GeneralizedParetoFactory

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../../_images/openturns-GeneralizedParetoFactory-1.png
class GeneralizedParetoFactory(*args)

Generalized Pareto factory.

Available constructor:
GeneralizedParetoFactory()

Notes

OpenTURNS proposes several estimators to build a GeneralizedPareto distribution from a scalar sample (see [Matthys2003] for the theory).

Moments based estimator:

Lets denote:

  • \displaystyle \overline{x}_n = \frac{1}{n} \sum_{i=1}^n x_i the empirical mean of the sample,
  • \displaystyle s_n^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x}_n)^2 its empirical variance,

Then we estimate (\hat{\sigma}, \hat{\xi}) using:

(1)\hat{\xi} &= \dfrac{1}{2}\left(\dfrac{\overline{x}_n^2}{s_n^2}-1\right) \\
\hat{\sigma} &= \dfrac{\overline{x}_n}{2}\left(\dfrac{\overline{x}_n^2}{s_n^2}+1\right)

this estimator is well-defined only if \hat{\xi}>-1/4, otherwise the second moment does not exist.

Probability weighted moments based estimator:

Lets denote:

  • \left(x_{(i)}\right)_{i\in\{1,\dots,n\}} the sample sorted in ascending order
  • m=\dfrac{1}{n}\sum_{i=1}^n\left(1-\dfrac{i-7/20}{n}\right)x_{(i)}
  • \rho=\dfrac{m}{\overline{x}_n}

Then we estimate (\hat{\sigma}, \hat{\xi}) using:

(2)\hat{\xi} &= \dfrac{1-4\rho}{1-2\rho} \\
\hat{\sigma} &= \dfrac{2\overline{x}_n}{1-2\rho}

this estimator is well-defined only if \hat{\xi}>-1/2, otherwise the first moment does not exist.

Exponential regression based estimator:

Lets denote:

  • y_{i}=i\log\left(\dfrac{x_{(n-i)}-x_{(1)}}{x_{(n-i-1)}-x_{(1)}}\right) for i\in\{1,n-3\}

Then we estimate (\hat{\sigma}, \hat{\xi}) using:

(3)\hat{\xi} &= \xi^* \\
\hat{\sigma} &= \dfrac{2\overline{x}_n}{1-2\rho}

Where \xi^* maximizes:

(4)\sum_{i=1}^{n-2}\log\left(\dfrac{1-(j/n)^{\xi}}{\xi}\right)-\dfrac{1-(j/n)^{\xi}}{\xi}y_i

under the constraint -1 \leq \xi \leq 1.

Methods

build(*args) Build the distribution.
buildAsGeneralizedPareto(*args) Build the distribution as a GeneralizedPareto type.
buildEstimator(*args) Build the distribution and the parameter distribution.
buildMethodOfExponentialRegression(sample) Build the distribution based on the exponential regression estimator.
buildMethodOfMoments(sample) Build the distribution based on the method of moments estimator.
buildMethodOfProbabilityWeightedMoments(sample) Build the distribution based on the probability weighted moments estimator.
getBootstrapSize() Accessor to the bootstrap size.
getClassName() Accessor to the object’s name.
getId() Accessor to the object’s id.
getName() Accessor to the object’s name.
getOptimizationAlgorithm() Accessor to the solver.
getShadowedId() Accessor to the object’s shadowed id.
getVisibility() Accessor to the object’s visibility state.
hasName() Test if the object is named.
hasVisibleName() Test if the object has a distinguishable name.
setBootstrapSize(bootstrapSize) Accessor to the bootstrap size.
setName(name) Accessor to the object’s name.
setOptimizationAlgorithm(solver) Accessor to the solver.
setShadowedId(id) Accessor to the object’s shadowed id.
setVisibility(visible) Accessor to the object’s visibility state.
__init__(*args)

x.__init__(…) initializes x; see help(type(x)) for signature

build(*args)

Build the distribution.

Available usages:

build()

build(sample)

build(param)

Parameters:

sample : 2-d sequence of float, of dimension 1

The sample from which the distribution parameters are estimated.

param : Collection of PointWithDescription

A vector of parameters of the distribution.

Notes

In the first usage, the default GeneralizedPareto distribution is built.

In the second usage, the parameters are evaluated according the following strategy:

  • If the sample size is less or equal to GeneralizedParetoFactory-SmallSize from ResourceMap, then the method of probability weighted moments is used. If it fails, the method of exponential regression is used.
  • Otherwise, the first method tried is the method of exponential regression, then the method of probability weighted moments if the first one fails.

In the fourth usage, a GeneralizedPareto distribution corresponding to the given parameters is built.

buildAsGeneralizedPareto(*args)

Build the distribution as a GeneralizedPareto type.

Available usages:

build()

build(sample)

build(param)

Parameters:

sample : 2-d sequence of float, of dimension 1

The sample from which the distribution parameters are estimated.

param : sequence of float

A vector of parameters of the distribution.

buildEstimator(*args)

Build the distribution and the parameter distribution.

Parameters:

sample : 2-d sequence of float

Sample from which the distribution parameters are estimated.

parameters : DistributionParameters

Optional, the parametrization.

Returns:

resDist : DistributionFactoryResult

The results.

Notes

According to the way the native parameters of the distribution are estimated, the parameters distribution differs:

  • Moments method: the asymptotic parameters distribution is normal and estimated by Bootstrap on the initial data;
  • Maximum likelihood method with a regular model: the asymptotic parameters distribution is normal and its covariance matrix is the inverse Fisher information matrix;
  • Other methods: the asymptotic parameters distribution is estimated by Bootstrap on the initial data and kernel fitting (see KernelSmoothing).

If another set of parameters is specified, the native parameters distribution is first estimated and the new distribution is determined from it:

  • if the native parameters distribution is normal and the transformation regular at the estimated parameters values: the asymptotic parameters distribution is normal and its covariance matrix determined from the inverse Fisher information matrix of the native parameters and the transformation;
  • in the other cases, the asymptotic parameters distribution is estimated by Bootstrap on the initial data and kernel fitting.

Examples

Create a sample from a Beta distribution:

>>> import openturns as ot
>>> sample = ot.Beta().getSample(10)
>>> ot.ResourceMap.SetAsUnsignedInteger('DistributionFactory-DefaultBootstrapSize', 100)

Fit a Beta distribution in the native parameters and create a DistributionFactory:

>>> fittedRes = ot.BetaFactory().buildEstimator(sample)

Fit a Beta distribution in the alternative parametrization (\mu, \sigma, a, b):

>>> fittedRes2 = ot.BetaFactory().buildEstimator(sample, ot.BetaMuSigma())
buildMethodOfExponentialRegression(sample)

Build the distribution based on the exponential regression estimator.

Parameters:

sample : 2-d sequence of float, of dimension 1

The sample from which the distribution parameters are estimated.

buildMethodOfMoments(sample)

Build the distribution based on the method of moments estimator.

Parameters:

sample : 2-d sequence of float, of dimension 1

The sample from which the distribution parameters are estimated.

buildMethodOfProbabilityWeightedMoments(sample)

Build the distribution based on the probability weighted moments estimator.

Parameters:

sample : 2-d sequence of float, of dimension 1

The sample from which the distribution parameters are estimated.

getBootstrapSize()

Accessor to the bootstrap size.

Returns:

size : integer

Size of the bootstrap.

getClassName()

Accessor to the object’s name.

Returns:

class_name : str

The object class name (object.__class__.__name__).

getId()

Accessor to the object’s id.

Returns:

id : int

Internal unique identifier.

getName()

Accessor to the object’s name.

Returns:

name : str

The name of the object.

getOptimizationAlgorithm()

Accessor to the solver.

Returns:

solver : OptimizationAlgorithm

The solver used for numerical optimization of the likelihood.

getShadowedId()

Accessor to the object’s shadowed id.

Returns:

id : int

Internal unique identifier.

getVisibility()

Accessor to the object’s visibility state.

Returns:

visible : bool

Visibility flag.

hasName()

Test if the object is named.

Returns:

hasName : bool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns:

hasVisibleName : bool

True if the name is not empty and not the default one.

setBootstrapSize(bootstrapSize)

Accessor to the bootstrap size.

Parameters:

size : integer

Size of the bootstrap.

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setOptimizationAlgorithm(solver)

Accessor to the solver.

Parameters:

solver : OptimizationAlgorithm

The solver used for numerical optimization of the likelihood.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters:

id : int

Internal unique identifier.

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:

visible : bool

Visibility flag.