BernsteinCopulaFactory¶

class BernsteinCopulaFactory(*args)¶

BernsteinCopula copula factory.

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

See also

DistributionFactory, EmpiricalBernsteinCopula

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.

__init__(*args)¶

static ComputeAMISEBinNumber(sample)¶

Compute the optimal AMISE number of bins.

Parameters

sample2-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

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

where $\lfloor x \rfloor$ is the largest integer less than or equal to $x$ , $n$ the sample size and $d$ the sample dimension.

static ComputeLogLikelihoodBinNumber(*args)¶

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

Parameters

sample2-d sequence of float, of dimension 1: The sample of size $n$ from which the optimal log-likelihood bin number is computed.
kFractionint, $0<kFraction<n$: 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

sample2-d sequence of float, of dimension 1: The sample of size $n$ from which the optimal AMISE bin number is computed.
fFunction: The function defining the Csiszar divergence of interest.
alphafloat, $\alpha\geq 0$: 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

sample2-d sequence of float, of dimension d: The sample of size $n>0$ from which the copula is estimated.
methodstr: The name of the bin number selection method. Possible choices are AMISE, LogLikelihood and PenalizedCsiszarDivergence. Default is LogLikelihood, given by the ‘BernsteinCopulaFactory-BinNumberSelection’ entry of ResourceMap.
mint: The number of sub-intervals in which all the edges of the unit cube $[0, 1]^d$ are regularly partitioned.

Returns

copulaDistribution: 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

sample2-d sequence of float, of dimension d: The sample of size $n>0$ from which the copula is estimated.
methodstr: The name of the bin number selection method. Possible choices are AMISE, LogLikelihood and PenalizedCsiszarDivergence. Default is LogLikelihood, given by the ‘BernsteinCopulaFactory-BinNumberSelection’ entry of ResourceMap.
mint: The number of sub-intervals in which all the edges of the unit cube $[0, 1]^d$ are regularly partitioned.

Returns

copulaEmpiricalBernsteinCopula: The estimated copula as an empirical Bernstein copula.

buildEstimator(*args)¶

Build the distribution and the parameter distribution.

Parameters

sample2-d sequence of float: Sample from which the distribution parameters are estimated.
parametersDistributionParameters: Optional, the parametrization.

Returns

resDistDistributionFactoryResult: 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.

getBootstrapSize()¶

Accessor to the bootstrap size.

Returns

sizeinteger: Size of the bootstrap.

getClassName()¶

Accessor to the object’s name.

Returns

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

getId()¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

getName()¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns

idint: Internal unique identifier.

getVisibility()¶

Accessor to the object’s visibility state.

Returns

visiblebool: Visibility flag.

hasName()¶

Test if the object is named.

Returns

hasNamebool: True if the name is not empty.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns

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

setBootstrapSize(bootstrapSize)¶

Accessor to the bootstrap size.

Parameters

sizeinteger: Size of the bootstrap.

setName(name)¶

Accessor to the object’s name.

Parameters

namestr: The name of the object.

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters

idint: Internal unique identifier.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters

visiblebool: Visibility flag.

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