# TensorizedCovarianceModel¶

class TensorizedCovarianceModel(*args)

Multivariate covariance function defined as a tensorization of covariance models.

Parameters: collsequence of CovarianceModelCollection of covariance models of dimension .

Notes

The tensorized covariance model defines a multivariate covariance model of dimension from the tensorization of a given covariance models.

We consider the stochastic process , where is an event, is a domain of .

Its covariance function is defined from the collection of covariance functions where , as follows:

The amplitude of the covariance function is and each model is parameterized by its scale .

The method updates the scale the following way. Let be the initial scale of the covariance model . After the update, has the scale where .

Examples

Create a tensorized covariance function from the tensorization of an absolute exponential function, a squared exponential function and an exponential function:

>>> import openturns as ot
>>> inputDimension = 2


Create the each covariance models:

>>> myCov1 = ot.AbsoluteExponential([3.0] * inputDimension)
>>> myCov2 = ot.SquaredExponential(inputDimension *[2.0])

>>> amplitude= [4.0, 2.0]
>>> scale = [1.0, 1.0]
>>> spatialCorrelation = ot.CorrelationMatrix(inputDimension)
>>> spatialCorrelation[1,0] = 0.3
>>> myCov3 = ot.ExponentialModel(scale, amplitude, spatialCorrelation)


Define the scale of the tensorized model:

>>> scale = [0.3, 0.8]


Create the tensorized model:

>>> covarianceModel = ot.TensorizedCovarianceModel([myCov1, myCov2, myCov3], scale)


Fix the same scale to each model:

>>> covarianceModel.setScale([1.0]*inputDimension)

Attributes: thisownThe membership flag

Methods

 __call__(*args) Evaluate the covariance function. computeAsScalar(s, t) Compute the covariance function for scalar model. computeStandardRepresentative(s, t) Compute the standard representative function of the covariance model. discretize(*args) Discretize the covariance function on a given mesh. discretizeAndFactorize(*args) Discretize and factorize the covariance function on a given mesh. discretizeAndFactorizeHMatrix(*args) Discretize and factorize the covariance function on a given mesh. discretizeHMatrix(*args) Discretize the covariance function on a given mesh using HMatrix result. discretizeRow(vertices, p) (TODO) draw(*args) Draw a specific component of the covariance model with input dimension 1. getActiveParameter() Accessor to the active parameter set. getAmplitude() Get the amplitude parameter of the covariance function. getClassName() Accessor to the object’s name. getFullParameter() Get the full parameters of the covariance function. getFullParameterDescription() Get the description full parameters of the covariance function. getId() Accessor to the object’s id. getInputDimension() Get the input dimension of the covariance function. getMarginal(index) Get the ith marginal of the model. getName() Accessor to the object’s name. getNuggetFactor() Accessor to the nugget factor. getOutputCorrelation() Get the spatial correlation matrix of the covariance function. getOutputDimension() Get the dimension of the covariance function. getParameter() Get the parameters of the covariance function. getParameterDescription() Get the description of the covariance function parameters. getScale() Get the scale parameter of the covariance function. 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. isDiagonal() Test whether the model is diagonal or not. isStationary() Test whether the model is stationary or not. parameterGradient(s, t) Compute the gradient according to the parameters. partialGradient(s, t) Compute the gradient of the covariance function. setActiveParameter(active) Accessor to the active parameter set. setAmplitude(amplitude) Set the amplitude parameter of the covariance function. setFullParameter(parameter) Set the full parameters of the covariance function. setName(name) Accessor to the object’s name. setNuggetFactor(nuggetFactor) Set the nugget factor for the regularization. setOutputCorrelation(correlation) Set the spatial correlation matrix of the covariance function. setParameter(parameter) Set the parameters of the covariance function. setScale(scale) Set the scale parameter of the covariance function. setShadowedId(id) Accessor to the object’s shadowed id. setVisibility(visible) Accessor to the object’s visibility state.
 getCollection
__init__(*args)

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

computeAsScalar(s, t)

Compute the covariance function for scalar model.

Available usages:

computeAsScalar(s, t)

computeAsScalar(tau)

Parameters: s, tsequences of floatMultivariate index tausequence of floatMultivariate index covariancefloatCovariance.

Notes

The method makes sense only if the dimension of the process is . It evaluates .

In the second usage, the covariance model must be stationary. Then we note for as this quantity does not depend on .

computeStandardRepresentative(s, t)

Compute the standard representative function of the covariance model.

Available usages:

computeStandardRepresentative(s, t)

computeStandardRepresentative(tau)

Parameters: s, tsequences of floatMultivariate index taufloat or sequence of floatMultivariate index rhofloatCorrelation model

Notes

It evaluates the scalar function or if the model is stationary.

discretize(*args)

Discretize the covariance function on a given mesh.

Parameters: meshOrGridMesh or time grid of size associated with the process. covarianceMatrixCovarianceMatrixCovariance matrix (if the process is of dimension

Notes

This method makes a discretization of the model on meshOrGrid composed of the vertices and returns the covariance matrix:

discretizeAndFactorize(*args)

Discretize and factorize the covariance function on a given mesh.

Parameters: meshOrGridMesh or time grid of size associated with the process. CholeskyMatrixTriangularMatrixCholesky factor of the covariance matrix (if the process is of dimension ).

Notes

This method makes a discretization of the model on meshOrGrid composed of the vertices thanks to the discretize method and returns its Cholesky factor.

discretizeAndFactorizeHMatrix(*args)

Discretize and factorize the covariance function on a given mesh.

This uses HMatrix.

Parameters: meshOrGridMesh or time grid of size associated with the process. nuggetFactor: floatNugget factor to be added to the discretized matrix hmatParamHMatrixParametersParameter values for the HMatrix HMatrixHMatrixCholesk matrix (if the process is of dimension ), stored in hierarchical format (H-Matrix)

Notes

This method si similar to the discretizeAndFactorize method. This method requires that OpenTURNS has been compiled with the hmat library. The method is helpfull for very large parameters (Mesh, grid, Sample) as its compress data.

discretizeHMatrix(*args)

Discretize the covariance function on a given mesh using HMatrix result.

Parameters: meshOrGridMesh or time grid of size associated with the process. nuggetFactor: floatNugget factor to be added to the discretized matrix hmatParamHMatrixParametersParameter values for the HMatrix HMatrixHMatrixCovariance matrix (if the process is of dimension ), stored in hierarchical format (H-Matrix)

Notes

This method si similar to the discretize method. This method requires that OpenTURNS has been compiled with the hmat library. The method is helpfull for very large parameters (Mesh, grid, Sample) as its compress data.

discretizeRow(vertices, p)

(TODO)

draw(*args)

Draw a specific component of the covariance model with input dimension 1.

Parameters: rowIndexint, The row index of the component to draw. Default value is 0. columnIndex: int, :math:0 leq columnIndex < dimensionThe column index of the component to draw. Default value is 0. tMinfloatThe lower bound of the range over which the model is plotted. Default value is CovarianceModel-DefaultTMin in ResourceMap. tMaxfloatThe upper bound of the range over which the model is plotted. Default value is CovarianceModel-DefaultTMax in ResourceMap. pointNumberint, The discretization of the range over which the model is plotted. Default value is CovarianceModel-DefaultPointNumber in class:~openturns.ResourceMap. asStationaryboolFlag to tell if the model has to be plotted as a stationary model, ie as a function of the lag if equals to True, or as a non-stationary model, ie as a function of if equals to False. Default value is True. correlationFlagboolFlag to tell if the model has to be plotted as a correlation function if equals to True or as a covariance function if equals to False. Default value is False. graphGraphA graph containing a unique curve if asStationary=True and if the model is actually a stationary model, or containing the iso-values of the model if asStationary=False or if the model is nonstationary.
getActiveParameter()

Accessor to the active parameter set.

Returns: activeIndicesIndices of the active parameters.
getAmplitude()

Get the amplitude parameter of the covariance function.

Returns: amplitudePointThe amplitude parameter of the covariance function.
getClassName()

Accessor to the object’s name.

Returns: class_namestrThe object class name (object.__class__.__name__).
getFullParameter()

Get the full parameters of the covariance function.

Returns: parameterPointList the full parameter of the covariance function i.e. scale parameter , the the amplitude parameter , the Spatial correlation parameter ; and potential other parameter depending on the model;
getFullParameterDescription()

Get the description full parameters of the covariance function.

Returns: descriptionDescriptionDescription of the full parameter of the covariance function.
getId()

Accessor to the object’s id.

Returns: idintInternal unique identifier.
getInputDimension()

Get the input dimension of the covariance function.

Returns: inputDimensionintSpatial dimension of the covariance function.
getMarginal(index)

Get the ith marginal of the model.

Returns: marginalint or sequence of intindex of marginal of the model.
getName()

Accessor to the object’s name.

Returns: namestrThe name of the object.
getNuggetFactor()

Accessor to the nugget factor.

This parameter allows smooth predictions from noisy data. The nugget is added to the diagonal of the assumed training covariance (thanks to discretize) and acts as a Tikhonov regularization in the problem.

Returns: nuggetFactorfloatNugget factor used for the regularization of the discretized covariance matrix.
getOutputCorrelation()

Get the spatial correlation matrix of the covariance function.

Returns: spatialCorrelationCorrelationMatrixCorrelation matrix .
getOutputDimension()

Get the dimension of the covariance function.

Returns: dintDimension such that This is the dimension of the process .
getParameter()

Get the parameters of the covariance function.

Returns: parametersPoint List of the scale parameter and the amplitude parameter of the covariance function. The other specific parameters are not included.
getParameterDescription()

Get the description of the covariance function parameters.

Returns: descriptionParamDescriptionDescription of the components of the parameters obtained with the getParameter method..
getScale()

Get the scale parameter of the covariance function.

Returns: scalePointThe scale parameter used in the covariance function.
getShadowedId()

Accessor to the object’s shadowed id.

Returns: idintInternal unique identifier.
getVisibility()

Accessor to the object’s visibility state.

Returns: visibleboolVisibility flag.
hasName()

Test if the object is named.

Returns: hasNameboolTrue if the name is not empty.
hasVisibleName()

Test if the object has a distinguishable name.

Returns: hasVisibleNameboolTrue if the name is not empty and not the default one.
isDiagonal()

Test whether the model is diagonal or not.

Returns: isDiagonalboolTrue if the model is diagonal.
isStationary()

Test whether the model is stationary or not.

Returns: isStationaryboolTrue if the model is stationary.

Notes

The covariance function is stationary when it is invariant by translation:

We note for .

parameterGradient(s, t)

Compute the gradient according to the parameters.

Parameters: s, tsequences of floatMultivariate index . gradientMatrixGradient of the function according to the parameters.
partialGradient(s, t)

Compute the gradient of the covariance function.

Parameters: s, tfloats or sequences of floatMultivariate index . gradientMatrixGradient of the covariance function.
setActiveParameter(active)

Accessor to the active parameter set.

Parameters: activesequence of intIndices of the active parameters.
setAmplitude(amplitude)

Set the amplitude parameter of the covariance function.

Parameters: amplitudePointThe amplitude parameter to be used in the covariance function. Its size must be equal to the dimension of the covariance function.
setFullParameter(parameter)

Set the full parameters of the covariance function.

Parameters: parameterPointList the full parameter of the covariance function i.e. scale parameter , the the amplitude parameter , the Spatial correlation parameter ; and potential other parameter depending on the model; Must be at least of dimension .
setName(name)

Accessor to the object’s name.

Parameters: namestrThe name of the object.
setNuggetFactor(nuggetFactor)

Set the nugget factor for the regularization.

Acts on the discretized covariance matrix.

Parameters: nuggetFactorfloatnugget factor to be used for the regularization of the discretized covariance matrix.
setOutputCorrelation(correlation)

Set the spatial correlation matrix of the covariance function.

Parameters: spatialCorrelationCorrelationMatrixCorrelation matrix .
setParameter(parameter)

Set the parameters of the covariance function.

Parameters: parametersPointList of the scale parameter and the amplitude parameter of the covariance function. Must be of dimension .
setScale(scale)

Set the scale parameter of the covariance function.

Parameters: scalePointThe scale parameter to be used in the covariance function. Its size must be equal to the input dimension of the covariance function.
setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters: idintInternal unique identifier.
setVisibility(visible)

Accessor to the object’s visibility state.

Parameters: visibleboolVisibility flag.
thisown

The membership flag