ProjectionStrategy¶

class
ProjectionStrategy
(*args)¶ Base class for the evaluation strategies of the approximation coefficients.
 Available constructors:
 ProjectionStrategy(projectionStrategy)
Parameters: projectionStrategy :
ProjectionStrategy
A projection strategy which is a
LeastSquaresStrategy
or anIntegrationStrategy
.Notes
Consider with , and with finite variance: .
The functional chaos expansion approximates using an isoprobabilistic transformation T and an orthonormal multivariate basis of . See
FunctionalChaosAlgorithm
to get more details.The meta model of , based on the functional chaos decomposition of writes:
where K is a non empty finite set of indices, whose cardinality is denoted by P.
We detail the case where .
The vector is equivalently defined by:
(1)¶
and:
(2)¶
where and the mean is evaluated with respect to the measure .
It corresponds to two points of view:
 relation (1) means that the coefficients
minimize the quadratic error between the model and
the polynomial approximation. Use
LeastSquaresStrategy
.  relation (2) means that is the scalar product of the
model with the kth element of the orthonormal basis .
Use
IntegrationStrategy
.
In both cases, the mean is approximated by a linear quadrature formula:
(3)¶
where f is a function in .
In the approximation (3), the set I, the points and the weights are evaluated from different methods implemented in OpenTURNS in the
WeightedExperiment
.The convergence criterion used to evaluate the coefficients is based on the residual value defined in the
FunctionalChaosAlgorithm
.Methods
computeCoefficients
(function, basis, ...[, ...])getClassName
()Accessor to the object’s name. getCoefficients
()Accessor to the coefficients. getExperiment
()Accessor to the experiments. getId
()Accessor to the object’s id. getImplementation
(*args)Accessor to the underlying implementation. getInputSample
()Accessor to the input sample. getMeasure
()Accessor to the measure. getName
()Accessor to the object’s name. getOutputSample
()Accessor to the output sample. getRelativeError
()Accessor to the relative error. getResidual
()Accessor to the residual. getWeights
()Accessor to the weights. setExperiment
(weightedExperiment)Accessor to the design of experiment. setMeasure
(measure)Accessor to the measure. setName
(name)Accessor to the object’s name. 
__init__
(*args)¶

getClassName
()¶ Accessor to the object’s name.
Returns: class_name : str
The object class name (object.__class__.__name__).

getCoefficients
()¶ Accessor to the coefficients.
Returns: coef :
NumericalPoint
Coefficients .

getExperiment
()¶ Accessor to the experiments.
Returns: exp :
WeightedExperiment
Weighted experiment used to evaluate the coefficients.

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

getImplementation
(*args)¶ Accessor to the underlying implementation.
Returns: impl : Implementation
The implementation class.

getInputSample
()¶ Accessor to the input sample.
Returns: X :
NumericalSample
Input Sample.

getMeasure
()¶ Accessor to the measure.
Returns: mu : Distribution
Measure defining the scalar product.

getName
()¶ Accessor to the object’s name.
Returns: name : str
The name of the object.

getOutputSample
()¶ Accessor to the output sample.
Returns: Y :
NumericalSample
Output Sample.

getRelativeError
()¶ Accessor to the relative error.
Returns: e : float
Relative error.

getResidual
()¶ Accessor to the residual.
Returns: er : float
Residual error.

getWeights
()¶ Accessor to the weights.
Returns: w :
NumericalPoint
Weights of the design of experiments.

setExperiment
(weightedExperiment)¶ Accessor to the design of experiment.
Parameters: exp :
WeightedExperiment
Weighted design of experiment.

setMeasure
(measure)¶ Accessor to the measure.
Parameters: m : Distribution
Measure defining the scalar product.

setName
(name)¶ Accessor to the object’s name.
Parameters: name : str
The name of the object.