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.

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.
`getMarginal`(indices)	Get the marginal orthogonal polynomials.
`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.
`isTensorProduct`()	Tell whether the basis is a tensor product
`setName`(name)	Accessor to the object's name.

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.

Examples

Create from a collection of orthogonal polynomials.

>>> import openturns as ot
>>> polyColl = [ot.HermiteFactory(), ot.LegendreFactory(), ot.LaguerreFactory()]
>>> productBasis = ot.OrthogonalProductPolynomialFactory(polyColl)

Easy way to create a multivariate orthonormal basis from a distribution.

>>> Xdist = ot.JointDistribution([ot.Normal(), ot.Uniform(), ot.Gamma(2.75, 1.0)])
>>> marginals = [Xdist.getMarginal(i) for i in range(Xdist.getDimension())]
>>> productBasis = ot.OrthogonalProductPolynomialFactory(marginals)

Set an enumerate function.

>>> polyColl = [ot.HermiteFactory(), ot.LegendreFactory(), ot.LaguerreFactory()]
>>> enumerateFunction = ot.LinearEnumerateFunction(3)
>>> productBasis = ot.OrthogonalProductPolynomialFactory(polyColl, enumerateFunction)

__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.

getMarginal(indices)¶

Get the marginal orthogonal polynomials.

Parameters:

indicessequence of int, $0 \leq i < \inputDim$: List of marginal indices of the input variables.

Returns:

polynomialFamilylistlist of OrthogonalUniVariatePolynomialFamily: The marginal orthogonal polynomials.

Examples

>>> import openturns as ot
>>> funcColl = [ot.LegendreFactory(), ot.LaguerreFactory(), ot.HermiteFactory()]
>>> productBasis = ot.OrthogonalProductPolynomialFactory(funcColl)
>>> marginalProduct = productBasis.getMarginal([0, 2])  # [ot.LegendreFactory(), ot.HermiteFactory()]

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.

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.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]