KrigingAlgorithm¶

(Source code, png)

class KrigingAlgorithm(*args)¶

Kriging algorithm.

Parameters:

inputSample, outputSample2-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.
covarianceModelCovarianceModel: Covariance model used for the underlying Gaussian process assumption.
basisBasis, optional: 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.

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 general linear model: it assumes that the sample $(\vect{y}_k)_{1 \leq k \leq N}$ is considered as the trace of a Gaussian process $\vect{Y}(\omega, \vect{x})$ on $(\vect{x}_k)_{1 \leq k \leq N}$ . The Gaussian 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 $\mu_l(\vect{x}) = \sum_{j=1}^{n_l} \beta_j^l \varphi_j^l(\vect{x})$ and $\varphi_j^l: \Rset^d \rightarrow \Rset$ the trend functions.

$W$ is a Gaussian 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 all parameters (the trend coefficients $\beta_j^l$ , the scale $\vect{\theta}$ and the amplitude $\vect{\sigma}$ ) are made by the GeneralLinearModelAlgorithm class.

The Kriging algorithm makes the general linear model interpolating 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
>>> 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.
`getClassName`()	Accessor to the object's name.
`getDistribution`()	Accessor to the joint probability density function of the physical input vector.
`getInputSample`()	Accessor to the input sample.
`getMethod`()	Linear algebra method accessor.
`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.
`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.
`getWeights`()	Return the weights of the input sample.
`hasName`()	Test if the object is named.
`run`()	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.
`setOptimizationAlgorithm`(solver)	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.

__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.

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 $\tau_k^2$ of each output value.

getOptimizationAlgorithm()¶

Accessor to solver used to optimize the covariance model parameters.

Returns:

solverOptimizationAlgorithm: Solver used to optimize the covariance model parameters.

getOptimizationBounds()¶

Accessor to the optimization bounds.

Returns:

optimizationBoundsInterval: 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 : $\vect{\theta}$ refers to the scale parameters and $\vect{\sigma}$ the amplitude. We can consider three situations here:

Output dimension is $\geq 2$ . In that case, we get the full log-likelihood function $\mathcal{L}(\vect{\theta}, \vect{\sigma})$ .

Output dimension is 1 and the GeneralLinearModelAlgorithm-UseAnalyticalAmplitudeEstimate key of ResourceMap is set to True. The amplitude parameter of the covariance model $\vect{\theta}$ is in the active set of parameters and thus we get the reduced log-likelihood function $\mathcal{L}(\vect{\theta})$ .

Output dimension is 1 and the GeneralLinearModelAlgorithm-UseAnalyticalAmplitudeEstimate key of ResourceMap is set to False. In that case, we get the full log-likelihood $\mathcal{L}(\vect{\theta}, \vect{\sigma})$ .

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 $\cM: \Rset \mapsto \Rset$ 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:

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

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.

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 $\tau_k^2$ of each output value.

setOptimizationAlgorithm(solver)¶

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

Parameters:

solverOptimizationAlgorithm: 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)