SoizeGhanemFactory

class SoizeGhanemFactory(*args)

SoizeGhanem orthonormal multivariate functional family.

For the any multivariate distribution with continuous copula.

Available constructor:

SoizeGhanemFactory()

SoizeGhanemFactory(measure, useCopula)

SoizeGhanemFactory(measure, phi, useCopula)

Parameters:

measure : Distribution

The measure defining the inner product of the factory.

phi : EnumerateFunction

The function mapping the index of the multivariate basis function to the multi-index of the marginal variables. Default is to use the LinearEnumerateFunction.

useCopula : bool

Flag to tell if the copula density has to be used directly or indirectly through the joint PDF of the measure. Default is True.

Notes

This class implements the multivariate orthonormal basis associated with an arbitrary multidimensional distribution with continuous copula and marginals with well-defined orthonormal polyomials of arbitrary order. The details are in [SoizeGhanem2004].

Examples

>>> import openturns as ot
>>> marginals = [ot.Uniform(-1.0, 1.0), ot.Normal(0.0, 1.0)]
>>> copula = ot.ClaytonCopula(1.0)
>>> distribution = ot.ComposedDistribution(marginals, copula)
>>> factory = ot.SoizeGhanemFactory(distribution)
>>> point = [0.5]*2
>>> for i in range(3):
...     value = factory.build(i)(point)
...     print('SoizeGhanem_' + str(i) + '(' + str(point) + ')=' + str(value))
SoizeGhanem_0([0.5, 0.5])=[0.870518]
SoizeGhanem_1([0.5, 0.5])=[0.753891]
SoizeGhanem_2([0.5, 0.5])=[0.435259]

Methods

add(elt) Add an element in the Basis.
build(index) Get the term of the basis collection at a given index.
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.
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.
isFunctional() Tell whether the basis is functional.
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.
__init__(*args)
add(elt)

Add an element in the Basis.

Parameters:

function : NumericalMathFunction

Function added in the Basis.

Examples

>>> import openturns as ot
>>> dimension = 3
>>> input = ['x0', 'x1', 'x2']
>>> functions = []
>>> for i in range(2):
...     functions.append(ot.NumericalMathFunction(input, ['y'], [input[i]]))
>>> basis = ot.Basis(functions)
>>> basis.add(ot.NumericalMathFunction(input, ['y'], [input[2]]))
build(index)

Get the term of the basis collection at a given index.

Parameters:

index : int

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.

Returns:

function : NumericalMathFunction

The term of the basis collection at the index index.

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)
getClassName()

Accessor to the object’s name.

Returns:

class_name : str

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

getDimension()

Get the dimension of the Basis.

Returns:

dimension : int

Dimension of the Basis.

getEnumerateFunction()

Return the enumerate function.

Returns:

enumerateFunction : EnumerateFunction

Enumerate function that translates unidimensional indices into multidimensional indices.

getId()

Accessor to the object’s id.

Returns:

id : int

Internal unique identifier.

getMeasure()

Get the measure upon which the basis is orthogonal.

Returns:

measure : Distribution

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:

name : str

The name of the object.

getShadowedId()

Accessor to the object’s shadowed id.

Returns:

id : int

Internal unique identifier.

getSize()

Get the size of the Basis.

Returns:

size : int

Size of the Basis.

getSubBasis(indices)

Get a sub-basis of the Basis.

Parameters:

indices : list of int

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

Returns:

subBasis : list of NumericalMathFunction

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.NumericalMathFunction(input, ['y'], [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:

visible : bool

Visibility flag.

hasName()

Test if the object is named.

Returns:

hasName : bool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns:

hasVisibleName : bool

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

isFunctional()

Tell whether the basis is functional.

Returns:

isOrthogonal : bool

True if the basis is functional i.e. if its terms are a solution to an equation (e.g. a basis made up of Legendre functions).

isOrthogonal()

Tell whether the basis is orthogonal.

Returns:

isOrthogonal : bool

True if the basis is orthogonal.

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters:

id : int

Internal unique identifier.

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:

visible : bool

Visibility flag.