FixedStrategy¶

class FixedStrategy(*args)¶

Fixed truncation strategy.

Parameters:

orthogonalBasisOrthogonalBasis: An OrthogonalBasis.
dimensionpositive int: Number of terms of the basis.

See also

AdaptiveStrategy, CleaningStrategy

Notes

The so-called fixed strategy simply consists in selecting the first $P$ 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:

$\widehat{h} (\uZ) = \sum_{j=0}^{P-1} \vect{a}_j \Psi_j (\uZ)$

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

$P = \binom{n_X + p}{p} = \frac{(n_X + p)!}{n_X!\,p!}$

where $n_X$ is the input dimension. This way the set of retained basis functions $\{\Psi_j, j = 0, \ldots, P-1\}$ gathers all the polynomials with total degree not greater than $p$ . The number of terms $P$ grows polynomially both in $n_X$ and $p$ 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

`computeInitialBasis`()	Compute initial basis for the approximation.
`getBasis`()	Accessor to the underlying orthogonal basis.
`getClassName`()	Accessor to the object's name.
`getMaximumDimension`()	Accessor to the maximum dimension of the orthogonal basis.
`getName`()	Accessor to the object's name.
`getPsi`()	Accessor to the selected orthogonal polynomials in the basis.
`hasName`()	Test if the object is named.
`involvesModelSelection`()	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.

See also

getPsi

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 $P$ polynomials.

Notes

The method computeInitialBasis() must be applied first.

Examples

>>> import openturns as ot
>>> productBasis = ot.OrthogonalProductPolynomialFactory([ot.HermiteFactory()])
>>> adaptiveStrategy = ot.FixedStrategy(productBasis, 3)
>>> adaptiveStrategy.computeInitialBasis()
>>> print(adaptiveStrategy.getPsi())
[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: