Apache Commons Math 3.6.1 · org.apache.commons.math3.distribution.LevyDistribution

Modifier and Type	Field and Description
private final double	c Scale parameter.
private final double	halfC Half of c (for calculations).
private final double	mu Location parameter.
private static final long	serialVersionUID Hides org.apache.commons.math3.distribution.AbstractRealDistribution.serialVersionUID. Serializable UID.

Access	Constructor and Description
public	LevyDistribution(final double location parameter mu, final double scale parameter c) Build a new instance.
public	LevyDistribution(final RandomGenerator random generator to be used for sampling rng, final double location mu, final double scale parameter c) Creates a LevyDistribution.

Method Summary

Modifier and Type	Method and Description
public double	cumulativeProbability(final double the point at which the CDF is evaluated x) Implements org.apache.commons.math3.distribution.RealDistribution.cumulativeProbability. For a random variable `X` whose values are distributed according to this distribution, this method returns `P(X <= x)`.
public double	density(final double the point at which the PDF is evaluated x) Implements org.apache.commons.math3.distribution.RealDistribution.density. Returns the probability density function (PDF) of this distribution evaluated at the specified point `x`.
public double	Returns: location parameter of the distribution getLocation() Get the location parameter of the distribution.
public double	getNumericalMean() Implements org.apache.commons.math3.distribution.RealDistribution.getNumericalMean. Use this method to get the numerical value of the mean of this distribution.
public double	getNumericalVariance() Implements org.apache.commons.math3.distribution.RealDistribution.getNumericalVariance. Use this method to get the numerical value of the variance of this distribution.
public double	Returns: scale parameter of the distribution getScale() Get the scale parameter of the distribution.
public double	getSupportLowerBound() Implements org.apache.commons.math3.distribution.RealDistribution.getSupportLowerBound. Access the lower bound of the support.
public double	getSupportUpperBound() Implements org.apache.commons.math3.distribution.RealDistribution.getSupportUpperBound. Access the upper bound of the support.
public double	inverseCumulativeProbability(final double the cumulative probability p) Overrides org.apache.commons.math3.distribution.AbstractRealDistribution.inverseCumulativeProbability. Implements org.apache.commons.math3.distribution.RealDistribution.inverseCumulativeProbability. Computes the quantile function of this distribution.
public boolean	isSupportConnected() Implements org.apache.commons.math3.distribution.RealDistribution.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.
public boolean	isSupportLowerBoundInclusive() Implements org.apache.commons.math3.distribution.RealDistribution.isSupportLowerBoundInclusive. Whether or not the lower bound of support is in the domain of the density function.
public boolean	isSupportUpperBoundInclusive() Implements org.apache.commons.math3.distribution.RealDistribution.isSupportUpperBoundInclusive. Whether or not the upper bound of support is in the domain of the density function.
public double	logDensity(double the point at which the PDF is evaluated x) Overrides org.apache.commons.math3.distribution.AbstractRealDistribution.logDensity. Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point `x`.

Inherited from org.apache.commons.math3.distribution.AbstractRealDistribution:: cumulativeProbability getSolverAbsoluteAccuracy probability probability reseedRandomGenerator sample sample

c	back to summary
private final double c Scale parameter.

halfC	back to summary
private final double halfC Half of c (for calculations).

mu	back to summary
private final double mu Location parameter.

serialVersionUID	back to summary
private static final long serialVersionUID Hides org.apache.commons.math3.distribution.AbstractRealDistribution.serialVersionUID. Serializable UID.

LevyDistribution	back to summary
public LevyDistribution(final double mu, final double c) Build a new instance. Note this constructor will implicitly create an instance of `Well19937c` as random generator to be used for sampling only (see `sample()` and `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. Parameters mu:double location parameter c:double scale parameter Since 3.4

LevyDistribution	back to summary
public LevyDistribution(final RandomGenerator rng, final double mu, final double c) Creates a LevyDistribution. Parameters rng:RandomGenerator random generator to be used for sampling mu:double location c:double scale parameter

Method Detail

cumulativeProbability back to summary

cumulativeProbability	back to summary
public double cumulativeProbability(final double x) Implements org.apache.commons.math3.distribution.RealDistribution.cumulativeProbability. Doc from org.apache.commons.math3.distribution.RealDistribution.cumulativeProbability. 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. From Wikipedia: the cumulative distribution function is f(x; u, c) = erfc (√ (c / 2 (x - u ))) Parameters x:double the point at which the CDF is evaluated Returns:double the probability that a random variable with this distribution takes a value less than or equal to `x`

public double cumulativeProbability(final double x)

Implements org.apache.commons.math3.distribution.RealDistribution.cumulativeProbability.

Doc from org.apache.commons.math3.distribution.RealDistribution.cumulativeProbability.

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.

From Wikipedia: the cumulative distribution function is

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

Parameters

x:double: the point at which the CDF is evaluated

Returns:double

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

density back to summary

density	back to summary
public double density(final double x) Implements org.apache.commons.math3.distribution.RealDistribution.density. Doc from org.apache.commons.math3.distribution.RealDistribution.density. 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. 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. Parameters x:double the point at which the PDF is evaluated Returns:double the value of the probability density function at point `x`

public double density(final double x)

Implements org.apache.commons.math3.distribution.RealDistribution.density.

Doc from org.apache.commons.math3.distribution.RealDistribution.density.

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.

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.

Parameters

x:double: the point at which the PDF is evaluated

Returns:double

the value of the probability density function at point x

getLocation	back to summary
public double getLocation() Get the location parameter of the distribution. Returns:double location parameter of the distribution

getLocation

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public double getLocation()

Get the location parameter of the distribution.

Returns:double: location parameter of the distribution

getNumericalMean back to summary

getNumericalMean	back to summary
public double getNumericalMean() Implements org.apache.commons.math3.distribution.RealDistribution.getNumericalMean. Doc from org.apache.commons.math3.distribution.RealDistribution.getNumericalMean. Use this method to get the numerical value of the mean of this distribution. Returns:double the mean or `Double.NaN` if it is not defined

public double getNumericalMean()

Implements org.apache.commons.math3.distribution.RealDistribution.getNumericalMean.

Doc from org.apache.commons.math3.distribution.RealDistribution.getNumericalMean.

Use this method to get the numerical value of the mean of this distribution.

Returns:double: the mean or Double.NaN if it is not defined

getNumericalVariance back to summary

getNumericalVariance	back to summary
public double getNumericalVariance() Implements org.apache.commons.math3.distribution.RealDistribution.getNumericalVariance. Doc from org.apache.commons.math3.distribution.RealDistribution.getNumericalVariance. Use this method to get the numerical value of the variance of this distribution. Returns:double the variance (possibly `Double.POSITIVE_INFINITY` as for certain cases in `TDistribution`) or `Double.NaN` if it is not defined

public double getNumericalVariance()

Implements org.apache.commons.math3.distribution.RealDistribution.getNumericalVariance.

Doc from org.apache.commons.math3.distribution.RealDistribution.getNumericalVariance.

Use this method to get the numerical value of the variance of this distribution.

Returns:double: the variance (possibly Double.POSITIVE_INFINITY as for certain cases in TDistribution) or Double.NaN if it is not defined

getScale	back to summary
public double getScale() Get the scale parameter of the distribution. Returns:double scale parameter of the distribution

getScale

back to summary

public double getScale()

Get the scale parameter of the distribution.

Returns:double: scale parameter of the distribution

getSupportLowerBound back to summary

getSupportLowerBound	back to summary
public double getSupportLowerBound() Implements org.apache.commons.math3.distribution.RealDistribution.getSupportLowerBound. Doc from org.apache.commons.math3.distribution.RealDistribution.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}`. Returns:double lower bound of the support (might be `Double.NEGATIVE_INFINITY`)

public double getSupportLowerBound()

Implements org.apache.commons.math3.distribution.RealDistribution.getSupportLowerBound.

Doc from org.apache.commons.math3.distribution.RealDistribution.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}.

Returns:double: lower bound of the support (might be Double.NEGATIVE_INFINITY)

getSupportUpperBound back to summary

getSupportUpperBound	back to summary
public double getSupportUpperBound() Implements org.apache.commons.math3.distribution.RealDistribution.getSupportUpperBound. Doc from org.apache.commons.math3.distribution.RealDistribution.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}`. Returns:double upper bound of the support (might be `Double.POSITIVE_INFINITY`)

public double getSupportUpperBound()

Implements org.apache.commons.math3.distribution.RealDistribution.getSupportUpperBound.

Doc from org.apache.commons.math3.distribution.RealDistribution.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}.

Returns:double: upper bound of the support (might be Double.POSITIVE_INFINITY)

inverseCumulativeProbability back to summary

inverseCumulativeProbability	back to summary
public double inverseCumulativeProbability(final double p) throws OutOfRangeException Overrides org.apache.commons.math3.distribution.AbstractRealDistribution.inverseCumulativeProbability. Implements org.apache.commons.math3.distribution.RealDistribution.inverseCumulativeProbability. Doc from org.apache.commons.math3.distribution.AbstractRealDistribution.inverseCumulativeProbability. 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 `getSupportLowerBound()` for `p = 0`, `getSupportUpperBound()` for `p = 1`. Parameters p:double Doc from org.apache.commons.math3.distribution.RealDistribution.inverseCumulativeProbability. the cumulative probability Returns:double Doc from org.apache.commons.math3.distribution.RealDistribution.inverseCumulativeProbability. the smallest `p`-quantile of this distribution (largest 0-quantile for `p = 0`) Annotations @Override Exceptions OutOfRangeException: Doc from org.apache.commons.math3.distribution.RealDistribution.inverseCumulativeProbability. if `p < 0` or `p > 1`

public double inverseCumulativeProbability(final double p) throws OutOfRangeException

Overrides org.apache.commons.math3.distribution.AbstractRealDistribution.inverseCumulativeProbability.

Implements org.apache.commons.math3.distribution.RealDistribution.inverseCumulativeProbability.

Doc from org.apache.commons.math3.distribution.AbstractRealDistribution.inverseCumulativeProbability.

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

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

Parameters

p:double

Doc from org.apache.commons.math3.distribution.RealDistribution.inverseCumulativeProbability.

the cumulative probability

Returns:double

Doc from org.apache.commons.math3.distribution.RealDistribution.inverseCumulativeProbability.

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

Annotations

@Override

Exceptions

OutOfRangeException:

Doc from org.apache.commons.math3.distribution.RealDistribution.inverseCumulativeProbability.

if p < 0 or p > 1

isSupportConnected	back to summary
public boolean isSupportConnected() Implements org.apache.commons.math3.distribution.RealDistribution.isSupportConnected. Doc from org.apache.commons.math3.distribution.RealDistribution.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. Returns:boolean whether the support is connected or not

isSupportConnected

back to summary

public boolean isSupportConnected()

Implements org.apache.commons.math3.distribution.RealDistribution.isSupportConnected.

Doc from org.apache.commons.math3.distribution.RealDistribution.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.

Returns:boolean: whether the support is connected or not

isSupportLowerBoundInclusive back to summary

isSupportLowerBoundInclusive	back to summary
public boolean isSupportLowerBoundInclusive() Implements org.apache.commons.math3.distribution.RealDistribution.isSupportLowerBoundInclusive. Doc from org.apache.commons.math3.distribution.RealDistribution.isSupportLowerBoundInclusive. Whether or not the lower bound of support is in the domain of the density function. Returns true iff `getSupporLowerBound()` is finite and `density(getSupportLowerBound())` returns a non-NaN, non-infinite value. Returns:boolean true if the lower bound of support is finite and the density function returns a non-NaN, non-infinite value there

public boolean isSupportLowerBoundInclusive()

Implements org.apache.commons.math3.distribution.RealDistribution.isSupportLowerBoundInclusive.

Doc from org.apache.commons.math3.distribution.RealDistribution.isSupportLowerBoundInclusive.

Whether or not the lower bound of support is in the domain of the density function. Returns true iff getSupporLowerBound() is finite and density(getSupportLowerBound()) returns a non-NaN, non-infinite value.

Returns:boolean: true if the lower bound of support is finite and the density function returns a non-NaN, non-infinite value there

isSupportUpperBoundInclusive back to summary

isSupportUpperBoundInclusive	back to summary
public boolean isSupportUpperBoundInclusive() Implements org.apache.commons.math3.distribution.RealDistribution.isSupportUpperBoundInclusive. Doc from org.apache.commons.math3.distribution.RealDistribution.isSupportUpperBoundInclusive. Whether or not the upper bound of support is in the domain of the density function. Returns true iff `getSupportUpperBound()` is finite and `density(getSupportUpperBound())` returns a non-NaN, non-infinite value. Returns:boolean true if the upper bound of support is finite and the density function returns a non-NaN, non-infinite value there

public boolean isSupportUpperBoundInclusive()

Implements org.apache.commons.math3.distribution.RealDistribution.isSupportUpperBoundInclusive.

Doc from org.apache.commons.math3.distribution.RealDistribution.isSupportUpperBoundInclusive.

Whether or not the upper bound of support is in the domain of the density function. Returns true iff getSupportUpperBound() is finite and density(getSupportUpperBound()) returns a non-NaN, non-infinite value.

Returns:boolean: true if the upper bound of support is finite and the density function returns a non-NaN, non-infinite value there

logDensity back to summary

logDensity	back to summary
public double logDensity(double x) Overrides org.apache.commons.math3.distribution.AbstractRealDistribution.logDensity. Doc from org.apache.commons.math3.distribution.AbstractRealDistribution.logDensity. Returns the natural logarithm of 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. Note that due to the floating point precision and under/overflow issues, this method will for some distributions be more precise and faster than computing the logarithm of `density(double)`. The default implementation simply computes the logarithm of `density(x)`. See documentation of `density(double)` for computation details. Parameters x:double the point at which the PDF is evaluated Returns:double the logarithm of the value of the probability density function at point `x` Annotations @Override

public double logDensity(double x)

Overrides org.apache.commons.math3.distribution.AbstractRealDistribution.logDensity.

Doc from org.apache.commons.math3.distribution.AbstractRealDistribution.logDensity.

Returns the natural logarithm of 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. Note that due to the floating point precision and under/overflow issues, this method will for some distributions be more precise and faster than computing the logarithm of density(double). The default implementation simply computes the logarithm of density(x). See documentation of density(double) for computation details.

Parameters

x:double: the point at which the PDF is evaluated

Returns:double

the logarithm of the value of the probability density function at point x

Annotations

@Override

public Class LevyDistribution

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