KrigingResult¶

class KrigingResult(*args)¶

Kriging result.

Available constructors:

KrigingResult(inputSample, outputSample, metaModel, residuals, relativeErrors, basis, trendCoefficients, covarianceModel, covarianceCoefficients)

KrigingResult(inputSample, outputSample, metaModel, residuals, relativeErrors, basis, trendCoefficients, covarianceModel, covarianceCoefficients, covarianceCholeskyFactor, covarianceHMatrix)

Parameters

inputSample, outputSample2-d sequence of float: The samples $(\vect{x}_k)_{1 \leq k \leq N} \in \Rset^d$ and $(\vect{y}_k)_{1 \leq k \leq N}\in \Rset^p$ .
metaModelFunction: The meta model: $\tilde{\cM}: \Rset^d \rightarrow \Rset^p$ , defined in (3).
residualsPoint: The residual errors.
relativeErrorsPoint: The relative errors.
basiscollection of Basis: Collection of the $p$ functional basis: $(\varphi_j^l)_{1 \leq j \leq n_l}$ for each $l \in [1, p]$ with $\varphi_j^l: \Rset^d \rightarrow \Rset$ . Its size must be equal to zero if the trend is not estimated.
trendCoefficientscollection of Point: The trend coeffient vectors $(\vect{\alpha}^1, \dots, \vect{\alpha}^p)$ .
covarianceModelCovarianceModel: Covariance function of the Gaussian process.
covarianceCoefficients2-d sequence of float: The $\vect{\gamma}$ defined in (2).
covarianceCholeskyFactorTriangularMatrix: The Cholesky factor $\mat{L}$ of $\mat{C}$ .
covarianceHMatrixHMatrix: The hmat implementation of $\mat{L}$ .

Notes

The Kriging meta model $\tilde{\cM}$ is defined by:

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

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

Equation (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

$\Cov{\vect{Y}(\omega, \vect{x}), (\vect{Y}(\omega, \vect{x}_1),\dots,\vect{Y}(\omega, \vect{x}_N))} = \left(\mat{C}(\vect{x},\vect{x}_1)|\dots|\mat{C}(\vect{x},\vect{x}_N)\right)\in \cM_{p,NP}(\Rset)$

and

(2)¶ $\vect{\gamma} = \mat{C}^{-1}(\vect{y}-\vect{m})$

At the end, the meta model writes:

(3)¶ $\tilde{\cM}(\vect{x}) = \vect{\mu}(\vect{x}) + \sum_{i=1}^N \gamma_i \mat{C}(\vect{x},\vect{x}_i)$

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]]
>>> sampleY = f(sampleX)

Create the algorithm:

>>> basis = ot.Basis([ot.SymbolicFunction(['x'], ['x']), ot.SymbolicFunction(['x'], ['x^2'])])
>>> covarianceModel = ot.GeneralizedExponential([2.0], 2.0)
>>> algoKriging = ot.KrigingAlgorithm(sampleX, sampleY, covarianceModel, basis)
>>> algoKriging.run()

Get the result:

>>> resKriging = algoKriging.getResult()

Get the meta model:

>>> metaModel = resKriging.getMetaModel()

Methods

`__call__`(self, \*args)	Call self as a function.
`getBasisCollection`(self)	Accessor to the collection of basis.
`getClassName`(self)	Accessor to the object’s name.
`getConditionalCovariance`(self, \*args)	Compute the expected covariance of the Gaussian process on a point (or several points).
`getConditionalMarginalCovariance`(self, \*args)	Compute the expected covariance of the Gaussian process on a point (or several points).
`getConditionalMarginalVariance`(self, \*args)	Compute the expected variance of the Gaussian process on a point (or several points).
`getConditionalMean`(self, \*args)	Compute the expected mean of the Gaussian process on a point or a sample of points.
`getCovarianceCoefficients`(self)	Accessor to the covariance coefficients.
`getCovarianceModel`(self)	Accessor to the covariance model.
`getId`(self)	Accessor to the object’s id.
`getInputSample`(self)	Accessor to the input sample.
`getMetaModel`(self)	Accessor to the metamodel.
`getModel`(self)	Accessor to the model.
`getName`(self)	Accessor to the object’s name.
`getOutputSample`(self)	Accessor to the output sample.
`getRelativeErrors`(self)	Accessor to the relative errors.
`getResiduals`(self)	Accessor to the residuals.
`getShadowedId`(self)	Accessor to the object’s shadowed id.
`getTransformation`(self)	Accessor to the normalizing transformation.
`getTrendCoefficients`(self)	Accessor to the trend coefficients.
`getVisibility`(self)	Accessor to the object’s visibility state.
`hasName`(self)	Test if the object is named.
`hasVisibleName`(self)	Test if the object has a distinguishable name.
`setMetaModel`(self, metaModel)	Accessor to the metamodel.
`setModel`(self, model)	Accessor to the model.
`setName`(self, name)	Accessor to the object’s name.
`setRelativeErrors`(self, relativeErrors)	Accessor to the relative errors.
`setResiduals`(self, residuals)	Accessor to the residuals.
`setShadowedId`(self, id)	Accessor to the object’s shadowed id.
`setTransformation`(self, transformation)	Accessor to the normalizing transformation.
`setVisibility`(self, visible)	Accessor to the object’s visibility state.

__init__(self, \*args)¶: Initialize self. See help(type(self)) for accurate signature.

getBasisCollection(self)¶

Accessor to the collection of basis.

Returns

basisCollectioncollection of Basis: Collection of the $p$ function basis: $(\varphi_j^l)_{1 \leq j \leq n_l}$ for each $l \in [1, p]$ with $\varphi_j^l: \Rset^d \rightarrow \Rset$ .

Notes

If the trend is not estimated, the collection is empty.

getClassName(self)¶

Accessor to the object’s name.

Returns

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

getConditionalCovariance(self, \*args)¶

Compute the expected covariance of the Gaussian process on a point (or several points).

Available usages:

getConditionalCovariance(x)

getConditionalCovariance(sampleX)

Parameters

xsequence of float: The point $\vect{x}$ where the conditional covariance of the output has to be evaluated.
sampleX2-d sequence of float: The sample $(\vect{\xi}_1, \dots, \vect{\xi}_M)$ where the conditional covariance of the output has to be evaluated (M can be equal to 1).

Returns

condCovCovarianceMatrix

The conditional covariance $\Cov{\vect{Y}(\omega, \vect{x})\, | \, \cC}$ at point $\vect{x}$ . Or the conditional covariance matrix at the sample $(\vect{\xi}_1, \dots, \vect{\xi}_M)$ :

$\left( \begin{array}{lcl} \Sigma_{11} & \dots & \Sigma_{1M} \\ \dots \\ \Sigma_{M1} & \dots & \Sigma_{MM} \end{array} \right)$

where $\Sigma_{ij} = \Cov{\vect{Y}(\omega, \vect{\xi}_i), \vect{Y}(\omega, \vect{\xi}_j)\, | \, \cC}$ .

getConditionalMarginalCovariance(self, \*args)¶

Compute the expected covariance of the Gaussian process on a point (or several points).

Available usages:

getConditionalMarginalCovariance(x)

getConditionalMarginalCovariance(sampleX)

Parameters

xsequence of float: The point $\vect{x}$ where the conditional marginal covariance of the output has to be evaluated.
sampleX2-d sequence of float: The sample $(\vect{\xi}_1, \dots, \vect{\xi}_M)$ where the conditional marginal covariance of the output has to be evaluated (M can be equal to 1).

Returns

condCovCovarianceMatrix: The conditional covariance $\Cov{\vect{Y}(\omega, \vect{x})\, | \, \cC}$ at point $\vect{x}$ .
condCovCovarianceMatrixCollection: The collection of conditional covariance matrices $\Cov{\vect{Y}(\omega, \vect{\xi})\, | \, \cC}$ at each point of the sample $(\vect{\xi}_1, \dots, \vect{\xi}_M)$ :

Notes

In case input parameter is a of type Sample, each element of the collection corresponds to the conditional covariance with respect to the input learning set (pointwise evaluation of the getConditionalCovariance).

getConditionalMarginalVariance(self, \*args)¶

Compute the expected variance of the Gaussian process on a point (or several points).

Available usages:

getConditionalMarginalVariance(x, marginalIndex)

getConditionalMarginalVariance(sampleX, marginalIndex)

getConditionalMarginalVariance(x, marginalIndices)

getConditionalMarginalVariance(sampleX, marginalIndices)

Parameters

xsequence of float: The point $\vect{x}$ where the conditional variance of the output has to be evaluated.
sampleX2-d sequence of float: The sample $(\vect{\xi}_1, \dots, \vect{\xi}_M)$ where the conditional variance of the output has to be evaluated (M can be equal to 1).
marginalIndexint: Marginal of interest (for multiple outputs). Default value is 0
marginalIndicessequence of int: Marginals of interest (for multiple outputs).

Returns

varfloat: Variance of interest. float if one point (x) and one marginal of interest (x, marginalIndex)
varPointsequence of float: The marginal variances

Notes

In case of fourth usage, the sequence of float is given as the concatenation of marginal variances for each point in sampleX.

getConditionalMean(self, \*args)¶

Compute the expected mean of the Gaussian process on a point or a sample of points.

Available usages:

getConditionalMean(x)

getConditionalMean(sampleX)

Parameters

xsequence of float: The point $\vect{x}$ where the conditional mean of the output has to be evaluated.
sampleX2-d sequence of float: The sample $(\vect{\xi}_1, \dots, \vect{\xi}_M)$ where the conditional mean of the output has to be evaluated (M can be equal to 1).

Returns

condMeanPoint: The conditional mean $\Expect{\vect{Y}(\omega, \vect{x})\, | \, \cC}$ at point $\vect{x}$ . Or the conditional mean matrix at the sample $(\vect{\xi}_1, \dots, \vect{\xi}_M)$ :

$\left( \begin{array}{l} \Expect{\vect{Y}(\omega, \vect{\xi}_1)\, | \, \cC}\\ \dots \\ \Expect{\vect{Y}(\omega, \vect{\xi}_M)\, | \, \cC} \end{array} \right)$

getCovarianceCoefficients(self)¶

Accessor to the covariance coefficients.

Returns

covCoeffSample: The $\vect{\gamma}$ defined in (2).

getCovarianceModel(self)¶

Accessor to the covariance model.

Returns

covModelCovarianceModel: The covariance model of the Gaussian process W with its optimized parameters.

getId(self)¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

getInputSample(self)¶

Accessor to the input sample.

Returns

inputSampleSample: The input sample.

getMetaModel(self)¶

Accessor to the metamodel.

Returns

metaModelFunction: Metamodel.

getModel(self)¶

Accessor to the model.

Returns

modelFunction: Physical model approximated by a metamodel.

getName(self)¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getOutputSample(self)¶

Accessor to the output sample.

Returns

outputSampleSample: The output sample.

getRelativeErrors(self)¶

Accessor to the relative errors.

Returns

relativeErrorsPoint: The relative errors defined as follows for each output of the model: $\displaystyle \frac{\sum_{i=1}^N (y_i - \hat{y_i})^2}{N \Var{\vect{Y}}}$ with $\vect{Y}$ the vector of the $N$ model’s values $y_i$ and $\hat{y_i}$ the metamodel’s values.

getResiduals(self)¶

Accessor to the residuals.

Returns

residualsPoint: The residual values defined as follows for each output of the model: $\displaystyle \frac{\sqrt{\sum_{i=1}^N (y_i - \hat{y_i})^2}}{N}$ with $y_i$ the $N$ model’s values and $\hat{y_i}$ the metamodel’s values.

getShadowedId(self)¶

Accessor to the object’s shadowed id.

Returns

idint: Internal unique identifier.

getTransformation(self)¶

Accessor to the normalizing transformation.

Returns

transformationFunction: The transformation T that normalizes the input sample.

getTrendCoefficients(self)¶

Accessor to the trend coefficients.

Returns

trendCoefcollection of Point: The trend coefficients vectors $(\vect{\alpha}^1, \dots, \vect{\alpha}^p)$

getVisibility(self)¶

Accessor to the object’s visibility state.

Returns

visiblebool: Visibility flag.

hasName(self)¶

Test if the object is named.

Returns

hasNamebool: True if the name is not empty.

hasVisibleName(self)¶

Test if the object has a distinguishable name.

Returns

hasVisibleNamebool: True if the name is not empty and not the default one.

setMetaModel(self, metaModel)¶

Accessor to the metamodel.

Parameters

metaModelFunction: Metamodel.

setModel(self, model)¶

Accessor to the model.

Parameters

modelFunction: Physical model approximated by a metamodel.

setName(self, name)¶

Accessor to the object’s name.

Parameters

namestr: The name of the object.

setRelativeErrors(self, relativeErrors)¶

Accessor to the relative errors.

Parameters

relativeErrorssequence of float: The relative errors defined as follows for each output of the model: $\displaystyle \frac{\sum_{i=1}^N (y_i - \hat{y_i})^2}{N \Var{\vect{Y}}}$ with $\vect{Y}$ the vector of the $N$ model’s values $y_i$ and $\hat{y_i}$ the metamodel’s values.

setResiduals(self, residuals)¶

Accessor to the residuals.

Parameters

residualssequence of float: The residual values defined as follows for each output of the model: $\displaystyle \frac{\sqrt{\sum_{i=1}^N (y_i - \hat{y_i})^2}}{N}$ with $y_i$ the $N$ model’s values and $\hat{y_i}$ the metamodel’s values.

setShadowedId(self, id)¶

Accessor to the object’s shadowed id.

Parameters

idint: Internal unique identifier.

setTransformation(self, transformation)¶

Accessor to the normalizing transformation.

Parameters

transformationFunction: The transformation T that normalizes the input sample.

setVisibility(self, visible)¶

Accessor to the object’s visibility state.

Parameters

visiblebool: Visibility flag.

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