KrigingAlgorithm

class KrigingAlgorithm(*args)

Kriging algorithm.

Available constructors:

KrigingAlgorithm(inputSample, outputSample, covarianceModel, basis, normalize=True)

KrigingAlgorithm(inputSample, inputTransformation, outputSample, covarianceModel, basis)

KrigingAlgorithm(inputSample, outputSample, covarianceModel, basisCollection, normalize=True)

KrigingAlgorithm(inputSample, inputTransformation, outputSample, covarianceModel, basisCollection)

Parameters:

inputSample, outputSample : 2-d sequence of float

The samples (\vect{x}_k)_{1 \leq k \leq N} \in \Rset^d and (\vect{y}_k)_{1 \leq k \leq N}\in \Rset^p upon which the meta-model is built.

inputTransformation : Function

Function T used to normalize the input sample.

If used, the meta model is built on the transformed data.

basis : Basis

Functional basis to estimate the trend (universal kriging): (\varphi_j)_{1 \leq j \leq n_1}: \Rset^d \rightarrow \Rset.

If p>1, the same basis is used for each marginal output.

covarianceModel : CovarianceModel

Covariance model used for the underlying Gaussian process assumption.

basisCollection : sequence of Basis

Collection of p functional basis: one basis for each marginal output: \left[(\varphi_j^1)_{1 \leq j \leq n_1}, \dots, (\varphi_j^p)_{1 \leq j \leq n_p}\right]. If the sequence is empty, no trend coefficient is estimated (simple kriging).

normalize : bool, optional

Indicates whether the input sample has to be normalized.

OpenTURNS uses the transformation fixed by the User in inputTransformation or the empirical mean and variance of the input sample. Default is set in resource map key GeneralLinearModelAlgorithm-NormalizeData

Notes

We suppose we have a sample (\vect{x}_k, \vect{y}_k)_{1 \leq k \leq N} where \vect{y}_k = \cM(\vect{x}_k) for all k, with \cM:\Rset^d \mapsto \Rset^p the model.

The meta model Kriging is based on the same principles as those of the generalized linear model: it assumes that the sample (\vect{y}_k)_{1 \leq k \leq N} is considered as the trace of a normal process \vect{Y}(\omega, \vect{x}) on (\vect{x}_k)_{1 \leq k \leq N}. The normal process \vect{Y}(\omega, \vect{x}) is defined by:

(1)\vect{Y}(\omega, \vect{x}) = \vect{\mu}(\vect{x}) + W(\omega, \vect{x})

where:

\vect{\mu}(\vect{x}) = \left(
  \begin{array}{l}
    \mu_1(\vect{x}) \\
    \dots  \\
    \mu_p(\vect{x}) 
   \end{array}
 \right)

with \mu_l(\vect{x}) = \sum_{j=1}^{n_l} \alpha_j^l \varphi_j^l(\vect{x}) and \varphi_j^l: \Rset^d \rightarrow \Rset the trend functions.

W is a normal process of dimension p with zero mean and covariance function C = C(\vect{\theta}, \vect{\sigma}, \mat{R}, \vect{\lambda}) (see CovarianceModel for the notations).

The estimation of the parameters \alpha_j^l, \vect{\theta}, \vect{\sigma} are made by the GeneralLinearModelAlgorithm class.

The Kriging algorithm makes the generalized linear model interpolary on the input samples. The Kriging meta model \tilde{\cM} is defined by:

\tilde{\cM}(\vect{x}) =  \vect{\mu}(\vect{x}) + \Expect{\vect{Y}(\omega, \vect{x})\, | \,  \cC}

where \cC is the condition \vect{Y}(\omega, \vect{x}_k) = \vect{y}_k for each k \in [1, N].

(1) writes:

\tilde{\cM}(\vect{x}) = \vect{\mu}(\vect{x}) + \Cov{\vect{Y}(\omega, \vect{x}), (\vect{Y}(\omega, \vect{x}_1), \dots, \vect{Y}(\omega, \vect{x}_N))} \vect{\gamma}

where \Cov{\vect{Y}(\omega, \vect{x}), (\vect{Y}(\omega, \vect{x}_1), \dots, \vect{Y}(\omega, \vect{x}_N))} = \left( \mat{C}( \vect{x},  \vect{x}_1) | \dots | \mat{C}( \vect{x},  \vect{x}_N)  \right) is a matrix in \cM_{p,NP}(\Rset) and \vect{\gamma} = \mat{C}^{-1}(\vect{y}-\vect{m}).

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

\hat{\vect{y}}_k = \vect{y}_k + \epsilon_k, \epsilon_k \sim \mathcal{N}(0, \tau_k^2)

Examples

Create the model \cM: \Rset \mapsto \Rset and the samples:

>>> import openturns as ot
>>> # use of Hmat implementation
>>> # ot.ResourceMap.Set('KrigingAlgorithm-LinearAlgebra', 'HMAT')
>>> f = ot.SymbolicFunction(['x'], ['x * sin(x)'])
>>> 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 resulting meta model:

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

Methods

getClassName() Accessor to the object’s name.
getDistribution() Accessor to the joint probability density function of the physical input vector.
getId() Accessor to the object’s id.
getInputSample() Accessor to the input sample.
getName() Accessor to the object’s name.
getNoise() Observation noise variance accessor.
getOptimizationAlgorithm() Accessor to solver used to optimize the covariance model parameters.
getOptimizationBounds() Accessor to the optimization bounds.
getOptimizationSolver()
getOptimizeParameters() Accessor to the covariance model parameters optimization flag.
getOutputSample() Accessor to the output sample.
getReducedLogLikelihoodFunction() Accessor to the reduced log-likelihood function that writes as argument of the covariance’s model parameters.
getResult() Get the results of the metamodel computation.
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.
run() Compute the response surface.
setDistribution(distribution) Accessor to the joint probability density function of the physical input vector.
setName(name) Accessor to the object’s name.
setNoise(noise) Observation noise variance accessor.
setOptimizationAlgorithm(solver) Accessor to the solver used to optimize the covariance model parameters.
setOptimizationBounds(optimizationBounds) Accessor to the optimization bounds.
setOptimizationSolver(solver)
setOptimizeParameters(optimizeParameters) Accessor to the covariance model parameters optimization flag.
setShadowedId(id) Accessor to the object’s shadowed id.
setVisibility(visible) Accessor to the object’s visibility state.
__init__(*args)
getClassName()

Accessor to the object’s name.

Returns:

class_name : str

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

getDistribution()

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

Returns:

distribution : Distribution

Joint probability density function of the physical input vector.

getId()

Accessor to the object’s id.

Returns:

id : int

Internal unique identifier.

getInputSample()

Accessor to the input sample.

Returns:

inputSample : Sample

The input sample (\vect{x}_k)_{1 \leq k \leq N}.

getName()

Accessor to the object’s name.

Returns:

name : str

The name of the object.

getNoise()

Observation noise variance accessor.

Returns:

noise : sequence of positive float

The noise variance \tau_k^2 of each output value.

getOptimizationAlgorithm()

Accessor to solver used to optimize the covariance model parameters.

Returns:

algorithm : OptimizationAlgorithm

Solver used to optimize the covariance model parameters.

getOptimizationBounds()

Accessor to the optimization bounds.

Returns:

problem : Interval

The bounds used for numerical optimization of the likelihood.

getOptimizeParameters()

Accessor to the covariance model parameters optimization flag.

Returns:

optimizeParameters : bool

Whether to optimize the covariance model parameters.

getOutputSample()

Accessor to the output sample.

Returns:

outputSample : Sample

The output sample (\vect{y}_k)_{1 \leq k \leq N} .

getReducedLogLikelihoodFunction()

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

Returns:

reducedLogLikelihood : Function

The reduced log-likelihood function as a function of (\vect{\theta}, \vect{\sigma}).

Notes

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

Examples

Create the model \cM: \Rset \mapsto \Rset 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:

result : KrigingResult

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

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.

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:

distribution : Distribution

Joint probability density function of the physical input vector.

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setNoise(noise)

Observation noise variance accessor.

Parameters:

noise : sequence of positive float

The noise variance \tau_k^2 of each output value.

setOptimizationAlgorithm(solver)

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

Parameters:

algorithm : OptimizationAlgorithm

Solver used to optimize the covariance model parameters.

Examples

Create the model \cM: \Rset \mapsto \Rset 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:

problem : Interval

The bounds used for numerical optimization of the likelihood.

setOptimizeParameters(optimizeParameters)

Accessor to the covariance model parameters optimization flag.

Parameters:

optimizeParameters : bool

Whether to optimize the covariance model parameters.

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.