FiniteOrthogonalFunctionFactory¶

class FiniteOrthogonalFunctionFactory(*args)¶

FiniteOrthogonalFunction orthonormal multivariate functional family.

Warning

This class is experimental and likely to be modified in future releases. To use it, import the openturns.experimental submodule.

Parameters:

functionssequence of Function: The finite set of functions orthogonal wrt the given measure.
measureDistribution: The measure defining the inner product of the factory.

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.
`getFunctionsCollection`()	Accessor to the collection of functions.
`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
`setFunctionsCollection`(functions)	Accessor to the collection of functions defining the basis.
`setName`(name)	Accessor to the object's name.

See also

openturns.StandardDistributionPolynomialFactory, openturns.OrthogonalProductPolynomialFactory, openturns.SoizeGhanemFactory

Notes

This class implements a user-defined multivariate orthonormal basis associated with an arbitrary multidimensional distribution.

Examples

>>> import openturns as ot
>>> import openturns.experimental as otexp
>>> mean = [0.0] * 2
>>> std = [1.0] * 2
>>> R = ot.CorrelationMatrix([[1.0, 0.8], [0.8, 1.0]])
>>> distribution = ot.Normal(mean, std, R)
>>> f0 = ot.SymbolicFunction(['x0', 'x1'], ['1.0'])
>>> f1 = ot.SymbolicFunction(['x0', 'x1'], ['x0'])
>>> f2 = ot.SymbolicFunction(['x0', 'x1'], ['-4.0 * x0 / 3.0 + 5.0 * x1 / 3.0'])
>>> f3 = ot.SymbolicFunction(['x0', 'x1'], ['-1.0 / sqrt(2.0) + x0^2 / sqrt(2)'])
>>> f4 = ot.SymbolicFunction(['x0', 'x1'], ['-4.0 / 3.0 - 1.885618083165693 * (-1.0 / sqrt(2.0) + x0^2 / sqrt(2.0)) + 5.0 * x0 * x1 / 3.0'])
>>> f5 = ot.SymbolicFunction(['x0', 'x1'], ['2.5141574442076222 + 16.0 / 9.0 * (-1.0 / sqrt(2.0) + x0^2 / sqrt(2.0)) - 3.142696805266918 * x0 * x1 + 25.0 / 9.0 * (-1.0 / sqrt(2.0) + x1^2 / sqrt(2.0))'])
>>> initialBasis = [f0, f1, f2, f3, f4, f5]
>>> factory = otexp.FiniteOrthogonalFunctionFactory(initialBasis, distribution)
>>> point = [0.5] * 2
>>> for i in range(len(initialBasis)):
...     value = factory.build(i)(point)
...     print(f'FiniteOrthogonalFunction_{i}({point})={value}')
FiniteOrthogonalFunction_0([0.5, 0.5])=[1]
FiniteOrthogonalFunction_1([0.5, 0.5])=[0.5]
FiniteOrthogonalFunction_2([0.5, 0.5])=[0.166667]
FiniteOrthogonalFunction_3([0.5, 0.5])=[-0.53033]
FiniteOrthogonalFunction_4([0.5, 0.5])=[0.0833333]
FiniteOrthogonalFunction_5([0.5, 0.5])=[-0.687465]

__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.

getFunctionsCollection()¶

Accessor to the collection of functions.

Returns:

fctCollFunctionCollection: The collection of scalar functions $(f_i)_{i = 1, \ldots, \ell}$ which defines the basis.

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 OrthogonalBasis: The marginal orthogonal 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.

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.

setFunctionsCollection(functions)¶

Accessor to the collection of functions defining the basis.

Parameters:

functionssequence of Function: The collection of functions $(f_i)_{i = 1, \ldots, \ell}$ .
.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

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