OrthogonalProductPolynomialFactory¶

class OrthogonalProductPolynomialFactory(*args)¶

Base class for orthogonal multivariate polynomials.

Available constructors:

OrthogonalProductPolynomialFactory(polynomials)

OrthogonalProductPolynomialFactory(polynomials, enumerateFunction)

OrthogonalProductPolynomialFactory(marginals)

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.

See also

StandardDistributionPolynomialFactory

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. To facilitate the construction of the basis it is recommanded to use the class StandardDistributionPolynomialFactory.

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.ComposedDistribution([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)

>>> # Easier way to construct the same multivariate orthonormal basis
>>> marginals = [Xdist.getMarginal(i) for i in range(Xdist.getDimension())]
>>> productBasis = ot.OrthogonalProductPolynomialFactory(marginals)

Methods

`build`(*args)	Get the term of the basis collection at a given index or multi-indices.
`getClassName`()	Accessor to the object’s name.
`getDimension`()	Get the dimension of the Basis.
`getEnumerateFunction`()	Return the enumerate function.
`getId`()	Accessor to the object’s id.
`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.
`getPolynomialFamilyCollection`()	Get the collection of univariate orthogonal polynomial families.
`getShadowedId`()	Accessor to the object’s shadowed id.
`getSize`()	Get the size of the Basis.
`getSubBasis`(indices)	Get a sub-basis of the Basis.
`getVisibility`()	Accessor to the object’s visibility state.
`hasName`()	Test if the object is named.
`hasVisibleName`()	Test if the object has a distinguishable name.
`isFinite`()	Tell whether the basis is finite.
`isOrthogonal`()	Tell whether the basis is orthogonal.
`setName`(name)	Accessor to the object’s name.
`setShadowedId`(id)	Accessor to the object’s shadowed id.
`setVisibility`(visible)	Accessor to the object’s visibility state.

add

__init__(*args)¶: Initialize self. See help(type(self)) for accurate signature.

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.OrthogonalBasis(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__).

getDimension()¶

Get the dimension of the Basis.

Returns

dimensionint: Dimension of the Basis.

getEnumerateFunction()¶

Return the enumerate function.

Returns

enumerateFunctionEnumerateFunction: Enumerate function that translates unidimensional indices into multidimensional indices.

getId()¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

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.OrthogonalBasis(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 positiv 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.

See also

OrthogonalUniVariatePolynomialFamily.getNodesAndWeights

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.ComposedDistribution([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]

getPolynomialFamilyCollection()¶

Get the collection of univariate orthogonal polynomial families.

Returns

polynomialFamilylist of OrthogonalUniVariatePolynomialFamily: List of orthogonal univariate polynomials families.

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns

idint: Internal unique identifier.

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]

getVisibility()¶

Accessor to the object’s visibility state.

Returns

visiblebool: Visibility flag.

hasName()¶

Test if the object is named.

Returns

hasNamebool: True if the name is not empty.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns

hasVisibleNamebool: True if the name is not empty and not the default one.

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.

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters

idint: Internal unique identifier.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters

visiblebool: Visibility flag.

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