Package dev.nm.stat.evt.evd.univariate
Class GeneralizedEVD
- java.lang.Object
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- dev.nm.stat.evt.evd.univariate.GeneralizedEVD
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- 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 areevd::dgev,evd::pgev,evd::qgev,evd::mtransform.
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Constructor Summary
Constructors Constructor 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.
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Method Summary
All Methods Instance Methods Concrete Methods Modifier and Type Method Description doublecdf(double x)Gets the cumulative probability F(x) = Pr(X ≤ x).doubledensity(double x)The density function, which, if exists, is the derivative of F.doubleentropy()Gets the entropy of this distribution.doublegetLocation()Get the location parameter.doublegetScale()Get the scale parameter.doublegetShape()Get the shape parameter.doublekurtosis()Gets the excess kurtosis of this distribution.doublelogDensity(double x)Get the logarithm of the probability density function at \(x\), that is, \(\log(f(x))\).doublemarginalInverseTransform(double x)Inverse of marginal transform.doublemarginalTransform(double x)Transform to exponential margins under the GEV model.doublemean()Gets the mean of this distribution.doublemedian()Gets the median of this distribution.doublemoment(double x)The moment generating function is the expected value of etX.doublequantile(double p)Gets the quantile, the inverse of the cumulative distribution function.doubleskew()Gets the skewness of this distribution.doublevariance()Gets the variance of this distribution.
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Constructor Detail
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GeneralizedEVD
public GeneralizedEVD()
Create an instance of generalized extreme value distribution with the default parameter values: location \(\mu=0\), scale \(\sigma=1\), shape \(\xi=0\).
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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 \(\xi\)
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Method Detail
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getLocation
public double getLocation()
Get the location parameter.- Returns:
- \(\mu\)
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getScale
public double getScale()
Get the scale parameter.- Returns:
- \(\sigma\)
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getShape
public double getShape()
Get the shape parameter.- Returns:
- \(\xi\)
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marginalTransform
public double marginalTransform(double x)
Transform to exponential margins under the GEV model. That is, /[ t(x) = \begin{cases} \big(1+(\tfrac{x-\mu}{\sigma})\xi\big)^{-1/\xi} \textrm{if}\ \xi\neq0 \\ e^{-(x-\mu)/\sigma} \textrm{if}\ \xi=0 \end{cases} /]- Parameters:
x- \(x\)- Returns:
- \(t(x)\)
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marginalInverseTransform
public double marginalInverseTransform(double x)
Inverse of marginal transform.- Parameters:
x- \(x\)- Returns:
- \(t^{-1}(x)\)
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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:
cdfin interfaceProbabilityDistribution- Parameters:
x- \(x\)- Returns:
- \(F(x)\)
- See Also:
- Wikipedia: Cumulative distribution function
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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:
densityin interfaceProbabilityDistribution- Parameters:
x- \(x\)- Returns:
- \(f(x)\)
- See Also:
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logDensity
public double logDensity(double x)
Description copied from interface:UnivariateEVDGet the logarithm of the probability density function at \(x\), that is, \(\log(f(x))\).- Specified by:
logDensityin interfaceUnivariateEVD- Parameters:
x- \(x\)- Returns:
- \(\log(f(x))\)
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quantile
public double quantile(double p)
Description copied from interface:ProbabilityDistributionGets the quantile, the inverse of the cumulative distribution function. It is the value below which random draws from the distribution would fall u×100 percent of the time.
This may not always exist.F-1(u) = x, such that Pr(X ≤ x) = u
- Specified by:
quantilein interfaceProbabilityDistribution- Parameters:
p-u, a quantile- Returns:
- F-1(u)
- See Also:
- Wikipedia: Quantile function
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mean
public double mean()
Description copied from interface:ProbabilityDistributionGets the mean of this distribution.- Specified by:
meanin interfaceProbabilityDistribution- Returns:
- the mean
- See Also:
- Wikipedia: Expected value
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moment
public double moment(double x)
Description copied from interface:ProbabilityDistributionThe moment generating function is the expected value of etX. That is,E(etX)
This may not always exist.- Specified by:
momentin interfaceProbabilityDistribution- Parameters:
x- t- Returns:
- E(exp(tX))
- See Also:
- Wikipedia: Moment-generating function
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skew
public double skew()
Description copied from interface:ProbabilityDistributionGets the skewness of this distribution.- Specified by:
skewin interfaceProbabilityDistribution- Returns:
- the skewness
- See Also:
- Wikipedia: Skewness
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variance
public double variance()
Description copied from interface:ProbabilityDistributionGets the variance of this distribution.- Specified by:
variancein interfaceProbabilityDistribution- Returns:
- the variance
- See Also:
- Wikipedia: Variance
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median
public double median()
Description copied from interface:ProbabilityDistributionGets the median of this distribution.- Specified by:
medianin interfaceProbabilityDistribution- Returns:
- the median
- See Also:
- Wikipedia: Median
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kurtosis
public double kurtosis()
Description copied from interface:ProbabilityDistributionGets the excess kurtosis of this distribution.- Specified by:
kurtosisin interfaceProbabilityDistribution- Returns:
- the excess kurtosis
- See Also:
- Wikipedia: Kurtosis
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entropy
public double entropy()
Description copied from interface:ProbabilityDistributionGets the entropy of this distribution.- Specified by:
entropyin interfaceProbabilityDistribution- Returns:
- the entropy
- See Also:
- Wikipedia: Entropy (information theory)
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