# KroneckerCovarianceModel¶

class KroneckerCovarianceModel(*args)

Multivariate stationary Kronecker covariance function.

Available constructors:

KroneckerCovarianceModel(inputDimension=1)

KroneckerCovarianceModel(rho, amplitude)

KroneckerCovarianceModel(rho, amplitude, outputCorrelation)

KroneckerCovarianceModel(rho, outputCovariance)

Parameters:
inputDimensionint

Input dimension . By default, the input dimension is deduced from the parameter. If this one is not specified, then .

rhoCovarianceModel

Covariance model

amplitudesequence of positive floats

Amplitude of the process .

outputCorrelationCorrelationMatrix

Correlation matrix By default, where the dimension is deduced from the amplitude .

outputCovarianceCovarianceMatrix

Covariance matrix .

Notes

We consider the stochastic process , where is a sample space and is a subdomain of .

The KroneckerCovarianceModel class builds a covariance function with output dimension from a correlation function with output dimension and a covariance matrix. The latter is to be, for any , the covariance matrix of the random vector .

This covariance function is defined by

where the output covariance matrix is given by

Examples

Create a Kronecker covariance model with identity output correlation matrix.

>>> import openturns as ot
>>> scale = [4.0, 5.0]
>>> rho = ot.GeneralizedExponential(scale, 1.0)
>>> amplitude = [1.0, 2.0]
>>> myCovarianceModel = ot.KroneckerCovarianceModel(rho, amplitude)


Create a Kronecker covariance model and specify the output correlation matrix.

>>> scale = [4.0, 5.0]
>>> rho = ot.GeneralizedExponential(scale,  1.0)
>>> amplitude = [1.0, 2.0]
>>> outputCorrelation = ot.CorrelationMatrix(2)
>>> outputCorrelation[1,0] = 0.8
>>> myCovarianceModel = ot.KroneckerCovarianceModel(rho, amplitude, outputCorrelation)


Create a Kronecker covariance model and specify the output covariance matrix.

>>> scale = [4.0, 5.0]
>>> rho = ot.GeneralizedExponential(scale, 1.0)
>>> outputCovariance = ot.CovarianceMatrix(2)
>>> outputCovariance[0,0] = 4.0
>>> outputCovariance[1,1] = 5.0
>>> outputCovariance[1,0] = 1.2
>>> myCovarianceModel = ot.KroneckerCovarianceModel(rho, outputCovariance)


Methods

 activateAmplitude(isAmplitudeActive) Activate/deactivate the amplitude parameter(s). activateNuggetFactor(isNuggetFactorActive) Activate/deactivate the nugget factor. activateScale(isScaleActive) Activate/deactivate the scale parameter(s). computeAsScalar(*args) Compute the covariance function for scalar model. Compute the covariance function on a given mesh. discretize(*args) Discretize the covariance function on a given mesh. Discretize and factorize the covariance function on a given mesh. 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. Accessor to the active parameter set. Get the amplitude parameter of the covariance function. Accessor to the object's name. Get the full parameters of the covariance function. Get the description full parameters of the covariance function. Get the input dimension of the covariance function. getMarginal(*args) Get the ith marginal of the model. Accessor to the object's name. Accessor to the nugget factor. Get the spatial correlation matrix of the covariance function. Get the dimension of the covariance function. Get the parameters of the covariance function. Get the description of the covariance function parameters. Get the scale parameter of the covariance function. Test if the object is named. Test whether the model is diagonal or not. Test whether the model is stationary or not. Compute the gradient according to the parameters. 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 variance of the observation error. 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.
__init__(*args)
activateAmplitude(isAmplitudeActive)

Activate/deactivate the amplitude parameter(s).

In the context of Kriging, defines whether amplitude parameters should be tuned.

Parameters:
isAmplitudeActivebool

If True, the amplitude parameters are all tuned. If False, none of them is tuned.

activateNuggetFactor(isNuggetFactorActive)

Activate/deactivate the nugget factor.

In the context of Kriging, defines whether the nugget factor should be tuned.

Parameters:
isNuggetFactorActivebool

If True (resp. False), the nugget factor is (resp. is not) tuned.

activateScale(isScaleActive)

Activate/deactivate the scale parameter(s).

In the context of Kriging, defines whether scale parameters should be tuned.

Parameters:
isScaleActivebool

If True, the scale parameters are all tuned. If False, none of them is tuned.

computeAsScalar(*args)

Compute the covariance function for scalar model.

Available usages:

computeAsScalar(s, t)

computeAsScalar(tau)

Parameters:
s, tfloats (if ) or sequences of floats (any )

Multivariate index

taufloat (if ) or sequence of floats (any )

Multivariate index

Returns:
covariancefloat

Covariance.

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 .

computeCrossCovariance(*args)

Compute the covariance function on a given mesh.

Parameters:
firstVertices

Container of the first discretization vertices

secondVertices

Container of the second discretization vertices

Returns:
MatrixMatrix

Cross covariance matrix

Notes

This method computes a cross-covariance matrix. The cross-covariance is the evaluation of the covariance model on both firstVertices and secondVertices.

If firstVertices contains points and secondVertices contains points, the method returns an matrix ( being the output dimension).

To make things easier, let us focus on the case. Let be the points of firstVertices and let be the points of secondVertices. The result is the matrix such that for any nonnegative integers and , .

discretize(*args)

Discretize the covariance function on a given mesh.

Parameters:
where

Container of the discretization vertices

Returns:
covarianceMatrixCovarianceMatrix

Covariance matrix (if the process is of dimension )

Notes

This method makes a discretization of the model on the given Mesh, RegularGrid or Sample composed of the vertices and returns the covariance matrix:

discretizeAndFactorize(*args)

Discretize and factorize the covariance function on a given mesh.

Parameters:
where

Container of the discretization vertices

Returns:
CholeskyMatrixTriangularMatrix

Cholesky factor of the covariance matrix (if the process is of dimension )

Notes

This method makes a discretization of the model on the given Mesh, RegularGrid or Sample 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:
where

Container of the discretization vertices

hmatParamHMatrixParameters

Parameter values for the HMatrix

Returns:
HMatrixHMatrix

Cholesk matrix (if the process is of dimension ), stored in hierarchical format (H-Matrix)

Notes

This method is similar to the discretizeAndFactorize() method. This method requires that requires that OpenTURNS has been compiled with the hmat library. The method is helpful for very large parameters (Mesh, grid, Sample) because it compresses data.

discretizeHMatrix(*args)

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

Parameters:
where

Container of the discretization vertices

hmatParamHMatrixParameters

Parameter values for the HMatrix

Returns:
HMatrixHMatrix

Covariance matrix (if the process is of dimension ), stored in hierarchical format (H-Matrix)

Notes

This method is similar to the discretize() method. This method requires that OpenTURNS has been compiled with the hmat library. The method is helpful for very large parameters (Mesh, grid, Sample) because it compresses 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 < dimension

The column index of the component to draw. Default value is 0.

tMinfloat

The lower bound of the range over which the model is plotted. Default value is CovarianceModel-DefaultTMin in ResourceMap.

tMaxfloat

The 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.

asStationarybool

Flag 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.

correlationFlagbool

Flag 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.

Returns:
graphGraph

A 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.

In the context of kriging, it allows one to choose which hyperparameters are tuned.

Returns:
activeIndices

Indices of the active parameters.

getAmplitude()

Get the amplitude parameter of the covariance function.

Returns:
amplitudePoint

The amplitude parameter of the covariance function.

getClassName()

Accessor to the object’s name.

Returns:
class_namestr

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

getFullParameter()

Get the full parameters of the covariance function.

Returns:
parameterPoint

List 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:
descriptionDescription

Description of the full parameter of the covariance function.

getInputDimension()

Get the input dimension of the covariance function.

Returns:
inputDimensionint

Spatial dimension of the covariance function.

getMarginal(*args)

Get the ith marginal of the model.

Returns:
marginalint or sequence of int

index of marginal of the model.

getName()

Accessor to the object’s name.

Returns:
namestr

The 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:
nuggetFactorfloat

Nugget factor used to model the observation error variance.

getOutputCorrelation()

Get the spatial correlation matrix of the covariance function.

Returns:
spatialCorrelationCorrelationMatrix

Correlation matrix .

getOutputDimension()

Get the dimension of the covariance function.

Returns:
dint

Dimension 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:
descriptionParamDescription

Description of the components of the parameters obtained with the getParameter method..

getScale()

Get the scale parameter of the covariance function.

Returns:
scalePoint

The scale parameter used in the covariance function.

hasName()

Test if the object is named.

Returns:
hasNamebool

True if the name is not empty.

isDiagonal()

Test whether the model is diagonal or not.

Returns:
isDiagonalbool

True if the model is diagonal.

isStationary()

Test whether the model is stationary or not.

Returns:
isStationarybool

True 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 float

Multivariate index .

Returns:
gradientMatrix

Gradient of the function according to the parameters.

partialGradient(s, t)

Compute the gradient of the covariance function.

Parameters:
s, tfloats or sequences of float

Multivariate index .

Returns:
gradientMatrix

Gradient of the covariance function.

setActiveParameter(active)

Accessor to the active parameter set.

In the context of kriging, it allows one to choose which hyperparameters are tuned.

Parameters:
activesequence of int

Indices of the active parameters.

setAmplitude(amplitude)

Set the amplitude parameter of the covariance function.

Parameters:
amplitudePoint

The 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:
parameterPoint

List 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:
namestr

The name of the object.

setNuggetFactor(nuggetFactor)

Set the nugget factor for the variance of the observation error.

Acts on the discretized covariance matrix.

Parameters:
nuggetFactorfloat

nugget factor to be used to model the variance of the observation error.

setOutputCorrelation(correlation)

Set the spatial correlation matrix of the covariance function.

Parameters:
spatialCorrelationCorrelationMatrix

Correlation matrix .

setParameter(parameter)

Set the parameters of the covariance function.

Parameters:
parametersPoint

List 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:
scalePoint

The scale parameter to be used in the covariance function. Its size must be equal to the input dimension of the covariance function.

## Examples using the class¶

Sample trajectories from a Gaussian Process with correlated outputs

Sample trajectories from a Gaussian Process with correlated outputs