GeneralLinearModelAlgorithm

(Source code, png, hires.png, pdf)

../../../_images/GeneralLinearModelAlgorithm.png
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, outputSampleSample or 2d-array

The samples (\vect{x}_k)_{1 \leq k \leq N} \in \Rset^n and (\vect{y}_k)_{1 \leq k \leq N}\in \Rset^d.

basisBasis

Functional basis to estimate the trend: (\varphi_j)_{1 \leq j \leq n_1}: \Rset^n \rightarrow \Rset.

If d>1, the same basis is used for each marginal output.

basisCollectioncollection of Basis

Collection of d functional basis: one basis for each marginal output.

An empty collection means that no trend is estimated.

covarianceModelCovarianceModel

Covariance model of the Gaussian process. See notes for the details.

normalizebool, optional

Indicates whether the input sample has to be normalized. If True, input sample is centered & reduced. Default is set in resource map key GeneralLinearModelAlgorithm-NormalizeData

keepCovariancebool, optional

Indicates whether the covariance matrix has to be stored in the result structure GeneralLinearModelResult. Default value is set in resource map key GeneralLinearModelAlgorithm-KeepCovariance

Notes

We suppose we have a sample (\vect{x}_k, \vect{y}_k)_{1 \leq k \leq N} where \vect{y}_k = \cM(\vect{x}_k) for all k, with \cM:\Rset^n \mapsto \Rset^d a given function.

The objective is to build a metamodel \tilde{\cM}, using a general linear model: the sample (\vect{y}_k)_{1 \leq k \leq N} is considered as the restriction of a Gaussian process \vect{Y}(\omega, \vect{x}) on (\vect{x}_k)_{1 \leq k \leq N}. The Gaussian process \vect{Y}(\omega, \vect{x}) is defined by:

\vect{Y}(\omega, \vect{x}) = \vect{\mu}(\vect{x}) + \vect{W}(\omega, \vect{x})

where:

\vect{\mu}(\vect{x}) = \left(
  \begin{array}{l}
    \mu_1(\vect{x}) \\
    \dots  \\
    \mu_d(\vect{x}) 
   \end{array}
 \right)

with \mu_\ell(\vect{x}) = \sum_{j=1}^{n_\ell} \beta_j^\ell \varphi_j^\ell(\vect{x}) and \varphi_j^\ell: \Rset^n \rightarrow \Rset the trend functions.

\vect{W} is a Gaussian process of dimension d with zero mean and covariance function C = C(\vect{\theta}, \vect{\sigma}, \mat{R}, \vect{\lambda}) (see CovarianceModel for the notations).

We note:

\vect{\beta}^\ell = \left(
  \begin{array}{l}
    \beta_1^\ell \\
    \dots  \\
    \beta_{n_\ell}^\ell
   \end{array}
 \right) \in \Rset^{n_\ell}
 \quad \mbox{ and } \quad 
 \vect{\beta} = \left(
  \begin{array}{l}
     \vect{\beta}^1\\
    \dots  \\
     \vect{\beta}^d
   \end{array}
 \right)\in \Rset^{\sum_{\ell=1}^p n_\ell}

The GeneralLinearModelAlgorithm class estimates the coefficients \beta_j^\ell, \vect{p} where \vect{p} is the vector of parameters of the covariance model (a subset of \vect{\theta}, \vect{\sigma}, \mat{R}, \vect{\lambda}) that has been declared as active (by default, the full vectors \vect{\theta} and \vect{\sigma}).

The estimation is done by maximizing the reduced log-likelihood of the model, see its expression below.

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

Estimation of the parameters \beta_j^\ell, \vect{p}

We note:

\vect{y} = \left(
     \begin{array}{l}
       \vect{y}_1 \\
       \dots  \\
       \vect{y}_N
      \end{array}
    \right) \in \Rset^{dN},
    \quad 
    \vect{m}_{\vect{\beta}} = \left(
     \begin{array}{l}
       \vect{\mu}(\vect{x}_1) \\
       \dots  \\
       \vect{\mu}(\vect{x}_N)
      \end{array}
    \right) \in \Rset^{dN}

and

\mat{C}_{\vect{p}} = \left(
  \begin{array}{lcl}
    \mat{C}_{11} & \dots &  \mat{C}_{1N}\\
    \dots & \dots & \\
    \mat{C}_{N1} & \dots &  \mat{C}_{NN}
   \end{array}
 \right) \in \cS_{dN}^+(\Rset)

where \mat{C}_{ij} = C_{\vect{p}}(\vect{x}_i, \vect{x}_j).

The model likelihood writes:

\cL(\vect{\beta}, \vect{p};(\vect{x}_k, \vect{y}_k)_{1 \leq k \leq N}) = \dfrac{1}{(2\pi)^{dN/2} |\det \mat{C}_{\vect{p}}|^{1/2}} \exp\left[ -\dfrac{1}{2}\Tr{\left( \vect{y}-\vect{m} \right)} \mat{C}_{\vect{p}}^{-1}  \left( \vect{y}-\vect{m} \right)  \right]

If \mat{L} is the Cholesky factor of \mat{C}, ie the lower triangular matrix with positive diagonal such that \mat{L}\,\Tr{\mat{L}} = \mat{C}, then:

(1)\log \cL(\vect{\beta}, \vect{p};(\vect{x}_k, \vect{y}_k)_{1 \leq k \leq N}) = cste - \log \det \mat{L}_{\vect{p}} -\dfrac{1}{2}  \| \mat{L}_{\vect{p}}^{-1}(\vect{y}-\vect{m}_{\vect{\beta}}) \|^2

The maximization of (1) leads to the following optimality condition for \vect{\beta}:

\vect{\beta}^*(\vect{p}^*)=\argmin_{\vect{\beta}} \| \mat{L}_{\vect{p}^*}^{-1}(\vect{y}-\vect{m}_{\vect{\beta}}) \|^2

This expression of \vect{\beta}^* as a function of \vect{p}^* is taken as a general relation between \vect{\beta} and \vect{p} and is substituted into (1), leading to a reduced log-likelihood function depending solely on \vect{p}.

In the particular case where d=\dim(\vect{\sigma})=1 and \sigma is a part of \vect{p}, then a further reduction is possible. In this case, if \vect{q} is the vector \vect{p} in which \sigma has been substituted by 1, then:

\| \mat{L}_{\vect{p}}^{-1}(\vect{y}-\vect{m}_{\vect{\beta}}) \|^2=\| \mat{L}_{\vect{q}}^{-1}(\vect{y}-\vect{m}_{\vect{\beta}}) \|^2/\sigma^2

showing that \vect{\beta}^* is a function of \vect{q}^* only, and the optimality condition for \sigma reads:

\vect{\sigma}^*(\vect{q}^*)=\dfrac{1}{N}\| \mat{L}_{\vect{q}^*}^{-1}(\vect{y}-\vect{m}_{\vect{\beta}^*(\vect{q}^*)}) \|^2

which leads to a further reduction of the log-likelihood function where both \vect{\beta} and \sigma are replaced by their expression in terms of \vect{q}.

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 GeneralLinearModelAlgorithm-DefaultOptimizationAlgorithm resource map key to one of the NLopt solver names.

It is also possible to proceed as follows:

  • ask for the reduced log-likelihood function of the GeneralLinearModelAlgorithm thanks to the getObjectiveFunction() method

  • optimize it with respect to the parameters \vect{\theta} and \vect{\sigma} 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(‘GeneralLinearModelAlgorithm-UseAnalyticalAmplitudeEstimate’, true) to use the reduction associated to \sigma. It has no effect if d>1 or if d=1 and \sigma is not part of \vect{p}

  • ResourceMap.SetAsBool(‘GeneralLinearModelAlgorithm-UnbiasedVariance’, true) allows to use the unbiased estimate of \sigma where \dfrac{1}{N} is replaced by \dfrac{1}{N-p} in the optimality condition for \sigma.

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 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 GeneralLinearModelAlgorithm-LinearAlgebra resource map key should be instancied to HMAT. Default value of the key is LAPACK.

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

\hat{\vect{y}}_k = \vect{y}_k + \epsilon_k, \epsilon_k \sim \mathcal{N}(0, \tau_k^2)

Examples

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

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x'], ['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:

>>> f1 = ot.SymbolicFunction(['x'], ['sin(x)'])
>>> f2 = ot.SymbolicFunction(['x'], ['x'])
>>> f3 = ot.SymbolicFunction(['x'], ['cos(x)'])
>>> basis = ot.Basis([f1,f2, f3])
>>> covarianceModel = ot.SquaredExponential([1.0])
>>> covarianceModel.setActiveParameter([])
>>> algo = ot.GeneralLinearModelAlgorithm(inputSample, outputSample, covarianceModel, basis)
>>> algo.run()

Get the resulting meta model:

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

Methods

getClassName(self)

Accessor to the object’s name.

getDistribution(self)

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

getId(self)

Accessor to the object’s id.

getInputSample(self)

Accessor to the input sample.

getInputTransformation(self)

Get the function normalizing the input.

getName(self)

Accessor to the object’s name.

getNoise(self)

Observation noise variance accessor.

getObjectiveFunction(self)

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

getOptimizationAlgorithm(self)

Accessor to solver used to optimize the covariance model parameters.

getOptimizationBounds(self)

Optimization bounds accessor.

getOptimizeParameters(self)

Accessor to the covariance model parameters optimization flag.

getOutputSample(self)

Accessor to the output sample.

getResult(self)

Get the results of the metamodel computation.

getShadowedId(self)

Accessor to the object’s shadowed id.

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.

run(self)

Compute the response surface.

setDistribution(self, distribution)

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

setInputTransformation(self, inputTransformation)

Set the function normalizing the input.

setName(self, name)

Accessor to the object’s name.

setNoise(self, noise)

Observation noise variance accessor.

setOptimizationAlgorithm(self, solver)

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

setOptimizationBounds(self, optimizationBounds)

Optimization bounds accessor.

setOptimizeParameters(self, optimizeParameters)

Accessor to the covariance model parameters optimization flag.

setShadowedId(self, id)

Accessor to the object’s shadowed id.

setVisibility(self, visible)

Accessor to the object’s visibility state.

__init__(self, *args)

Initialize self. See help(type(self)) for accurate signature.

getClassName(self)

Accessor to the object’s name.

Returns
class_namestr

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

getDistribution(self)

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

Returns
distributionDistribution

Joint probability density function of the physical input vector.

getId(self)

Accessor to the object’s id.

Returns
idint

Internal unique identifier.

getInputSample(self)

Accessor to the input sample.

Returns
inputSampleSample

The input sample (\vect{x}_k)_{1 \leq k \leq N}.

getInputTransformation(self)

Get the function normalizing the input.

Returns
transformationFunction

Function T that normalizes the input.

getName(self)

Accessor to the object’s name.

Returns
namestr

The name of the object.

getNoise(self)

Observation noise variance accessor.

Parameters
noisesequence of positive float

The noise variance \tau_k^2 of each output value.

getObjectiveFunction(self)

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

Returns
logLikelihoodFunction

The log-likelihood function degined in (1) as a function of (\vect{\theta}, \vect{\sigma}).

Notes

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

Examples

Create the model \cM: \Rset \mapsto \Rset 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 log-likelihood function:

>>> likelihoodFunction = algo.getObjectiveFunction()
getOptimizationAlgorithm(self)

Accessor to solver used to optimize the covariance model parameters.

Returns
algorithmOptimizationAlgorithm

Solver used to optimize the covariance model parameters. Default optimizer is TNC

getOptimizationBounds(self)

Optimization bounds accessor.

Returns
boundsInterval

Bounds for covariance model parameter optimization.

getOptimizeParameters(self)

Accessor to the covariance model parameters optimization flag.

Returns
optimizeParametersbool

Whether to optimize the covariance model parameters.

getOutputSample(self)

Accessor to the output sample.

Returns
outputSampleSample

The output sample (\vect{y}_k)_{1 \leq k \leq N} .

getResult(self)

Get the results of the metamodel computation.

Returns
resultGeneralLinearModelResult

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

getShadowedId(self)

Accessor to the object’s shadowed id.

Returns
idint

Internal unique identifier.

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.

run(self)

Compute the response surface.

Notes

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

setDistribution(self, distribution)

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

Parameters
distributionDistribution

Joint probability density function of the physical input vector.

setInputTransformation(self, inputTransformation)

Set the function normalizing the input.

Parameters
transformationFunction

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(self, name)

Accessor to the object’s name.

Parameters
namestr

The name of the object.

setNoise(self, noise)

Observation noise variance accessor.

Parameters
noisesequence of positive float

The noise variance \tau_k^2 of each output value.

setOptimizationAlgorithm(self, solver)

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

Parameters
algorithmOptimizationAlgorithm

Solver used to optimize the covariance model parameters.

setOptimizationBounds(self, optimizationBounds)

Optimization bounds accessor.

Parameters
boundsInterval

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 10^{-2} and defined thanks to the GeneralLinearModelAlgorithm-DefaultOptimizationLowerBound resource map key.

For scale parameters, default upper bounds are set as 2 times the difference between the max and min values of X for each coordinate, X being the (transformed) input sample. The value 2 is defined in resource map (GeneralLinearModelAlgorithm-DefaultOptimizationScaleFactor).

Finally for other parameters (amplitude,…), default upper bound is set to 100 (corresponding resource map key is GeneralLinearModelAlgorithm-DefaultOptimizationUpperBound)

setOptimizeParameters(self, optimizeParameters)

Accessor to the covariance model parameters optimization flag.

Parameters
optimizeParametersbool

Whether to optimize the covariance model parameters.

setShadowedId(self, id)

Accessor to the object’s shadowed id.

Parameters
idint

Internal unique identifier.

setVisibility(self, visible)

Accessor to the object’s visibility state.

Parameters
visiblebool

Visibility flag.