GeneralizedEVD (NMDEV 1.2.9 API)

java.lang.Object
- dev.nm.stat.evt.evd.univariate.GeneralizedEVD

All Implemented Interfaces:

ProbabilityDistribution, UnivariateEVD

Direct Known Subclasses:

FrechetDistribution, GumbelDistribution, ReversedWeibullDistribution
```
public class GeneralizedEVD
extends Object
implements UnivariateEVD
```
Generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. By the extreme value theorem the GEV distribution is the limit distribution of properly normalized maxima of a sequence of independent and identically distributed (IID) random variables.
The R equivalent functions are evd::dgev, evd::pgev, evd::qgev, evd::mtransform.
Calling the GEV model

Constructor arguments are location, strictly positive scale, and shape, in that order. Shape zero gives the Gumbel case. For nonzero shape, the interior support satisfies 1 + shape * (x - location) / scale > 0. The quantile method requires a probability strictly between zero and one.

The CDF is zero at or below a finite lower endpoint (positive shape), and one at or above a finite upper endpoint (negative shape). The Gumbel case has support on the entire real line. A NaN argument produces NaN.

Example
```
 import dev.nm.stat.evt.evd.univariate.GeneralizedEVD;

 GeneralizedEVD gev = new GeneralizedEVD(0.0, 1.0, 0.0);
 double p = 0.95;
 double cutoff = gev.quantile(p);
 assert Math.abs(gev.cdf(cutoff) - p) < 1e-12;
 assert Math.abs(cutoff + Math.log(-Math.log(p))) < 1e-12;
 
```
Probabilities outside finite support

Negative shape bounds maxima from above; positive shape bounds them from below. Cumulative probability keeps its endpoint value outside support.
```
 import dev.nm.stat.evt.evd.univariate.GeneralizedEVD;
 GeneralizedEVD upperBounded = new GeneralizedEVD(0, 1, -0.5);
 assert upperBounded.cdf(2) == 1.0;
 assert upperBounded.cdf(3) == 1.0;
 GeneralizedEVD lowerBounded = new GeneralizedEVD(0, 1, 0.5);
 assert lowerBounded.cdf(-2) == 0.0;
 assert lowerBounded.cdf(-3) == 0.0;
 
```
See Also:

Wikipedia: Generalized extreme value distribution, Wikipedia: Fisher-Tippett-Gnedenko theorem

Constructor Summary

Constructors
Constructor and Description
`GeneralizedEVD()` Create an instance of generalized extreme value distribution with the default parameter values: location \(\mu=0\), scale \(\sigma=1\), shape \(\xi=0\).
`GeneralizedEVD(double location, double scale, double shape)` Create an instance of generalized extreme value distribution with the given parameters.

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`double`	`cdf(double x)` Gets the cumulative probability F(x) = Pr(X ≤ x).
`double`	`density(double x)` The density function, which, if exists, is the derivative of F.
`double`	`entropy()` Gets the entropy of this distribution.
`double`	`getLocation()` Get the location parameter.
`double`	`getScale()` Get the scale parameter.
`double`	`getShape()` Get the shape parameter.
`double`	`kurtosis()` Gets the excess kurtosis of this distribution.
`double`	`logDensity(double x)` Get the logarithm of the probability density function at \(x\), that is, \(\log(f(x))\).
`double`	`marginalInverseTransform(double x)` Inverse of marginal transform.
`double`	`marginalTransform(double x)` Transform to exponential margins under the GEV model.
`double`	`mean()` Gets the mean of this distribution.
`double`	`median()` Gets the median of this distribution.
`double`	`moment(double x)` The moment generating function is the expected value of \(e^{tX}\).
`double`	`quantile(double p)` Returns the GEV quantile for an interior probability.
`double`	`skew()` Gets the skewness of this distribution.
`double`	`variance()` Gets the variance of this distribution.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Constructor Detail
  - GeneralizedEVD
```
public GeneralizedEVD()
```
    Create an instance of generalized extreme value distribution with the default parameter values: location \(\mu=0\), scale \(\sigma=1\), shape \(\xi=0\).
  - GeneralizedEVD
```
public GeneralizedEVD(double location,
                      double scale,
                      double shape)
```
    Create an instance of generalized extreme value distribution with the given parameters.
    
    Parameters:
    
    location - the location parameter \(\mu\)
    
    scale - the scale parameter \(\sigma > 0\)
    
    shape - the shape parameter \(\xi\)
- Method Detail
  - getLocation
```
public double getLocation()
```
    Get the location parameter.
    
    Returns:
    
    \(\mu\)
  - getScale
```
public double getScale()
```
    Get the scale parameter.
    
    Returns:
    
    \(\sigma\)
  - getShape
```
public double getShape()
```
    Get the shape parameter.
    
    Returns:
    
    \(\xi\)
  - marginalTransform
```
public double marginalTransform(double x)
```
    Transform to exponential margins under the GEV model. That is, \[ t(x) = \begin{cases} \left(1+\xi\frac{x-\mu}{\sigma}\right)^{-1/\xi}, & \xi\ne0, \\ \exp\!\left(-\frac{x-\mu}{\sigma}\right), & \xi=0. \end{cases} \]
    
    Parameters:
    
    x - \(x\)
    
    Returns:
    
    \(t(x)\)
  - marginalInverseTransform
```
public double marginalInverseTransform(double x)
```
    Inverse of marginal transform.
    
    Parameters:
    
    x - \(x\)
    
    Returns:
    
    \(t^{-1}(x)\)
  - cdf
```
public double cdf(double x)
```
    Gets the cumulative probability F(x) = Pr(X ≤ x). The cumulative distribution function of GEV distribution is \[ F(x;\mu,\sigma,\xi) = \exp\left\{-\left[1+\xi\left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi}\right\} \] for \(1+\xi(x-\mu)/\sigma>0\).
    
    Specified by:
    
    cdf in interface ProbabilityDistribution
    
    Parameters:
    
    x - \(x\)
    
    Returns:
    
    \(F(x)\)
    
    See Also:
    
    Wikipedia: Cumulative distribution function
  - density
```
public double density(double x)
```
    The density function, which, if exists, is the derivative of F. It describes the density of probability at each point in the sample space. \[ f(x) = dF(X) / dx \] This may not always exist.
    For the discrete cases, this is the probability mass function. It gives the probability that a discrete random variable is exactly equal to some value. The probability density function of GEV distribution is \[ f(x;\mu,\sigma,\xi) = \frac{1}{\sigma}\left[1+\xi\left(\frac{x-\mu}{\sigma}\right)\right]^{(-1/\xi)-1} \exp\left\{-\left[1+\xi\left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi}\right\} \] for \(1+\xi(x-\mu)/\sigma>0\).
    Specified by:
    
    density in interface ProbabilityDistribution
    
    Parameters:
    
    x - \(x\)
    
    Returns:
    
    \(f(x)\)
    
    See Also:
    Wikipedia: Probability density function
    
    Wikipedia: Probability mass function
  - logDensity
```
public double logDensity(double x)
```
    Description copied from interface: UnivariateEVD
    
    Get the logarithm of the probability density function at \(x\), that is, \(\log(f(x))\).
    
    Specified by:
    
    logDensity in interface UnivariateEVD
    
    Parameters:
    
    x - \(x\)
    
    Returns:
    
    \(\log(f(x))\)
  - quantile
```
public double quantile(double p)
```
    Returns the GEV quantile for an interior probability.
    
    Specified by:
    
    quantile in interface ProbabilityDistribution
    
    Parameters:
    
    p - probability strictly between zero and one
    
    Returns:
    
    the quantile in the same units as location and scale
    
    Throws:
    
    IllegalArgumentException - if p is outside the open interval (0, 1)
    
    See Also:
    
    Wikipedia: Quantile function
  - mean
```
public double mean()
```
    Description copied from interface: ProbabilityDistribution
    
    Gets the mean of this distribution.
    
    Specified by:
    
    mean in interface ProbabilityDistribution
    
    Returns:
    
    the mean
    
    See Also:
    
    Wikipedia: Expected value
  - moment
```
public double moment(double x)
```
    Description copied from interface: ProbabilityDistribution
    
    The moment generating function is the expected value of \(e^{tX}\). That is, \[ E(e^{tX}) \] This may not always exist.
    
    Specified by:
    
    moment in interface ProbabilityDistribution
    
    Parameters:
    
    x - t
    
    Returns:
    
    E(exp(tX))
    
    See Also:
    
    Wikipedia: Moment-generating function
  - skew
```
public double skew()
```
    Description copied from interface: ProbabilityDistribution
    
    Gets the skewness of this distribution.
    
    Specified by:
    
    skew in interface ProbabilityDistribution
    
    Returns:
    
    the skewness
    
    See Also:
    
    Wikipedia: Skewness
  - variance
```
public double variance()
```
    Description copied from interface: ProbabilityDistribution
    
    Gets the variance of this distribution.
    
    Specified by:
    
    variance in interface ProbabilityDistribution
    
    Returns:
    
    the variance
    
    See Also:
    
    Wikipedia: Variance
  - median
```
public double median()
```
    Description copied from interface: ProbabilityDistribution
    
    Gets the median of this distribution.
    
    Specified by:
    
    median in interface ProbabilityDistribution
    
    Returns:
    
    the median
    
    See Also:
    
    Wikipedia: Median
  - kurtosis
```
public double kurtosis()
```
    Description copied from interface: ProbabilityDistribution
    
    Gets the excess kurtosis of this distribution.
    
    Specified by:
    
    kurtosis in interface ProbabilityDistribution
    
    Returns:
    
    the excess kurtosis
    
    See Also:
    
    Wikipedia: Kurtosis
  - entropy
```
public double entropy()
```
    Description copied from interface: ProbabilityDistribution
    
    Gets the entropy of this distribution.
    
    Specified by:
    
    entropy in interface ProbabilityDistribution
    
    Returns:
    
    the entropy
    
    See Also:
    
    Wikipedia: Entropy (information theory)

Class GeneralizedEVD

Calling the GEV model

Example

Probabilities outside finite support

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

GeneralizedEVD

GeneralizedEVD

Method Detail

getLocation

getScale

getShape

marginalTransform

marginalInverseTransform

cdf

density

logDensity

quantile

mean

moment

skew

variance

median

kurtosis

entropy