GeneralLinearModelAlgorithm¶

class
GeneralLinearModelAlgorithm
(*args)¶ Algorithm for the evaluation of general linear models.
 Available constructors:
GeneralLinearModelAlgorithm(inputSample, outputSample, covarianceModel, basis, normalize=True, keepCovariance=True)
GeneralLinearModelAlgorithm(inputSample, outputSample, covarianceModel, basisCollection, normalize=True, keepCovariance=True)
Parameters:  inputSample, outputSample :
Sample
or 2darray The samples and .
 inputTransformation :
Function
Function used 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.
 basisCollection : collection of
Basis
Collection of functional basis: one basis for each marginal output.
An empty collection means that no trend is estimated.
 covarianceModel :
CovarianceModel
Covariance model of the Gaussian process. See notes for the details.
 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 GeneralLinearModelAlgorithmNormalizeData
 keepCovariance : bool, optional
Indicates whether the covariance matrix has to be stored in the result structure GeneralLinearModelResult. Default is set in resource map key GeneralLinearModelAlgorithmKeepCovariance
Notes
We suppose we have a sample where for all , with a given function.
The objective is to build a metamodel , using a general linear model: the sample is considered as the restriction of a Gaussian process on . The Gaussian process is defined by:
where:
with and the trend functions.
is a Gaussian process of dimension with zero mean and covariance function (see
CovarianceModel
for the notations).We note:
The GeneralLinearModelAlgorithm class estimates the coefficients where is the vector of parameters of the covariance model (a subset of ) that has been declared as active (by default, the full vectors and ).
The estimation is done by maximizing the reduced loglikelihood of the model, see its expression below.
If a normalizing transformation has been used, the meta model is built on the inputs .
Estimation of the parameters
We note:
where .
The model likelihood writes:
If is the Cholesky factor of , ie the lower triangular matrix with positive diagonal such that , then:
(1)¶
The maximization of (1) leads to the following optimality condition for :
This expression of as a function of is taken as a general relation between and and is substituted into (1), leading to a reduced loglikelihood function depending solely on .
In the particular case where and is a part of , then a further reduction is possible. In this case, if is the vector in which has been substituted by 1, then:
showing that is a function of only, and the optimality condition for reads:
which leads to a further reduction of the loglikelihood function where both and are replaced by their expression in terms of .
The default optimizer is
TNC
and can be changed thanks to the setOptimizationAlgorithm method. User could also change the default optimization solver by setting the GeneralLinearModelAlgorithmDefaultOptimizationAlgorithm resource map key to one of theNLopt
solver names.It is also possible to proceed as follows:
 ask for the reduced loglikelihood function of the GeneralLinearModelAlgorithm thanks to the getObjectiveFunction() method
 optimize it with respect to the parameters and using any optimization algorithms (that can take into account some additional constraints if needed)
 set the optimal parameter value into the covariance model used in the GeneralLinearModelAlgorithm
 tell the algorithm not to optimize the parameter using setOptimizeParameters
 The behaviour of the reduction is controlled by the following keys in
ResourceMap
:  ResourceMap.SetAsBool(‘GeneralLinearModelAlgorithmUseAnalyticalAmplitudeEstimate’, true) to use the reduction associated to . It has no effect if or if and is not part of
 ResourceMap.SetAsBool(‘GeneralLinearModelAlgorithmUnbiasedVariance’, true) allows to use the unbiased estimate of where is replaced by in the optimality condition for .
With huge samples, the hierarchical matrix implementation could be used if OpenTURNS had been compiled with hmatoss support.
This implementation, which is based on a compressed 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 GeneralLinearModelAlgorithmLinearAlgebra 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.SymbolicFunction(['x'], ['x * sin(x)']) >>> inputSample = ot.Sample([[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.GeneralLinearModelAlgorithm(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 loglikelihood function that writes as argument of the covariance’s model parameters. getOptimizationAlgorithm
()Accessor to solver used to optimize the covariance model parameters. getOptimizationBounds
()Optimization bounds accessor. 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. setOptimizationAlgorithm
(solver)Accessor to the solver used to optimize the covariance model parameters. setOptimizationBounds
(optimizationBounds)Optimization bounds accessor. 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)¶ Initialize self. See help(type(self)) for accurate signature.

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.
 distribution :

getId
()¶ Accessor to the object’s id.
Returns:  id : int
Internal unique identifier.

getInputTransformation
()¶ Get the function normalizing the input.
Returns:  transformation :
Function
Function T that normalizes the input.
 transformation :

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 loglikelihood function that writes as argument of the covariance’s model parameters.
Returns: Notes
The loglikelihood 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.SymbolicFunction(['x0'], ['x0 * sin(x0)']) >>> inputSample = ot.Sample([[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.GeneralLinearModelAlgorithm(inputSample, outputSample, covarianceModel, basis) >>> algo.run()
Get the loglikelihood function:
>>> likelihoodFunction = algo.getObjectiveFunction()

getOptimizationAlgorithm
()¶ Accessor to solver used to optimize the covariance model parameters.
Returns:  algorithm :
OptimizationAlgorithm
Solver used to optimize the covariance model parameters. Default optimizer is
TNC
 algorithm :

getOptimizationBounds
()¶ Optimization bounds accessor.
Returns:  bounds :
Interval
Bounds for covariance model parameter optimization.
 bounds :

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 :
Sample
The output sample .
 outputSample :

getResult
()¶ Get the results of the metamodel computation.
Returns:  result :
GeneralLinearModelResult
Structure containing all the results obtained after computation and created by the method
run()
.
 result :

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
GeneralLinearModelResult
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.
 distribution :

setInputTransformation
(inputTransformation)¶ Set the function normalizing the input.
Parameters:  transformation :
Function
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
 transformation :

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.

setOptimizationAlgorithm
(solver)¶ Accessor to the solver used to optimize the covariance model parameters.
Parameters:  algorithm :
OptimizationAlgorithm
Solver used to optimize the covariance model parameters.
 algorithm :

setOptimizationBounds
(optimizationBounds)¶ Optimization bounds accessor.
Parameters:  bounds :
Interval
Bounds for covariance model parameter optimization.
Notes
Parameters involved by this method are:
 Scale parameters,
 Amplitude parameters if output dimension is greater than one or analytical sigma disabled,
 Additional parameters.
Lower & upper bounds are defined in resource map. Default lower upper bounds value for all parameters is and defined thanks to the GeneralLinearModelAlgorithmDefaultOptimizationLowerBound resource map key.
For scale parameters, default upper bounds are set as times the difference between the max and min values of X for each coordinate, X being the (transformed) input sample. The value is defined in resource map (GeneralLinearModelAlgorithmDefaultOptimizationScaleFactor).
Finally for other parameters (amplitude,…), default upper bound is set to (corresponding resource map key is GeneralLinearModelAlgorithmDefaultOptimizationUpperBound)
 bounds :

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.