# FixedStrategy¶

class FixedStrategy(*args)

Fixed truncation strategy.

Parameters:
orthogonalBasisOrthogonalBasis

An OrthogonalBasis.

dimensionpositive int

Number of terms of the basis.

Notes

The so-called fixed strategy simply consists in selecting the first elements of the PC basis, the latter being ordered according to a given EnumerateFunction (hyperbolic or not). The retained set is built in a single pass. The truncated PC expansion is given by:

In case of a LinearEnumerateFunction, for a given natural integer representing the total polynomial degree, the number of coefficients is equal to:

where is the input dimension. This way the set of retained basis functions gathers all the polynomials with total degree not greater than . The number of terms grows polynomially both in and though, which may lead to difficulties in terms of computational efficiency and memory requirements when dealing with high-dimensional problems.

Examples

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> # Define the model
>>> inputDim = 1
>>> model = ot.SymbolicFunction(['x'], ['x*sin(x)'])
>>> # Create the input distribution
>>> distribution = ot.JointDistribution([ot.Uniform()]*inputDim)
>>> # Construction of the multivariate orthonormal basis
>>> polyColl = [0.0]*inputDim
>>> for i in range(distribution.getDimension()):
...     polyColl[i] = ot.StandardDistributionPolynomialFactory(distribution.getMarginal(i))
>>> enumerateFunction = ot.LinearEnumerateFunction(inputDim)
>>> productBasis = ot.OrthogonalProductPolynomialFactory(polyColl, enumerateFunction)
>>> # Truncature strategy of the multivariate orthonormal basis
>>> # We choose all the polynomials of degree <= 4
>>> degree = 4
>>> indexMax = enumerateFunction.getStrataCumulatedCardinal(degree)
>>> print(indexMax)
5
>>> # We keep all the polynomials of degree <= 4
>>> # which corresponds to the 5 first ones
>>> adaptiveStrategy = ot.FixedStrategy(productBasis, indexMax)


Methods

 Compute initial basis for the approximation. Accessor to the underlying orthogonal basis. Accessor to the object's name. Accessor to the maximum dimension of the orthogonal basis. Accessor to the object's name. Accessor to the selected orthogonal polynomials in the basis. Test if the object is named. Get the model selection flag. setMaximumDimension(maximumDimension) Accessor to the maximum dimension of the orthogonal basis. setName(name) Accessor to the object's name. updateBasis(alpha_k, residual, relativeError) Update the basis for the next iteration of approximation.
__init__(*args)
computeInitialBasis()

Compute initial basis for the approximation.

getBasis()

Accessor to the underlying orthogonal basis.

Returns:
basisOrthogonalBasis

Orthogonal basis of which the adaptive strategy is based.

getClassName()

Accessor to the object’s name.

Returns:
class_namestr

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

getMaximumDimension()

Accessor to the maximum dimension of the orthogonal basis.

Returns:
maximumDimensionint

Maximum dimension of the truncated basis.

getName()

Accessor to the object’s name.

Returns:
namestr

The name of the object.

getPsi()

Accessor to the selected orthogonal polynomials in the basis.

The value returned by this method depends on the specific choice of adaptive strategy and the previous calls to the updateBasis() method.

Returns:
polynomialslist of polynomials

Sequence of polynomials.

Notes

The method computeInitialBasis() must be applied first.

Examples

>>> import openturns as ot
>>> productBasis = ot.OrthogonalProductPolynomialFactory([ot.HermiteFactory()])
>>> adaptiveStrategy = ot.FixedStrategy(productBasis, 3)
[1,x0,-0.707107 + 0.707107 * x0^2]

hasName()

Test if the object is named.

Returns:
hasNamebool

True if the name is not empty.

involvesModelSelection()

Get the model selection flag.

A model selection method can be used to select the coefficients of the decomposition which enable to best predict the output. Model selection can lead to a sparse functional chaos expansion.

Returns:
involvesModelSelection: bool

True if the method involves a model selection method.

setMaximumDimension(maximumDimension)

Accessor to the maximum dimension of the orthogonal basis.

Parameters:
maximumDimensionint

Maximum dimension of the truncated basis.

setName(name)

Accessor to the object’s name.

Parameters:
namestr

The name of the object.

updateBasis(alpha_k, residual, relativeError)

Update the basis for the next iteration of approximation.

No changes are made to the basis in the fixed strategy. Hence, the number of functions in the basis is equal to dimension, whatever the number of calls to the updateBasis method.

Parameters:
alpha_ksequence of floats

The coefficients of the expansion at this step.

residualfloat

The current value of the residual.

relativeErrorfloat

The relative error.

## Examples using the class¶

Mixture of experts

Mixture of experts

Compute grouped indices for the Ishigami function

Compute grouped indices for the Ishigami function

Validate a polynomial chaos

Validate a polynomial chaos

Create a full or sparse polynomial chaos expansion

Create a full or sparse polynomial chaos expansion

Create a polynomial chaos metamodel by integration on the cantilever beam

Create a polynomial chaos metamodel by integration on the cantilever beam

Advanced polynomial chaos construction

Advanced polynomial chaos construction

Create a polynomial chaos for the Ishigami function: a quick start guide to polynomial chaos

Create a polynomial chaos for the Ishigami function: a quick start guide to polynomial chaos

Polynomial chaos expansion cross-validation

Polynomial chaos expansion cross-validation

Polynomial chaos is sensitive to the degree

Polynomial chaos is sensitive to the degree

Create a sparse chaos by integration

Create a sparse chaos by integration

Compute Sobol’ indices confidence intervals

Compute Sobol' indices confidence intervals

Metamodel of a field function

Metamodel of a field function

Use the ANCOVA indices

Use the ANCOVA indices

Compute leave-one-out error of a polynomial chaos expansion

Compute leave-one-out error of a polynomial chaos expansion