BernsteinCopulaFactory

class BernsteinCopulaFactory(*args)

BernsteinCopula copula factory.

Available constructors:
BernsteinCopulaFactory()

Methods

build(*args) Build the nonparametric Bernstein copula estimator based on the empirical copula.
buildEstimator(*args) Build the distribution and the parameter distribution.
computeBinNumber(sample) Compute the optimal AMISE number of bins.
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.
isParallel()
setBootstrapSize(bootstrapSize) Accessor to the bootstrap size.
setName(name) Accessor to the object’s name.
setParallel(flag)
setShadowedId(id) Accessor to the object’s shadowed id.
setVisibility(visible) Accessor to the object’s visibility state.
__init__(*args)
build(*args)

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

Available usages:

build(sample)

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.

m : int

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

Notes

If not given, 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.

Then, the copula is estimated by a mixture of products of Beta distributions, ie its density function \hat{c} is given by:

\forall (u_1,\hdots,u_d)\in[0,1]^d,\quad\hat{c}(u_1,\hdots,u_d) = \frac{1}{n}\sum_{i=1}^n\prod_{j=1}^d\beta_{\nu^i_j+1, m+1}(u_j)

where \nu_j=\left\lfloor \dfrac{m}{n}\mathrm{rank}(X^i_j)\right\rfloor is the index of the bin to which the normalized rank of the component j of the observation j belongs and \beta_{r,t} is the density of the beta distribution supported by [0,1], see Beta.

This estimator is called the Bernstein estimator because of the fact that, for integer shape parameters r and t, the density function of the beta distribution is equal to the Bernstein polynomial P_{r,t}:

\beta_{r+1,t+1}(u)=(t+1)\binom{t}{r}u^r(1-u)^{t-r-1}=P_{r,t-r-1}(u)

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())
computeBinNumber(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.

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.