# KrigingAlgorithm¶

class KrigingAlgorithm(*args)

Kriging algorithm.

Refer to Kriging.

Parameters:
inputSample, outputSample2-d sequence of float

The samples and upon which the meta-model is built.

covarianceModelCovarianceModel

Covariance model used for the underlying Gaussian process assumption.

basisBasis, optional

Functional basis to estimate the trend (universal kriging): . If , the same basis is used for each marginal output.

Notes

We suppose we have a sample where for all k, with the model.

The meta model Kriging is based on the same principles as those of the general linear model: it assumes that the sample is considered as the trace of a Gaussian process on . The Gaussian process is defined by:

(1)

where:

with and the trend functions.

is a Gaussian process of dimension p with zero mean and covariance function (see CovarianceModel for the notations).

The estimation of the all parameters (the trend coefficients , the scale and the amplitude ) are made by the GeneralLinearModelAlgorithm class.

The Kriging algorithm makes the general linear model interpolating on the input samples. The Kriging meta model is defined by:

where is the condition for each .

(1) writes:

where is a matrix in and .

A known centered gaussian observation noise can be taken into account with setNoise():

Examples

Create the model and the samples:

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x'],  ['x * sin(x)'])
>>> sampleX = [[1.0], [2.0], [3.0], [4.0], [5.0], [6.0], [7.0], [8.0]]
>>> sampleY = f(sampleX)


Create the algorithm:

>>> basis = ot.Basis([ot.SymbolicFunction(['x'], ['x']), ot.SymbolicFunction(['x'], ['x^2'])])
>>> covarianceModel = ot.SquaredExponential([1.0])
>>> covarianceModel.setActiveParameter([])
>>> algo = ot.KrigingAlgorithm(sampleX, sampleY, covarianceModel, basis)
>>> algo.run()


Get the resulting meta model:

>>> result = algo.getResult()
>>> metamodel = result.getMetaModel()


Methods

 BuildDistribution(inputSample) Recover the distribution, with metamodel performance in mind. Accessor to the object's name. Accessor to the joint probability density function of the physical input vector. Accessor to the object's id. Accessor to the input sample. Linear algebra method accessor. Accessor to the object's name. Observation noise variance accessor. Accessor to solver used to optimize the covariance model parameters. Accessor to the optimization bounds. Accessor to the covariance model parameters optimization flag. Accessor to the output sample. Accessor to the reduced log-likelihood function that writes as argument of the covariance's model parameters. Get the results of the metamodel computation. Accessor to the object's shadowed id. Accessor to the object's visibility state. Return the weights of the input sample. Test if the object is named. Test if the object has a distinguishable name. Compute the response surface. setDistribution(distribution) Accessor to the joint probability density function of the physical input vector. setMethod(method) Linear algebra method set accessor. setName(name) Accessor to the object's name. setNoise(noise) Observation noise variance accessor. Accessor to the solver used to optimize the covariance model parameters. setOptimizationBounds(optimizationBounds) Accessor to the optimization bounds. setOptimizeParameters(optimizeParameters) Accessor to the covariance model parameters optimization flag. Accessor to the object's shadowed id. setVisibility(visible) Accessor to the object's visibility state.
__init__(*args)
static BuildDistribution(inputSample)

Recover the distribution, with metamodel performance in mind.

For each marginal, find the best 1-d continuous parametric model else fallback to the use of a nonparametric one.

The selection is done as follow:

• We start with a list of all parametric models (all factories)

• For each model, we estimate its parameters if feasible.

• We check then if model is valid, ie if its Kolmogorov score exceeds a threshold fixed in the MetaModelAlgorithm-PValueThreshold ResourceMap key. Default value is 5%

• We sort all valid models and return the one with the optimal criterion.

For the last step, the criterion might be BIC, AIC or AICC. The specification of the criterion is done through the MetaModelAlgorithm-ModelSelectionCriterion ResourceMap key. Default value is fixed to BIC. Note that if there is no valid candidate, we estimate a non-parametric model (KernelSmoothing or Histogram). The MetaModelAlgorithm-NonParametricModel ResourceMap key allows selecting the preferred one. Default value is Histogram

One each marginal is estimated, we use the Spearman independence test on each component pair to decide whether an independent copula. In case of non independence, we rely on a NormalCopula.

Parameters:
sampleSample

Input sample.

Returns:
distributionDistribution

Input distribution.

getClassName()

Accessor to the object’s name.

Returns:
class_namestr

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

getDistribution()

Accessor to the joint probability density function of the physical input vector.

Returns:
distributionDistribution

Joint probability density function of the physical input vector.

getId()

Accessor to the object’s id.

Returns:
idint

Internal unique identifier.

getInputSample()

Accessor to the input sample.

Returns:
inputSampleSample

Input sample of a model evaluated apart.

getMethod()

Linear algebra method accessor.

Returns:
methodstr

Used linear algebra method.

getName()

Accessor to the object’s name.

Returns:
namestr

The name of the object.

getNoise()

Observation noise variance accessor.

Returns:
noisesequence of positive float

The noise variance of each output value.

getOptimizationAlgorithm()

Accessor to solver used to optimize the covariance model parameters.

Returns:
algorithmOptimizationAlgorithm

Solver used to optimize the covariance model parameters.

getOptimizationBounds()

Accessor to the optimization bounds.

Returns:
problemInterval

The bounds used for numerical optimization of the likelihood.

getOptimizeParameters()

Accessor to the covariance model parameters optimization flag.

Returns:
optimizeParametersbool

Whether to optimize the covariance model parameters.

getOutputSample()

Accessor to the output sample.

Returns:
outputSampleSample

Output sample of a model evaluated apart.

getReducedLogLikelihoodFunction()

Accessor to the reduced log-likelihood function that writes as argument of the covariance’s model parameters.

Returns:
reducedLogLikelihoodFunction

The potentially reduced log-likelihood function.

Notes

We use the same notations as in CovarianceModel and GeneralLinearModelAlgorithm : refers to the scale parameters and the amplitude. We can consider three situations here:

• Output dimension is . In that case, we get the full log-likelihood function .

• Output dimension is 1 and the GeneralLinearModelAlgorithm-UseAnalyticalAmplitudeEstimate key of ResourceMap is set to True. The amplitude parameter of the covariance model is in the active set of parameters and thus we get the reduced log-likelihood function .

• Output dimension is 1 and the GeneralLinearModelAlgorithm-UseAnalyticalAmplitudeEstimate key of ResourceMap is set to False. In that case, we get the full log-likelihood .

The reduced log-likelihood function may be useful for some pre/postprocessing: vizualisation of the maximizer, use of an external optimizers to maximize the reduced log-likelihood etc.

Examples

Create the model and the samples:

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x0'], ['x0 * sin(x0)'])
>>> inputSample = ot.Sample([[1.0], [3.0], [5.0], [6.0], [7.0], [8.0]])
>>> outputSample = f(inputSample)


Create the algorithm:

>>> basis = ot.ConstantBasisFactory().build()
>>> covarianceModel = ot.SquaredExponential(1)
>>> algo = ot.KrigingAlgorithm(inputSample, outputSample, covarianceModel, basis)
>>> algo.run()


Get the reduced log-likelihood function:

>>> reducedLogLikelihoodFunction = algo.getReducedLogLikelihoodFunction()

getResult()

Get the results of the metamodel computation.

Returns:
resultKrigingResult

Structure containing all the results obtained after computation and created by the method run().

Accessor to the object’s shadowed id.

Returns:
idint

Internal unique identifier.

getVisibility()

Accessor to the object’s visibility state.

Returns:
visiblebool

Visibility flag.

getWeights()

Return the weights of the input sample.

Returns:
weightssequence of float

The weights of the points in the input sample.

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.

run()

Compute the response surface.

Notes

It computes the kriging response surface and creates a KrigingResult structure containing all the results.

setDistribution(distribution)

Accessor to the joint probability density function of the physical input vector.

Parameters:
distributionDistribution

Joint probability density function of the physical input vector.

setMethod(method)

Linear algebra method set accessor.

Parameters:
methodstr

Used linear algebra method. Value should be LAPACK or HMAT

Notes

The setter update the implementation and require new evaluation. We might also use the ResourceMap key to set the method when instantiating the algorithm. For that purpose, we can use ResourceMap.SetAsString(GeneralLinearModelAlgorithm-LinearAlgebra, key) with key being HMAT or LAPACK.

setName(name)

Accessor to the object’s name.

Parameters:
namestr

The name of the object.

setNoise(noise)

Observation noise variance accessor.

Parameters:
noisesequence of positive float

The noise variance of each output value.

setOptimizationAlgorithm(solver)

Accessor to the solver used to optimize the covariance model parameters.

Parameters:
algorithmOptimizationAlgorithm

Solver used to optimize the covariance model parameters.

Examples

Create the model and the samples:

>>> import openturns as ot
>>> input_data = ot.Uniform(-1.0, 2.0).getSample(10)
>>> model = ot.SymbolicFunction(['x'], ['x-1+sin(pi_*x/(1+0.25*x^2))'])
>>> output_data = model(input_data)


Create the Kriging algorithm with the optimizer option:

>>> basis = ot.Basis([ot.SymbolicFunction(['x'], ['0.0'])])
>>> thetaInit = 1.0
>>> covariance = ot.GeneralizedExponential([thetaInit], 2.0)
>>> bounds = ot.Interval(1e-2,1e2)
>>> algo = ot.KrigingAlgorithm(input_data, output_data, covariance, basis)
>>> algo.setOptimizationBounds(bounds)

setOptimizationBounds(optimizationBounds)

Accessor to the optimization bounds.

Parameters:
boundsInterval

The bounds used for numerical optimization of the likelihood.

Notes

See GeneralLinearModelAlgorithm class for more details, particularly setOptimizationBounds().

setOptimizeParameters(optimizeParameters)

Accessor to the covariance model parameters optimization flag.

Parameters:
optimizeParametersbool

Whether to optimize the covariance model parameters.

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.

## Examples using the class¶

Kriging: propagate uncertainties

Kriging: propagate uncertainties

Kriging : multiple input dimensions

Kriging : multiple input dimensions

Kriging : draw the likelihood

Kriging : draw the likelihood

Kriging : cantilever beam model

Kriging : cantilever beam model

Kriging: choose an arbitrary trend

Kriging: choose an arbitrary trend

Kriging the cantilever beam model using HMAT

Kriging the cantilever beam model using HMAT

Example of multi output Kriging on the fire satellite model

Example of multi output Kriging on the fire satellite model

Kriging : generate trajectories from a metamodel

Kriging : generate trajectories from a metamodel

Kriging: choose a polynomial trend on the beam model

Kriging: choose a polynomial trend on the beam model

Kriging with an isotropic covariance function

Kriging with an isotropic covariance function

Kriging: metamodel of the Branin-Hoo function

Kriging: metamodel of the Branin-Hoo function

Kriging : quick-start

Kriging : quick-start

Sequentially adding new points to a kriging

Sequentially adding new points to a kriging

Kriging :configure the optimization solver

Kriging :configure the optimization solver

Kriging: choose a polynomial trend

Kriging: choose a polynomial trend

Kriging: metamodel with continuous and categorical variables

Kriging: metamodel with continuous and categorical variables

EfficientGlobalOptimization examples

EfficientGlobalOptimization examples