GaussianProcessRegressionResult¶

class GaussianProcessRegressionResult(*args)¶

Gaussian process regression (aka kriging) result.

Warning

This class is experimental and likely to be modified in future releases. To use it, import the openturns.experimental submodule.

Parameters:

gpfResultGaussianProcessFitterResult: Structure result of a gaussian process fitter.
covarianceCoefficients2-d sequence of float: The $\vect{\gamma}$ defined in (2).

Methods

`getBasis`()	Accessor to the collection of basis.
`getClassName`()	Accessor to the object's name.
`getCovarianceCoefficients`()	Accessor to the covariance coefficients.
`getCovarianceModel`()	Accessor to the covariance model.
`getInputSample`()	Accessor to the input sample.
`getLinearAlgebraMethod`()	Accessor to the used linear algebra method to fit.
`getMetaModel`()	Accessor to the metamodel.
`getName`()	Accessor to the object's name.
`getNoise`()	Accessor to the Gaussian process.
`getOptimalLogLikelihood`()	Accessor to the optimal log-likelihood of the model.
`getOutputSample`()	Accessor to the output sample.
`getRegressionMatrix`()	Accessor to the regression matrix.
`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 Gaussian Process Regression (aka 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 $k \in [1, \sampleSize]$ .

Equation (1) writes:

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

where

$\Cov{\vect{Y}(\omega, \vect{x}), (\vect{Y}(\omega, \vect{x}_1),\dots,\vect{Y}(\omega, \vect{x}_{\sampleSize}))} = \left(\mat{C}(\vect{x},\vect{x}_1)|\dots|\mat{C}(\vect{x},\vect{x}_{\sampleSize})\right)\in \cM_{\outputDim,\sampleSize \times \outputDim}(\Rset)$

and

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

At the end, the meta model writes:

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

Examples

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

>>> import openturns as ot
>>> import openturns.experimental as otexp
>>> g = ot.SymbolicFunction(['x'],  ['x * sin(x)'])
>>> sampleX = [[1.0], [2.0], [3.0], [4.0], [5.0], [6.0]]
>>> sampleY = g(sampleX)

Create the algorithm:

>>> basis = ot.Basis([ot.SymbolicFunction(['x'], ['x']), ot.SymbolicFunction(['x'], ['x^2'])])
>>> covarianceModel = ot.GeneralizedExponential([2.0], 2.0)

>>> fit_algo = otexp.GaussianProcessFitter(sampleX, sampleY, covarianceModel, basis)
>>> fit_algo.run()

>>> algo = otexp.GaussianProcessRegression(fit_algo.getResult())
>>> algo.run()

Get the result:

>>> result = algo.getResult()

Get the meta model:

>>> metaModel = result.getMetaModel()

__init__(*args)¶

getBasis()¶

Accessor to the collection of basis.

Returns:

basisBasis: Functional basis of size $b$ : $(\varphi^\ell: \Rset^{\inputDim} \rightarrow \Rset^{\outputDim})$ for each $l \in [1, b]$ .

Notes

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

getClassName()¶

Accessor to the object’s name.

Returns:

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

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.

getInputSample()¶

Accessor to the input sample.

Returns:

inputSampleSample: The input sample.

getLinearAlgebraMethod()¶

Accessor to the used linear algebra method to fit.

Returns:

linAlgMethodint

The used linear algebra method to fit the model:

otexp.GaussianProcessFitterResult.LAPACK or 0: using LAPACK to fit the model,
otexp.GaussianProcessFitterResult.HMAT or 1: using HMAT to fit the model.

getMetaModel()¶

Accessor to the metamodel.

Returns:

metaModelFunction: Metamodel.

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getNoise()¶

Accessor to the Gaussian process.

Returns:

processProcess: Returns the Gaussian process $W$ with the optimized parameters.

getOptimalLogLikelihood()¶

Accessor to the optimal log-likelihood of the model.

Returns:

optimalLogLikelihoodfloat: The value of the log-likelihood corresponding to the model.

getOutputSample()¶

Accessor to the output sample.

Returns:

outputSampleSample: The output sample.

getRegressionMatrix()¶

Accessor to the regression matrix.

Returns:

processMatrix: Returns the regression matrix.

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:

trendCoefsequence of float: The trend coefficients vectors $(\vect{\alpha}^1, \dots, \vect{\alpha}^{\outputDim})$ 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.