BernsteinCopulaFactory

class BernsteinCopulaFactory(*args)

BernsteinCopula copula factory.

Available constructors:
BernsteinCopulaFactory()

This class allows to estimate a copula in a nonparametric way as an EmpiricalBernsteinCopula.

Attributes:
thisown

The membership flag

Methods

ComputeAMISEBinNumber(sample) Compute the optimal AMISE number of bins.
ComputeLogLikelihoodBinNumber(*args) Compute the optimal log-likelihood number of bins by cross-validation.
ComputePenalizedCsiszarDivergenceBinNumber(*args) Compute the optimal penalized Csiszar divergence number of bins.
build(*args) Build the nonparametric Bernstein copula estimator based on the empirical copula.
buildAsEmpiricalBernsteinCopula(*args) Build the nonparametric Bernstein copula estimator based on the empirical copula.
buildEstimator(*args) Build the distribution and the parameter distribution.
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.
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.
setShadowedId(id) Accessor to the object’s shadowed id.
setVisibility(visible) Accessor to the object’s visibility state.
BuildCrossValidationSamples  
__init__(*args)

Initialize self. See help(type(self)) for accurate signature.

static ComputeAMISEBinNumber(sample)

Compute the optimal AMISE number of bins.

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

The sample from which the optimal AMISE bin number is computed.

Notes

The number of bins is computed by minimizing the asymptotic mean integrated squared error (AMISE), leading to

m = 1+\left\lfloor n^{\dfrac{2}{4+n}} \right\rfloor

where \lfloor x \rfloor is the largest integer less than or equal to x.

static ComputeLogLikelihoodBinNumber(*args)

Compute the optimal log-likelihood number of bins by cross-validation.

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

The sample of size n from which the optimal log-likelihood bin number is computed.

kFraction : int, 0<kFraction<n

The fraction of the sample used for the validation.

Notes

Let \cE=\left\{\vect{X}_1,\dots,\vect{X}_n\right\} be the given sample. If kFraction=1, the bin number m is given by:

m = \argmin_{M\in\{1,\dots,n\}}\dfrac{1}{n}\sum_{\vect{X}_i\in\cE}-\log c^{\cE}_{M}(\vect{X}_i)

where c_M^{\cE} is the density function of the EmpiricalBernsteinCopula associated to the sample \cE and the bin number M.

If kFraction>1, the bin number m is given by:

m = \argmin_{M\in\{1,\dots,n\}}\dfrac{1}{kFraction}\sum_{k=0}^{kFraction-1}\dfrac{1}{n}\sum_{\vect{X}_i\in\cE^V_k}-\log c^{\cE^L_k}_{M}(\vect{X}_i)

where \cE^V_k=\left\{\vect{X}_i\in\cE\,|\,i\equiv k \mod kFraction\right\} and \cE^L_k=\cE \backslash \cE^V_k

static ComputePenalizedCsiszarDivergenceBinNumber(*args)

Compute the optimal penalized Csiszar divergence number of bins.

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

The sample of size n from which the optimal AMISE bin number is computed.

f : Function

The function defining the Csiszar divergence of interest.

alpha : float, \alpha\geq 0

The penalization factor.

Notes

Let \cE=\left\{\vect{X}_1,\dots,\vect{X}_n\right\} be the given sample. The bin number m is given by:

m = \argmin_{M\in\{1,\dots,n\}}\left[\hat{D}_f(c^{\cE}_{M})-\dfrac{1}{n}\sum_{\vect{X}_i\in\cE}f\left(\dfrac{1}{c^{\cE}_{M}(\vect{X}_i)}\right)\right]^2-[\rho_S(c^{\cE}_{M})-\rho_S({\cE}_{M})]^2

where c_M^{\cE} is the density function of the EmpiricalBernsteinCopula associated to the sample \cE and the bin number M, \hat{D}_f(c^{\cE}_{M})=\dfrac{1}{N}\sum_{j=1}^Nf\left(\dfrac{1}{\vect{U}_j}\right) a Monte Carlo estimate of the Csiszar f divergence, \rho_S(c^{\cE}_{M}) the exact Spearman correlation of the empirical Bernstein copula c^{\cE}_{M} and \rho_S({\cE}_{M}) the empirical Spearman correlation of the sample {\cE}_{M}.

The parameter N is controlled by the ‘BernsteinCopulaFactory-SamplingSize’ key in ResourceMap.

build(*args)

Build the nonparametric Bernstein copula estimator based on the empirical copula.

Available usages:

build()

build(sample)

build(sample, method, objective)

build(sample, m)

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

The sample of size n>0 from which the copula is estimated.

method : string

The name of thebin number selection method. Possible choices are AMISE, LogLikelihood and PenalizedCsiszarDivergence. Default is LogLikelihood, given by the ‘BernsteinCopulaFactory-BinNumberSelection’ entry of ResourceMap.

m : int

The number of sub-intervals in which all the edges of the unit cube [0, 1]^d are regularly partitioned.

Returns:
copula : Distribution

The estimated copula as a generic distribution.

buildAsEmpiricalBernsteinCopula(*args)

Build the nonparametric Bernstein copula estimator based on the empirical copula.

Available usages:

buildAsEmpiricalBernsteinCopula()

buildAsEmpiricalBernsteinCopula(sample)

buildAsEmpiricalBernsteinCopula(sample, method, objective)

buildAsEmpiricalBernsteinCopula(sample, m)

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

The sample of size n>0 from which the copula is estimated.

method : string

The name of thebin number selection method. Possible choices are AMISE, LogLikelihood and PenalizedCsiszarDivergence. Default is LogLikelihood, given by the ‘BernsteinCopulaFactory-BinNumberSelection’ entry of ResourceMap.

m : int

The number of sub-intervals in which all the edges of the unit cube [0, 1]^d are regularly partitioned.

Returns:
copula : EmpiricalBernsteinCopula

The estimated copula as an empirical Bernstein copula.

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())
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.

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.

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.

thisown

The membership flag