EnumerateFunction¶

class EnumerateFunction(*args)¶

Enumerate function.

See also

LinearEnumerateFunction, HyperbolicAnisotropicEnumerateFunction, NormInfEnumerateFunction

Notes

EnumerateFunction is a bijection $\vect{\alpha}$ from $\Nset$ to $\Nset^{n_X}$ . This bijection is based on a particular enumeration rule of the set of multi-indices. For any integer $i \in \Nset$ , the value of the enumerate function is:

$\vect{\alpha}(i) = (\alpha_1(i),\dots, \alpha_{n_X}(i)) \in {\Nset}^{n_X}.$

The first multi-index is:

$\vect{\alpha}(0) = (0, 0, \dots, 0).$

Let $i, j \in \Nset$ be any pair of indices. If $|i - j|\leq 1$ then:

$\left|\sum_{k=1}^{n_X} \left[ \alpha_k(i) - \alpha_k(j) \right]\right| \leq 1$

where $\alpha_k(i)$ is the k-th component of the multi-index $\vect{\alpha}(i)$ . This provides a necessary but insufficient condition for the construction of the bijection: a supplementary hypothesis must be made.

For example, consider the dimension $\textrm{dim}=2$ . The following mapping is an enumerate function:

$\vect{\alpha}(0) &= (0, \; 0) \\ \vect{\alpha}(1) &= (1, \; 0) \\ \vect{\alpha}(2) &= (0, \; 1) \\ \vect{\alpha}(3) &= (2, \; 0) \\ \vect{\alpha}(4) &= (1, \; 1) \\ \vect{\alpha}(5) &= (0, \; 2) \\ \vect{\alpha}(6) &= (3, \; 0)$

For the functional expansion (respectively polynomial chaos expansion), the multi-index $\vect{\alpha}$ is the collection of marginal 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 defines the marginal degrees of the univariate functions (respectively univariate polynomials). The multivariate tensor product function is based on the product of marginal univariate functions:

$\psi_{\vect{\alpha}}(\vect{x}) = \prod_{i = 1}^{n_X} \pi_{\alpha_i}^{(i)}(x_i)$

for any point $\vect{x}$ where $\pi_{\alpha_i}^{(i)}$ is the polynomial of degree $\alpha_i$ for the i-th marginal. The total degree of the polynomial is equal to the sum of the integers in the multi-index:

$\operatorname{degree} \left(\psi_{\vect{\alpha}} \right) = \sum_{i = 1}^{n_X} \alpha_i = \|\vect{\alpha}\|_1.$

For example, consider the multi-index $\vect{\alpha} = (2, 0, 3)$ . This multi-index defines a multivariate polynomial which has a input vector of dimension 3. The marginal degree of the polynomial of the first dimension is equal to 2, the marginal degree of the polynomial of the second dimension is equal to 0 and the marginal degree of the polynomial of the third dimension is equal to 3.

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

`getBasisSizeFromTotalDegree`(maximumDegree)	Get the basis size corresponding to a total degree.
`getClassName`()	Accessor to the object's name.
`getDimension`()	Return the dimension of the EnumerateFunction.
`getId`()	Accessor to the object's id.
`getImplementation`()	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.
`getName`()	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.
`getUpperBound`()	Accessor to the upper bound.
`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.
`setUpperBound`(upperBound)	Accessor to the upper bound.

__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 $\leq \max_{deg}$ .

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, $dim \geq 0$: 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 $\leq \max_{deg}$ .

Notes

In the specific context of a linear enumeration (LinearEnumerateFunction) this is also the cumulated cardinal of stratas of index $\leq \max_{deg}$ .

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 $\leq \max_{deg}$ .

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

getUpperBound()¶

Accessor to the upper bound.

Returns:

ubsequence of int: Upper bound of the indices (inclusive).

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