OrthogonalProductPolynomialFactory

class OrthogonalProductPolynomialFactory(*args)

Base class for orthogonal multivariate polynomials.

Available constructors:

OrthogonalProductPolynomialFactory(polynomials)

OrthogonalProductPolynomialFactory(polynomials, enumerateFunction)

OrthogonalProductPolynomialFactory(marginals)

OrthogonalProductPolynomialFactory(marginals, enumerateFunction)

Parameters:
polynomialssequence of OrthogonalUniVariatePolynomialFamily

List of orthogonal univariate polynomials factories with the same dimension as the orthogonal basis.

enumerateFunctionEnumerateFunction

Associates to an integer its multi-index image in the \Nset^d dimension, which is the dimension of the basis. This multi-index represents the collection of degrees of the univariate polynomials.

marginalssequence of Distribution

List of physical space marginals.

Notes

OrthogonalProductPolynomialFactory is a particular case of implementation of the OrthogonalBasis in the case of polynomial chaos expansion. It provides to the OrthogonalBasis the persistent types of the univariate orthogonal polynomials (e.g. Hermite, Legendre, Laguerre and Jacobi) needed to determine the distribution measure of projection of the input variable. Let’s note that the exact hessian and gradient have been implemented for the product of polynomials.

Examples

>>> import openturns as ot
>>> # Define the model
>>> myModel = ot.SymbolicFunction(['x1','x2','x3'], ['1+x1*x2 + 2*x3^2'])
>>> # Create a distribution of dimension 3
>>> Xdist = ot.JointDistribution([ot.Normal(), ot.Uniform(), ot.Gamma(2.75, 1.0)])
>>> # Create the multivariate orthonormal basis
>>> polyColl = [ot.HermiteFactory(), ot.LegendreFactory(), ot.LaguerreFactory(2.75)]
>>> enumerateFunction = ot.LinearEnumerateFunction(3)
>>> productBasis = ot.OrthogonalProductPolynomialFactory(polyColl, enumerateFunction)
>>> # Easier way to create the same multivariate orthonormal basis
>>> marginals = [Xdist.getMarginal(i) for i in range(Xdist.getDimension())]
>>> productBasis = ot.OrthogonalProductPolynomialFactory(marginals)

Methods

add(elt)

Add a function.

build(*args)

Get the term of the basis collection at a given index or multi-indices.

getClassName()

Accessor to the object's name.

getEnumerateFunction()

Return the enumerate function.

getInputDimension()

Get the input dimension of the Basis.

getMeasure()

Get the measure upon which the basis is orthogonal.

getName()

Accessor to the object's name.

getNodesAndWeights(degrees)

Get the nodes and the weights.

getOutputDimension()

Get the output dimension of the Basis.

getPolynomialFamilyCollection()

Get the collection of univariate orthogonal polynomial families.

getSize()

Get the size of the Basis.

getSubBasis(indices)

Get a sub-basis of the Basis.

hasName()

Test if the object is named.

isFinite()

Tell whether the basis is finite.

isOrthogonal()

Tell whether the basis is orthogonal.

setName(name)

Accessor to the object's name.

__init__(*args)
add(elt)

Add a function.

Parameters:
functiona Function

Function to be added.

build(*args)

Get the term of the basis collection at a given index or multi-indices.

Parameters:
indexint

Indicates the term of the basis which must be constructed. In other words, index is used by a bijection from \Nset to \Nset^d (with d the dimension of the basis). The bijection is detailed in EnumerateFunction.

indicessequence of int

Indicates the term of the basis which must be constructed. In other words, indices is used by a bijection from \Nset^d to \Nset (with d the dimension of the basis). The bijection is the inverse of EnumerateFunction.

Returns:
functionFunction

The term of the basis collection at the index index or the inverse of indices.

Examples

>>> import openturns as ot
>>> # Create an orthogonal basis
>>> polynomialCollection = [ot.LegendreFactory(), ot.LaguerreFactory(), ot.HermiteFactory()]
>>> productBasis =  ot.OrthogonalProductPolynomialFactory(polynomialCollection)
>>> termBasis = productBasis.build(4)
>>> print(termBasis.getEvaluation())
-1.11803 + 3.3541 * x0^2
>>> termBasis = productBasis.build(5)
>>> print(termBasis.getEvaluation())
1.73205 * x0 * (-1 + x1)
>>> termBasis2 = productBasis.build([1, 1, 0])
>>> print(termBasis2.getEvaluation())
1.73205 * x0 * (-1 + x1)
getClassName()

Accessor to the object’s name.

Returns:
class_namestr

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

getEnumerateFunction()

Return the enumerate function.

Returns:
enumerateFunctionEnumerateFunction

Enumerate function that translates unidimensional indices into multidimensional indices.

getInputDimension()

Get the input dimension of the Basis.

Returns:
inDimint

Input dimension of the functions.

getMeasure()

Get the measure upon which the basis is orthogonal.

Returns:
measureDistribution

Measure upon which the basis is orthogonal.

Examples

>>> import openturns as ot
>>> # Create an orthogonal basis
>>> polynomialCollection = [ot.LegendreFactory(), ot.LaguerreFactory(), ot.HermiteFactory()]
>>> productBasis = ot.OrthogonalProductPolynomialFactory(polynomialCollection)
>>> measure = productBasis.getMeasure()
>>> print(measure.getMarginal(0))
Uniform(a = -1, b = 1)
>>> print(measure.getMarginal(1))
Gamma(k = 1, lambda = 1, gamma = 0)
>>> print(measure.getMarginal(2))
Normal(mu = 0, sigma = 1)
getName()

Accessor to the object’s name.

Returns:
namestr

The name of the object.

getNodesAndWeights(degrees)

Get the nodes and the weights.

Parameters:
degreeslist of positive int (k_1, \dots, k_n)

List of n polynomial orders associated with the n univariate polynomials of the basis.

Returns:
nodesSample
weightsPoint

Nodes and weights of the multivariate polynomial associated with the marginal degrees (k_1, \dots, k_n) as the tensor product of the marginal orthogonal univariate polynomials, to build multivariate quadrature rules.

Examples

>>> import openturns as ot
>>> # Define the model
>>> myModel = ot.SymbolicFunction(['x1','x2','x3'], ['1+x1*x2 + 2*x3^2'])
>>> # Create a distribution of dimension 3
>>> Xdist = ot.JointDistribution([ot.Normal(), ot.Uniform(), ot.Gamma(2.75, 1.0)])
>>> # Construct the multivariate orthonormal basis
>>> polyColl = [ot.HermiteFactory(), ot.LegendreFactory(), ot.LaguerreFactory(2.75)]
>>> enumerateFunction = ot.LinearEnumerateFunction(3)
>>> productBasis = ot.OrthogonalProductPolynomialFactory(polyColl, enumerateFunction)
>>> nodes, weights = productBasis.getNodesAndWeights([2, 3, 1])
>>> print(nodes[:2])
    [ v0        v1        v2        ]
0 : [ -1        -0.774597  3.75     ]
1 : [  1        -0.774597  3.75     ]
>>> print(weights[:2])
[0.138889,0.138889]
getOutputDimension()

Get the output dimension of the Basis.

Returns:
outDimint

Output dimension of the functions.

getPolynomialFamilyCollection()

Get the collection of univariate orthogonal polynomial families.

Returns:
polynomialFamilylist of OrthogonalUniVariatePolynomialFamily

List of orthogonal univariate polynomials families.

getSize()

Get the size of the Basis.

Returns:
sizeint

Size of the Basis.

getSubBasis(indices)

Get a sub-basis of the Basis.

Parameters:
indiceslist of int

Indices of the terms of the Basis put in the sub-basis.

Returns:
subBasislist of Function

Functions defining a sub-basis.

Examples

>>> import openturns as ot
>>> dimension = 3
>>> input = ['x0', 'x1', 'x2']
>>> functions = []
>>> for i in range(dimension):
...     functions.append(ot.SymbolicFunction(input, [input[i]]))
>>> basis = ot.Basis(functions)
>>> subbasis = basis.getSubBasis([1])
>>> print(subbasis[0].getEvaluation())
[x0,x1,x2]->[x1]
hasName()

Test if the object is named.

Returns:
hasNamebool

True if the name is not empty.

isFinite()

Tell whether the basis is finite.

Returns:
isFinitebool

True if the basis is finite.

isOrthogonal()

Tell whether the basis is orthogonal.

Returns:
isOrthogonalbool

True if the basis is orthogonal.

setName(name)

Accessor to the object’s name.

Parameters:
namestr

The name of the object.

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

Kriging: choose an arbitrary trend

Kriging: choose an arbitrary trend

Metamodel of a field function

Metamodel of a field function

Use the ANCOVA indices

Use the ANCOVA indices

Create multivariate functions

Create multivariate functions

Create a multivariate basis of functions from scalar multivariable functions

Create a multivariate basis of functions from scalar multivariable functions

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

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