NonStationaryCovarianceModelFactory

(Source code, png)

../../_images/NonStationaryCovarianceModelFactory.png
class NonStationaryCovarianceModelFactory

Estimation of a non stationary covariance model.

Refer to Estimation of a non stationary covariance model.

Notes

We consider X: \Omega \times \cD \rightarrow \Rset^d be a multivariate process of dimension d where \cD \in \Rset^n. We denote (\vect{t}_0, \dots, \vect{t}_{N-1}) the vertices of the mesh \cM \in \cD.

X is supposed to be a second order process and we note C : \cD \times \cD \rightarrow \mathcal{M}_{d \times d}(\mathbb{R}) its covariance function. X may be stationary or non stationary as well.

We suppose that we have K fields and we note (\vect{x}_0^k, \dots, \vect{x}_{N-1}^k) the values of the field k on the mesh \cM.

We recall that the covariance function C writes:

\forall (\vect{s}, \vect{t}) \in \cD \times \cD, \quad C(\vect{s}, \vect{t}) = \Expect{\left(X_{\vect{s}}-m(\vect{s})\right)\Tr{\left(X_{\vect{t}}-m(\vect{t})\right)}}

where the mean function m: \cD \rightarrow \Rset^d is defined by:

\forall \vect{t}\in \cD , \quad m(\vect{t}) = \Expect{X_{\vect{t}}}

First, we estimate the covariance function C on the vertices of the mesh \cM using the empirical mean estimator:

\begin{eqnarray*} & \forall \vect{t}_i \in \cM, \quad m(\vect{t}_i) \simeq \frac{1}{K} \sum_{k=1}^{K} \vect{x}_i^k \\ & \forall (\vect{t}_i, \vect{t}_j) \in \cD \times \cD, \quad C(\vect{t}_i, \vect{t}_j) \simeq \frac{1}{K} \sum_{k=1}^{K} \left( \vect{x}_i^k - m(\vect{t}_i) \right) \Tr{\left( \vect{x}_j^k - m(\vect{t}_j) \right)} \end{eqnarray*}

Then, we build a covariance function defined on \cD \times \cD which is a piecewise constant function defined on \cD \times \cD by:

\forall (\vect{s}, \vect{t}) \in \cD \times \cD, \, C(\vect{s}, \vect{t}) = C(\vect{t}_k, \vect{t}_l)

where k is such that \vect{t}_k is the vertex of \cM the nearest to \vect{s} and \vect{t}_l the nearest to \vect{t}.

Methods

build(*args)

Estimate the covariance model.

buildAsCovarianceMatrix(sample[, isCentered])

Estimate the covariance model as a covariance matrix.

buildAsUserDefinedCovarianceModel(sample[, ...])

Estimate the covariance model as a User defined covariance model.

getClassName()

Accessor to the object's name.

getName()

Accessor to the object's name.

hasName()

Test if the object is named.

setName(name)

Accessor to the object's name.
__init__()
build(*args)

Estimate the covariance model.

Parameters:
sampleFieldsProcessSample

The fields used to estimate the covariance model which is not supposed to be stationary.

Returns:
covEstCovarianceModel

The estimated covariance model.

Examples

Create the covariance model, a mesh and a process:

>>> import openturns as ot
>>> myModel = ot.AbsoluteExponential([0.1]*2)
>>> myMesh = ot.IntervalMesher([10]*2).build(ot.Interval([0.0]*2, [1.0]*2))
>>> myProcess = ot.GaussianProcess(myModel, myMesh)

Generate 10 fields:

>>> mySample = myProcess.getSample(10)

Estimate the covariance model without supposing the stationarity:

>>> myEstCov = ot.NonStationaryCovarianceModelFactory().build(mySample)
buildAsCovarianceMatrix(sample, isCentered=False)

Estimate the covariance model as a covariance matrix.

Parameters:
sampleFieldsProcessSample

The fields used to estimate the covariance model.

isCenteredbool, optional

Flag telling if the given sample is from a centered process or if it has to be centered by the empirical mean. Default value is False.

Returns:
covEstCovarianceMatrix

The unbiased estimation of the discretization of the covariance function over the mesh defining the given sample.

buildAsUserDefinedCovarianceModel(sample, isCentered=False)

Estimate the covariance model as a User defined covariance model.

Parameters:
sampleFieldsProcessSample

The fields used to estimate the covariance model.

isCenteredbool, optional

Flag telling if the given sample is from a centered process or if it has to be centered by the empirical mean. Default value is False.

Returns:
covEstUserDefinedCovarianceModel

The estimated covariance model that can be used as a UserDefinedCovarianceModel.

getClassName()

Accessor to the object’s name.

Returns:
class_namestr

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

getName()

Accessor to the object’s name.

Returns:
namestr

The name of the object.

hasName()

Test if the object is named.

Returns:
hasNamebool

True if the name is not empty.

setName(name)

Accessor to the object’s name.

Parameters:
namestr

The name of the object.

Examples using the class

Estimate a non stationary covariance function

Estimate a non stationary covariance function