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.
basisBasis: Functional basis of size $b$ : $(\varphi^l: \Rset^d \rightarrow \Rset^p)$ for each $l \in [1, b]$ . Its size should be equal to zero if the trend is not estimated.
trendCoefficientsPoint: The trend coefficients vectors $(\vect{\alpha}^1, \dots, \vect{\alpha}^p)$ stored as a Point.
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 $\mat{L}$ .

Methods

`getBasis`()	Accessor to the functional basis.
`getClassName`()	Accessor to the object's name.
`getConditionalCovariance`(*args)	Compute the conditional covariance of the Gaussian process on a point (or several points).
`getConditionalMarginalCovariance`(*args)	Compute the conditional covariance of the Gaussian process on a point (or several points).
`getConditionalMarginalVariance`(*args)	Compute the conditional variance of the Gaussian process on a point (or several points).
`getConditionalMean`(*args)	Compute the conditional 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.
`getInputSample`()	Accessor to the input sample.
`getMetaModel`()	Accessor to the metamodel.
`getName`()	Accessor to the object's name.
`getOutputSample`()	Accessor to the output sample.
`getRelativeErrors`()	Accessor to the relative errors.
`getResiduals`()	Accessor to the residuals.
`getTrendCoefficients`()	Accessor to the trend coefficients.
`hasName`()	Test if the object is named.
`setInputSample`(sampleX)	Accessor to the input sample.
`setMetaModel`(metaModel)	Accessor to the metamodel.
`setName`(name)	Accessor to the object's name.
`setOutputSample`(sampleY)	Accessor to the output sample.
`setRelativeErrors`(relativeErrors)	Accessor to the relative errors.
`setResiduals`(residuals)	Accessor to the residuals.

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()

__init__(*args)¶

getBasis()¶

Accessor to the functional basis.

Returns:

basisBasis: Functional basis of size $b$ : $(\varphi^l: \Rset^d \rightarrow \Rset^p)$ for each $l \in [1, b]$ .

Notes

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

getClassName()¶

Accessor to the object’s name.

Returns:

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

getConditionalCovariance(*args)¶

Compute the conditional 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(*args)¶

Compute the conditional 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(*args)¶

Compute the conditional 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(*args)¶

Compute the conditional 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()¶

Accessor to the covariance coefficients.

Returns:

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

getCovarianceModel()¶

Accessor to the covariance model.

Returns:

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

getInputSample()¶

Accessor to the input sample.

Returns:

inputSampleSample: The input sample.

getMetaModel()¶

Accessor to the metamodel.

Returns:

metaModelFunction: Metamodel.

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getOutputSample()¶

Accessor to the output sample.

Returns:

outputSampleSample: The output sample.

getRelativeErrors()¶

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()¶

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.

getTrendCoefficients()¶

Accessor to the trend coefficients.

Returns:

trendCoefPoint: The trend coefficients vectors $(\vect{\alpha}^1, \dots, \vect{\alpha}^p)$ stored as a Point.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

setInputSample(sampleX)¶

Accessor to the input sample.

Parameters:

inputSampleSample: The input sample.

setMetaModel(metaModel)¶

Accessor to the metamodel.

Parameters:

metaModelFunction: Metamodel.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setOutputSample(sampleY)¶

Accessor to the output sample.

Parameters:

outputSampleSample: The output sample.

setRelativeErrors(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(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.