameters.

Returns:

lower bound of the support (always Double.NEGATIVE_INFINITY)

getSupportUpperBound

public double getSupportUpperBound()

Access the upper bound of the support. This method must return the same value as inverseCumulativeProbability(1). In other words, this method must return

inf {x in R | P(X <= x) = 1}.

The upper bound of the support is always positive infinity no matter the parameters.

Returns:

upper bound of the support (always Double.POSITIVE_INFINITY)

isSupportLowerBoundInclusive

public boolean isSupportLowerBoundInclusive()

Use this method to get information about whether the lower bound of the support is inclusive or not.

Returns:

whether the lower bound of the support is inclusive or not

isSupportUpperBoundInclusive

public boolean isSupportUpperBoundInclusive()

Use this method to get information about whether the upper bound of the support is inclusive or not.

Returns:

whether the upper bound of the support is inclusive or not

isSupportConnected

public boolean isSupportConnected()

Use this method to get information about whether the support is connected, i.e. whether all values between the lower and upper bound of the support are included in the support. The support of this distribution is connected.

Returns:

true