# EnumerateFunction¶

class EnumerateFunction(*args)

Enumerate function.

Notes

EnumerateFunction represents a bijection from to . This bijection is based on a particular enumeration rule of the set of multi-indices.

For every integer , we associate a multi-index

The first multi-index is

Let be any pair of indices. If then

where represents the k-th component of the multi-index .

This provides a necessary but insufficient condition for the construction of the bijection: a supplementary hypothesis must be made.

For example, consider the dimension . The following mapping is an enumerate function:

For the functional expansion (respectively polynomial chaos expansion), the multi-index represents the collection of degrees of the selected orthogonal functions (respectively orthogonal polynomials). More precisely, after the selection of the type of orthogonal functions (respectively orthogonal polynomials) for the construction of the orthogonal basis, the EnumerateFunction characterizes the term of the basis by providing the degrees of the univariate functions (respectively univariate polynomials).

The total degree of the polynomial of the multivariate basis is equal to the sum of all the integers given in the list.

Examples

>>> import openturns as ot
>>> dim = 2
>>> enum_func = ot.LinearEnumerateFunction(dim)
>>> for i in range(6):
...     print(enum_func(i))
[0,0]
[1,0]
[0,1]
[2,0]
[1,1]
[0,2]


Methods

 __call__(index) Call self as a function. getBasisSizeFromTotalDegree(maximumDegree) Get the basis size corresponding to a total degree. Accessor to the object's name. Return the dimension of the EnumerateFunction. Accessor to the object's id. Accessor to the underlying implementation. getMaximumDegreeCardinal(maximumDegree) Get the number of multi-indices of total degree lower or equal to a threshold. getMaximumDegreeStrataIndex(maximumDegree) Get the largest index of the strata containing multi-indices lower or equal to the given maximum degree. Accessor to the object's name. getStrataCardinal(strataIndex) Get the number of multi-indices in the basis inside a given strata. getStrataCumulatedCardinal(strataIndex) Get the number of multi-indices in the basis inside a range of stratas. inverse(indices) Get the antecedent of a indices list in the EnumerateFunction. setDimension(dimension) Set the dimension of the EnumerateFunction. setName(name) Accessor to the object's name.
 getUpperBound setUpperBound
__init__(*args)
getBasisSizeFromTotalDegree(maximumDegree)

Get the basis size corresponding to a total degree.

Parameters:
max_degint

Maximum total degree.

Returns:
sizeint

Number of multi-indices in the basis of total degree .

Notes

In the specific context of a linear enumeration (LinearEnumerateFunction) this is also the cumulated cardinal of stratas up to max_deg.

Examples

>>> import openturns as ot
>>> dim = 2
>>> enum_func = ot.LinearEnumerateFunction(dim)
>>> enum_func.getBasisSizeFromTotalDegree(3)
10
>>> enum_func.getStrataCumulatedCardinal(3)
10

getClassName()

Accessor to the object’s name.

Returns:
class_namestr

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

getDimension()

Return the dimension of the EnumerateFunction.

Returns:
dimint,

Dimension of the EnumerateFunction.

getId()

Accessor to the object’s id.

Returns:
idint

Internal unique identifier.

getImplementation()

Accessor to the underlying implementation.

Returns:
implImplementation

A copy of the underlying implementation object.

getMaximumDegreeCardinal(maximumDegree)

Get the number of multi-indices of total degree lower or equal to a threshold.

Parameters:
max_degint

Maximum total degree.

Returns:
cardinalint

Number of multi-indices in the basis of total degree .

Notes

In the specific context of a linear enumeration (LinearEnumerateFunction) this is also the cumulated cardinal of stratas of index .

Examples

>>> import openturns as ot
>>> dim = 2
>>> enum_func = ot.LinearEnumerateFunction(dim)
>>> enum_func.getMaximumDegreeCardinal(2)
6

getMaximumDegreeStrataIndex(maximumDegree)

Get the largest index of the strata containing multi-indices lower or equal to the given maximum degree.

Parameters:
max_degint

Maximum total degree.

Returns:
indexint

Index of the last strata that contains multi-indices of total degree .

Notes

In the specific context of a linear enumeration (LinearEnumerateFunction) this is the strata of index max_deg.

Examples

>>> import openturns as ot
>>> dim = 2
>>> enum_func = ot.LinearEnumerateFunction(dim)
>>> enum_func.getMaximumDegreeStrataIndex(2)
2

getName()

Accessor to the object’s name.

Returns:
namestr

The name of the object.

getStrataCardinal(strataIndex)

Get the number of multi-indices in the basis inside a given strata.

Parameters:
strataIndexint

Index of the strata of the tensorized basis.

Returns:
cardinalint

Number of multi-indices in the basis inside the strata strataIndex.

Notes

In the specific context of a linear enumeration (LinearEnumerateFunction) the strata strataIndex consists of a hyperplane of all the multi-indices of total degree strataIndex, and its cardinal is strataIndex + 1.

Examples

>>> import openturns as ot
>>> dim = 2
>>> enum_func = ot.LinearEnumerateFunction(dim)
>>> enum_func.getStrataCardinal(2)
3

getStrataCumulatedCardinal(strataIndex)

Get the number of multi-indices in the basis inside a range of stratas.

Parameters:
strataIndexint

Index of the strata of the tensorized basis.

Returns:
cardinalint

Number of multi-indices in the basis inside the stratas of index lower or equal to strataIndex.

Notes

The number of multi-indices is the total of multi-indices inside the stratas. In the specific context of a linear enumeration (LinearEnumerateFunction) this returns the number of multi-indices of maximal total degree strataIndex.

Examples

>>> import openturns as ot
>>> dim = 2
>>> enum_func = ot.LinearEnumerateFunction(dim)
>>> enum_func.getStrataCumulatedCardinal(2)
6
>>> sum([enum_func.getStrataCardinal(i) for i in range(3)])
6

inverse(indices)

Get the antecedent of a indices list in the EnumerateFunction.

Parameters:
multiIndexsequence of int

List of indices.

Returns:
antecedentint

Represents the antecedent of the multiIndex in the EnumerateFunction.

Examples

>>> import openturns as ot
>>> dim = 2
>>> enum_func = ot.LinearEnumerateFunction(dim)
>>> for i in range(6):
...     print(str(i)+' '+str(enum_func(i)))
0 [0,0]
1 [1,0]
2 [0,1]
3 [2,0]
4 [1,1]
5 [0,2]
>>> print(enum_func.inverse([1,1]))
4

setDimension(dimension)

Set the dimension of the EnumerateFunction.

Parameters:
dimint,

Dimension of the EnumerateFunction.

setName(name)

Accessor to the object’s name.

Parameters:
namestr

The name of the object.

## Examples using the class¶

Compute grouped indices for the Ishigami function

Compute grouped indices for the Ishigami function

Validate a polynomial chaos

Validate a polynomial chaos

Create a polynomial chaos metamodel by integration on the cantilever beam

Create a polynomial chaos metamodel by integration on the cantilever beam

Create a polynomial chaos for the Ishigami function: a quick start guide to polynomial chaos

Create a polynomial chaos for the Ishigami function: a quick start guide to polynomial chaos

Create a sparse chaos by integration

Create a sparse chaos by integration

Configuring an arbitrary trend in Kriging

Configuring an arbitrary trend in Kriging

Compute leave-one-out error of a polynomial chaos expansion

Compute leave-one-out error of a polynomial chaos expansion