OrthogonalProductFunctionFactory¶

class OrthogonalProductFunctionFactory(*args)¶

Base class for orthogonal multivariate functions.

Parameters:

functionslist of OrthogonalUniVariateFunctionFamily: List of orthogonal univariate function factories with the same dimension as the orthogonal basis.
enumerateFunctionEnumerateFunction, optional: 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.

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.
`getFunctionFamilyCollection`()	Get the collection of univariate orthogonal function families.
`getInputDimension`()	Get the input dimension of the Basis.
`getMarginal`(indices)	Get the marginal orthogonal functions.
`getMeasure`()	Get the measure upon which the basis is orthogonal.
`getName`()	Accessor to the object's name.
`getOutputDimension`()	Get the output dimension of the Basis.
`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

OrthogonalProductPolynomialFactory

Notes

OrthogonalProductFunctionFactory 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 recommended to use the class StandardDistributionPolynomialFactory.

Examples

Create from a list of orthogonal functions.

>>> import openturns as ot
>>> funcColl = [ot.HaarWaveletFactory(), ot.FourierSeriesFactory()]
>>> productBasis = ot.OrthogonalProductFunctionFactory(funcColl)

Set an enumerate function.

>>> funcColl = [ot.HaarWaveletFactory(), ot.FourierSeriesFactory()]
>>> inputDimension = len(funcColl)
>>> enumerateFunction = ot.LinearEnumerateFunction(inputDimension)
>>> productBasis = ot.OrthogonalProductFunctionFactory(funcColl, 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.

getFunctionFamilyCollection()¶

Get the collection of univariate orthogonal function families.

Returns:

polynomialFamilylist of OrthogonalUniVariateFunctionFamily: List of orthogonal univariate function families.

getInputDimension()¶

Get the input dimension of the Basis.

Returns:

inDimint: Input dimension of the functions.

getMarginal(indices)¶

Get the marginal orthogonal functions.

Parameters:

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

Returns:

functionFamilylistlist of OrthogonalUniVariateFunctionFamily: The marginal orthogonal functions.

Examples

>>> import openturns as ot
>>> funcColl = [ot.HaarWaveletFactory(), ot.FourierSeriesFactory(), ot.HaarWaveletFactory()]
>>> productBasis = ot.OrthogonalProductFunctionFactory(funcColl)
>>> marginalProduct = productBasis.getMarginal([0, 2])  #  [ot.HaarWaveletFactory(), ot.HaarWaveletFactory()]

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.

getOutputDimension()¶

Get the output dimension of the Basis.

Returns:

outDimint: Output dimension of the functions.

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.

isTensorProduct()¶

Tell whether the basis is a tensor product

Returns:

isTensorProductbool: True if the basis is a tensor product.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

Examples using the class¶

Create multivariate functions

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OrthogonalProductFunctionFactory¶

Examples using the class¶