GeneralLinearModelResult¶

class GeneralLinearModelResult(*args)¶

General linear model result.

Available constructors:

GeneralLinearModelResult(inputSample, outputSample, metaModel, residuals, relativeErrors, basis, trendCoefficients, covarianceModel, optimalLogLikelihood)

GeneralLinearModelResult(inputSample, outputSample, metaModel, residuals, relativeErrors, basis, trendCoefficients, covarianceModel, covarianceCholeskyFactor, covarianceHMatrix, optimalLogLikelihood)

Parameters

inputSample, outputSampleSample: 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 :eq:metaModel.
residualsPoint: The residual errors.
relativeErrorsPoint: The relative errors.
basiscollection of Basis: Collection of the $p$ functional basis: $(\varphi_j^l: \Rset^d \rightarrow \Rset)_{1 \leq j \leq n_l}$ for each $l \in [1, p]$ . Its size should be equal to zero if the trend is not estimated.
trendCoefficientscollection of Point: The trend coefficients vectors $(\vect{\alpha}^1, \dots, \vect{\alpha}^p)$ .
covarianceModelCovarianceModel: Covariance function of the Gaussian process with its optimized parameters.
covarianceCholeskyFactorTriangularMatrix: The Cholesky factor $\mat{L}$ of $\mat{C}$ .
covarianceHMatrixHMatrix: The hmat implementation of $\mat{L}$ .
optimalLogLikelihoodfloat: The maximum log-likelihood corresponding to the model.

Notes

The structure is usually created by the method run() of a GeneralLinearModelAlgorithm, and obtained thanks to the getResult() method.

The meta model $\tilde{\cM}: \Rset^d \rightarrow \Rset^p$ is defined by:

(1)¶ $\tilde{\cM}(\vect{x}) = \left( \begin{array}{l} \mu_1(\vect{x}) \\ \dots \\ \mu_p(\vect{x}) \end{array} \right)$

where $\mu_l(\vect{x}) = \sum_{j=1}^{n_l} \alpha_j^l \varphi_j^l(\vect{x})$ and $\varphi_j^l: \Rset^d \rightarrow \Rset$ are the trend functions.

If a normalizing transformation T has been used, the meta model is built on the inputs $\vect{z}_k = T(\vect{x}_k)$ and the meta model writes:

(2)¶ $\tilde{\cM}(\vect{x}) = \left( \begin{array}{l} \mu_1\circ T(\vect{x}) \\ \dots \\ \mu_p\circ T(\vect{x}) \end{array} \right)$

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)
>>> algo = ot.GeneralLinearModelAlgorithm(sampleX, sampleY, covarianceModel, basis)
>>> algo.run()

Get the result:

>>> result = algo.getResult()

Get the meta model:

>>> metaModel = result.getMetaModel()
>>> graph = metaModel.draw(0.0, 7.0)
>>> cloud = ot.Cloud(sampleX, sampleY)
>>> cloud.setPointStyle('fcircle')
>>> graph = ot.Graph()
>>> graph.add(cloud)
>>> graph.add(f.draw(0.0, 7.0))
>>> graph.setColors(['black', 'blue', 'red'])

Methods

`getBasisCollection`(self)	Accessor to the collection of basis.
`getClassName`(self)	Accessor to the object’s name.
`getCovarianceModel`(self)	Accessor to the covariance model.
`getId`(self)	Accessor to the object’s id.
`getMetaModel`(self)	Accessor to the metamodel.
`getModel`(self)	Accessor to the model.
`getName`(self)	Accessor to the object’s name.
`getNoise`(self)	Accessor to the Gaussian process.
`getOptimalLogLikelihood`(self)	Accessor to the optimal log-likelihood of the model.
`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)	Set 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: $(\varphi_j^l: \Rset^d \rightarrow \Rset)_{1 \leq j \leq n_l}$ for each $l \in [1, p]$ .

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__).

getCovarianceModel(self)¶

Accessor to the covariance model.

Returns

covModelCovarianceModel: The covariance model of the Gaussian process W.

getId(self)¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

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.

getNoise(self)¶

Accessor to the Gaussian process.

Returns

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

getOptimalLogLikelihood(self)¶

Accessor to the optimal log-likelihood of the model.

Returns

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

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)¶

Set 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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