# GeneralizedLinearModelAlgorithm¶

class GeneralizedLinearModelAlgorithm(*args)

Algorithm for the evaluation of general linear models.

Available constructors:

GeneralizedLinearModelAlgorithm(inputSample, outputSample, covarianceModel, basis, normalize=True, keepCovariance=True)

GeneralizedLinearModelAlgorithm(inputSample, inputTransformation, outputSample, covarianceModel, basis, keepCovariance=True)

GeneralizedLinearModelAlgorithm(inputSample, outputSample, covarianceModel, multivariateBasis, normalize=True, keepCovariance=True)

GeneralizedLinearModelAlgorithm(inputSample, inputTransformation, outputSample, covarianceModel, multivariateBasis, keepCovariance=True)

Parameters: inputSample, outputSample : NumericalSample or 2d-array The samples and . inputTransformation : NumericalMathFunction Function T that helps to normalize the input sample. If used, the meta model is built on the transformed data. basis : Basis Functional basis to estimate the trend: . If , the same basis is used for each marginal output. multivariateBasis : collection of Basis Collection of functional basis: one basis for each marginal output. If the trend is not estimated, the collection must be empty. covarianceModel : CovarianceModel Covariance model of the normal process. normalize : bool, optional Indicates whether the input sample has to be normalized. OpenTURNS uses the transformation fixed by the User in inputTransformation or the empirical mean and variance of the input sample. Default is set in resource map key GeneralizedLinearModelAlgorithm-NormalizeData keepCovariance : bool, optional Indicates whether the covariance matrix has to be stored in the result structure GeneralizedLinearModelResult. Default is set in resource map key GeneralizedLinearModelAlgorithm-KeepCovariance

Notes

We suppose we have a sample where for all k, with the model.

The objective is to build a metamodel , using a generalized linear model: the sample is considered as the trace of a normal process on . The normal process is defined by:

where:

with and the trend functions.

is a normal process of dimension p with zero mean and covariance function (see CovarianceModel for the notations).

We note:

The GeneralizedLinearModelAlgorithm class estimates the coefficients by maximizing the likelihood of the model. The values of fixed by the User at the creation of the class are used as initial values for the optimization algorithm.

Note that the parameters are not optimized: they are fixed to the values specified by the User.

If a normalizing transformation T has been used, the meta model is built on the inputs .

Estimation of the parameters

We note:

where .

The model likelihood writes:

With the inferior triangular matrix such that , then:

(1)

OpenTURNS proceeds as follows:

• maximization of (1) with respect to where are fixed: it is equivalent to minimizing the term (the optimization problem is a least square problem). Then ;
• maximization of (1) with respect to where has been replaced by .
• then, we obtain the optimal values and .

In the particular case where , the first step introduces the additional relation . Thus, the second step is optimized with respect to the only parameter and the optimized values write and .

Default optimizer is TNC and can be changed thanks to the setOptimizationSolver method. User could also change the default optimization solver by setting the GeneralizedLinearModelAlgorithm-DefaultOptimizationSolver resource map key to NELDER-MEAD or LBFGS respectively for Nelder-Mead and LBFGS-B solvers.

It is also possible to proceed as follows:

• ask for the log-likelihood function of the GeneralizedLinearModelAlgorithm thanks to the getObjectiveFunction() method
• optimize it with respect to the parameters and using any optimisation algorithms (that can take into account some additional constraints if needed)
• fulfill the GeneralizedLinearModelAlgorithm with the optimized value of these parameters.

With huge samples, the hierarchical matrix implementation could be used if OpenTURNS had been compiled with hmat-oss support.

This implementation, which is based on a sparse representation of an approximated covariance matrix (and its Cholesky factor), has a better complexity both in terms of memory requirements and floating point operations. To use it, the GeneralizedLinearModelAlgorithm-LinearAlgebra resource map key should be instancied to HMAT. Default value of the key is LAPACK.

A known centered gaussian observation noise can be taken into account with setNoise():

Examples

Create the model and the samples:

>>> import openturns as ot
>>> f = ot.NumericalMathFunction(['x'], ['x * sin(x)'])
>>> inputSample = ot.NumericalSample([[1.0], [3.0], [5.0], [6.0], [7.0], [8.0]])
>>> outputSample = f(inputSample)


Create the algorithm:

>>> basis = ot.ConstantBasisFactory().build()
>>> covarianceModel = ot.SquaredExponential(1)
>>> algo = ot.GeneralizedLinearModelAlgorithm(inputSample, outputSample, covarianceModel, basis)
>>> algo.run()


Get the resulting meta model:

>>> result = algo.getResult()
>>> metamodel = result.getMetaModel()


Methods

 getClassName() Accessor to the object’s name. getDistribution() Accessor to the joint probability density function of the physical input vector. getId() Accessor to the object’s id. getInputSample() Accessor to the input sample. getInputTransformation() Get the function normalizing the input. getName() Accessor to the object’s name. getNoise() Observation noise variance accessor. getObjectiveFunction() Accessor to the log-likelihood function that writes as argument of the covariance’s model parameters. getOptimizationSolver() Accessor to solver used to optimize the covariance model parameters. getOptimizeParameters() Accessor to the covariance model parameters optimization flag. getOutputSample() Accessor to the output sample. getResult() Get the results of the metamodel computation. getShadowedId() Accessor to the object’s shadowed id. getVisibility() Accessor to the object’s visibility state. hasName() Test if the object is named. hasVisibleName() Test if the object has a distinguishable name. run() Compute the response surface. setDistribution(distribution) Accessor to the joint probability density function of the physical input vector. setInputTransformation(inputTransformation) Set the function normalizing the input. setName(name) Accessor to the object’s name. setNoise(noise) Observation noise variance accessor. setOptimizationSolver(solver) Accessor to the solver used to optimize the covariance model parameters. setOptimizeParameters(optimizeParameters) Accessor to the covariance model parameters optimization flag. setShadowedId(id) Accessor to the object’s shadowed id. setVisibility(visible) Accessor to the object’s visibility state.
__init__(*args)
getClassName()

Accessor to the object’s name.

Returns: class_name : str The object class name (object.__class__.__name__).
getDistribution()

Accessor to the joint probability density function of the physical input vector.

Returns: distribution : Distribution Joint probability density function of the physical input vector.
getId()

Accessor to the object’s id.

Returns: id : int Internal unique identifier.
getInputSample()

Accessor to the input sample.

Returns: inputSample : NumericalSample The input sample .
getInputTransformation()

Get the function normalizing the input.

Returns: transformation : NumericalMathFunction Function T that normalizes the input.
getName()

Accessor to the object’s name.

Returns: name : str The name of the object.
getNoise()

Observation noise variance accessor.

Parameters: noise : sequence of positive float The noise variance of each output value.
getObjectiveFunction()

Accessor to the log-likelihood function that writes as argument of the covariance’s model parameters.

Returns: logLikelihood : NumericalMathFunction The log-likelihood function degined in (1) as a function of .

Notes

The log-likelihood function may be useful for some postprocessing: maximization using external optimizers for example.

Examples

Create the model and the samples:

>>> import openturns as ot
>>> f = ot.NumericalMathFunction(['x0'], ['f0'], ['x0 * sin(x0)'])
>>> inputSample = ot.NumericalSample([[1.0], [3.0], [5.0], [6.0], [7.0], [8.0]])
>>> outputSample = f(inputSample)


Create the algorithm:

>>> basis = ot.ConstantBasisFactory().build()
>>> covarianceModel = ot.SquaredExponential(1)
>>> algo = ot.GeneralizedLinearModelAlgorithm(inputSample, outputSample, covarianceModel, basis)
>>> algo.run()


Get the log-likelihood function:

>>> likelihoodFunction = algo.getObjectiveFunction()

getOptimizationSolver()

Accessor to solver used to optimize the covariance model parameters.

Returns: algorithm : OptimizationSolver Solver used to optimize the covariance model parameters. Default optimizer is TNC
getOptimizeParameters()

Accessor to the covariance model parameters optimization flag.

Returns: optimizeParameters : bool Whether to optimize the covariance model parameters.
getOutputSample()

Accessor to the output sample.

Returns: outputSample : NumericalSample The output sample .
getResult()

Get the results of the metamodel computation.

Returns: Structure containing all the results obtained after computation and created by the method run().
getShadowedId()

Accessor to the object’s shadowed id.

Returns: id : int Internal unique identifier.
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.
run()

Compute the response surface.

Notes

It computes the response surface and creates a GeneralizedLinearModelResult structure containing all the results.

setDistribution(distribution)

Accessor to the joint probability density function of the physical input vector.

Parameters: distribution : Distribution Joint probability density function of the physical input vector.
setInputTransformation(inputTransformation)

Set the function normalizing the input.

Parameters: transformation : NumericalMathFunction Function that normalizes the input. The input dimension should be the same as input’s sample dimension, output dimension should be output sample’s dimension
setName(name)

Accessor to the object’s name.

Parameters: name : str The name of the object.
setNoise(noise)

Observation noise variance accessor.

Parameters: noise : sequence of positive float The noise variance of each output value.
setOptimizationSolver(solver)

Accessor to the solver used to optimize the covariance model parameters.

Parameters: algorithm : OptimizationSolver Solver used to optimize the covariance model parameters.
setOptimizeParameters(optimizeParameters)

Accessor to the covariance model parameters optimization flag.

Parameters: optimizeParameters : bool Whether to optimize the covariance model parameters.
setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters: id : int Internal unique identifier.
setVisibility(visible)

Accessor to the object’s visibility state.

Parameters: visible : bool Visibility flag.