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, outputSample : 2-d sequence of float

The samples $(\vect{x}_k)_{1 \leq k \leq N} \in \Rset^d$ and $(\vect{y}_k)_{1 \leq k \leq N}\in \Rset^p$ .

metaModel : Function

The meta model: $\tilde{\cM}: \Rset^d \rightarrow \Rset^p$ , defined in (3).

residuals : Point

The residual errors.

relativeErrors : Point

The relative errors.

basis : collection 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.

trendCoefficients : collection of Point

The trend coeffient vectors $(\vect{\alpha}^1, \dots, \vect{\alpha}^p)$ .

covarianceModel : CovarianceModel

Covariance function of the normal process.

covarianceCoefficients : 2-d sequence of float

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

covarianceCholeskyFactor : TriangularMatrix

The Cholesky factor $\mat{L}$ of $\mat{C}$ .

covarianceHMatrix : HMatrix

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

__init__(*args)¶

getBasisCollection()¶

Accessor to the collection of basis.

Returns:

basisCollection : collection 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()¶

Accessor to the object’s name.

Returns:

class_name : str

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

getConditionalCovariance(*args)¶

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

Available usages:

getConditionalCovariance(x)

getConditionalCovariance(sampleX)

Parameters:

x : sequence of float

The point $\vect{x}$ where the conditional mean of the output has to be evaluated.

sampleX : 2-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:

condCov : CovarianceMatrix

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

getConditionalMean(*args)¶

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

Available usages:

getConditionalMean(x)

getConditionalMean(sampleX)

Parameters:

x : sequence of float

The point $\vect{x}$ where the conditional mean of the output has to be evaluated.

sampleX : 2-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:

condMean : Point

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()¶

Accessor to the covariance coefficients.

Returns:

covCoeff : Sample

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

getCovarianceModel()¶

Accessor to the covariance model.

Returns:

covModel : CovarianceModel

The covariance model of the Normal process W with its optimized parameters.

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.

getMetaModel()¶

Accessor to the metamodel.

Returns:

metaModel : Function

Metamodel.

getModel()¶

Accessor to the model.

Returns:

model : Function

Physical model approximated by a metamodel.

getName()¶

Accessor to the object’s name.

Returns:

name : str

The name of the object.

getOutputSample()¶

Accessor to the output sample.

Returns:

outputSample : Sample

The output sample.

getRelativeErrors()¶

Accessor to the relative errors.

Returns:

relativeErrors : Point

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()¶

Accessor to the residuals.

Returns:

residuals : Point

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()¶

Accessor to the object’s shadowed id.

Returns:

id : int

Internal unique identifier.

getTransformation()¶

Accessor to the normalizing transformation.

Returns:

transformation : Function

The transformation T that normalizes the input sample.

getTrendCoefficients()¶

Accessor to the trend coefficients.

Returns:

trendCoef : collection of Point

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

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.

setMetaModel(metaModel)¶

Accessor to the metamodel.

Parameters:

metaModel : Function

Metamodel.

setModel(model)¶

Accessor to the model.

Parameters:

model : Function

Physical model approximated by a metamodel.

setName(name)¶

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setRelativeErrors(relativeErrors)¶

Accessor to the relative errors.

Parameters:

relativeErrors : sequence 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(residuals)¶

Accessor to the residuals.

Parameters:

residuals : sequence 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(id)¶

Accessor to the object’s shadowed id.

Parameters:

id : int

Internal unique identifier.

setTransformation(transformation)¶

Accessor to the normalizing transformation.

Parameters:

transformation : Function

The transformation T that normalizes the input sample.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters:

visible : bool

Visibility flag.

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An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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