HyperbolicAnisotropicEnumerateFunction¶

class HyperbolicAnisotropicEnumerateFunction(*args)¶

Hyperbolic and anisotropic enumerate function.

Available constructors:

HyperbolicAnisotropicEnumerateFunction(dim)

HyperbolicAnisotropicEnumerateFunction(dim, q)

HyperbolicAnisotropicEnumerateFunction(weight)

HyperbolicAnisotropicEnumerateFunction(weight, q)

Parameters

diminteger: Dimension of the EnumerateFunction. dim must be equal to the dimension of the OrthogonalBasis.
qfloat: Correspond to the q-quasi norm parameter. If not precised, $q = 0.4$ .
weightsequence of float: Weights of the indices in each dimension. If not precised, all weights are equals to $w_i = 1$ .

See also

EnumerateFunction, LinearEnumerateFunction

Notes

The hyperbolic truncation strategy is inspired by the so-called sparsity-of- effects principle, which states that most models are principally governed by main effects and low-order interactions. Accordingly, one wishes to define an enumeration strategy which first selects those multi-indices related to main effects, i.e. with a reasonably small number of nonzero components, prior to selecting those associated with higher-order interactions.

For any real number $q \in ]0, 1]$ , one defines the anisotropic hyperbolic norm of a multi-index $\vect{\alpha}$ by:

$\| \vect{\alpha} \|_{\vect{w}, q} = \left( \sum_{i=1}^{n_X} w_i \alpha_i^q \right)^{1/q}$

where $n_X$ is the number of input variables and $(w_1, \dots , w_{n_X})$ is a sequence of real positive numbers called weights. Functions of input variables with smaller weights are selected first for the functional basis.

Examples

In the following example, we create an hyperbolic enumerate function in 2 dimension with a quasi-norm equal to 0.5. Notice, for example, that the function with multi-index [3,0] come before [1,1], although the sum of marginal indices is lower: this is the result of the hyperbolic quasi-norm.

>>> import openturns as ot
>>> enumerateFunction = ot.HyperbolicAnisotropicEnumerateFunction(2, 0.5)
>>> for i in range(10):
...     print(enumerateFunction(i))
[0,0]
[1,0]
[0,1]
[2,0]
[0,2]
[3,0]
[0,3]
[1,1]
[4,0]
[0,4]

In the following example, we create an hyperbolic enumerate function in 3 dimensions based on the weights [1,2,4]. Notice that the first marginal index, with weight equal to 1, comes first in the enumeration.

>>> import openturns as ot
>>> enumerateFunction = ot.HyperbolicAnisotropicEnumerateFunction([1, 2, 4])
>>> for i in range(20):
...     print('i=', i, 'enum=', enumerateFunction(i))
i= 0 enum= [0,0,0]
i= 1 enum= [1,0,0]
i= 2 enum= [0,1,0]
i= 3 enum= [2,0,0]
i= 4 enum= [3,0,0]
i= 5 enum= [0,0,1]
i= 6 enum= [0,2,0]
i= 7 enum= [4,0,0]
i= 8 enum= [5,0,0]
i= 9 enum= [0,3,0]
i= 10 enum= [6,0,0]
i= 11 enum= [7,0,0]
i= 12 enum= [0,0,2]
i= 13 enum= [0,4,0]
i= 14 enum= [8,0,0]
i= 15 enum= [1,1,0]
i= 16 enum= [9,0,0]
i= 17 enum= [0,5,0]
i= 18 enum= [10,0,0]
i= 19 enum= [11,0,0]

Methods

`__call__`(index)	Call self as a function.
`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.
`getMaximumDegreeCardinal`(maximumDegree)	Get the cardinal of indices of degree inferior or equal to a given value.
`getMaximumDegreeStrataIndex`(maximumDegree)	Get the largest index of the strata of degree lower or equal to a given value.
`getName`()	Accessor to the object's name.
`getQ`()	Accessor to the norm.
`getShadowedId`()	Accessor to the object's shadowed id.
`getStrataCardinal`(strataIndex)	Get the number of members of the basis associated to a given strata.
`getStrataCumulatedCardinal`(strataIndex)	Get the cardinal of the cumulated strata above or equal to the given strata.
`getVisibility`()	Accessor to the object's visibility state.
`getWeight`()	Accessor to the weights.
`hasName`()	Test if the object is named.
`hasVisibleName`()	Test if the object has a distinguishable name.
`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.
`setQ`(q)	Accessor to the norm.
`setShadowedId`(id)	Accessor to the object's shadowed id.
`setVisibility`(visible)	Accessor to the object's visibility state.
`setWeight`(weight)	Accessor to the weights.

__init__(*args)¶

getBasisSizeFromTotalDegree(maximumDegree)¶

Get the basis size corresponding to a total degree.

Parameters

maximumDegreeint: Degree.

Returns

sizeint: Number of members of the basis of degree $max \leq maximumDegree$ .

Examples

>>> import openturns as ot
>>> ef = ot.LinearEnumerateFunction(2)
>>> size = ef.getBasisSizeFromTotalDegree(3)

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.

getMaximumDegreeCardinal(maximumDegree)¶

Get the cardinal of indices of degree inferior or equal to a given value.

Parameters

maximumDegreeint: Degree.

Returns

cardinalint: Cardinal of indices of degree $max \leq maximumDegree$ .

Examples

>>> import openturns as ot
>>> enumerateFunction = ot.LinearEnumerateFunction(2)
>>> for i in range(6):
...     indices = enumerateFunction(i)
...     degree = sum(indices)
...     print(str(int(degree))+' '+str(indices))
0 [0,0]
1 [1,0]
1 [0,1]
2 [2,0]
2 [1,1]
2 [0,2]
>>> print(enumerateFunction.getMaximumDegreeCardinal(2))
6

getMaximumDegreeStrataIndex(maximumDegree)¶

Get the largest index of the strata of degree lower or equal to a given value.

Parameters

maximumDegreeint: Degree.

Returns

indexint: Index of the last strata of degree $max \leq maximumDegree$ .

Examples

>>> import openturns as ot
>>> enumerateFunction = ot.LinearEnumerateFunction(2)
>>> for i in [1, 2]:
...     indices = enumerateFunction(i)
...     strataIndex = sum(indices) + 1
...     print(str(int(strataIndex))+' '+str(indices))
2 [1,0]
2 [0,1]
>>> print(enumerateFunction.getMaximumDegreeStrataIndex(2))
2

getName()¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getQ()¶

Accessor to the norm.

Returns

qfloat: q-quasi norm parameter.

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns

idint: Internal unique identifier.

getStrataCardinal(strataIndex)¶

Get the number of members of the basis associated to a given strata.

Parameters

strataIndexint: Index of the strata in the hierarchical basis. In the context of product of polynomial basis, this is the total polynom degree.

Returns

cardinalint: Number of members of the basis associated to the strata strataIndex. In the context of product of polynomial basis, this is the number of polynoms of the basis which total degree is strataIndex.

Examples

>>> import openturns as ot
>>> enumerateFunction = ot.LinearEnumerateFunction(2)
>>> for i in [3, 4, 5]:
...     indices = enumerateFunction(i)
...     degree = sum(indices)
...     print(str(int(degree))+' '+str(indices))
2 [2,0]
2 [1,1]
2 [0,2]
>>> print(enumerateFunction.getStrataCardinal(2))
3

getStrataCumulatedCardinal(strataIndex)¶

Get the cardinal of the cumulated strata above or equal to the given strata.

Parameters

strataIndexint: Index of the strata in the hierarchical basis. In the context of product of polynomial basis, this is the total polynomial degree.

Returns

cardinalint: Number of members of the basis associated to the strata of index inferior or equal to strataIndex. In the context of product of polynomial basis, this is the number of polynomials of the basis which total degree is inferior or equal to strataIndex.

Examples

>>> import openturns as ot
>>> enumerateFunction = ot.LinearEnumerateFunction(2)
>>> for i in range(6):
...     indices = enumerateFunction(i)
...     degree = sum(indices)
...     print(str(int(degree))+' '+str(indices))
0 [0,0]
1 [1,0]
1 [0,1]
2 [2,0]
2 [1,1]
2 [0,2]
>>> print(enumerateFunction.getStrataCumulatedCardinal(2))
6

getVisibility()¶

Accessor to the object’s visibility state.

Returns

visiblebool: Visibility flag.

getWeight()¶

Accessor to the weights.

Returns

wPoint: Weights of the indices in each dimension.

hasName()¶

Test if the object is named.

Returns

hasNamebool: True if the name is not empty.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns

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

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

setDimension(dimension)¶

Set the dimension of the EnumerateFunction.

Parameters

dimint, $dim \geq 0$: Dimension of the EnumerateFunction.

setName(name)¶

Accessor to the object’s name.

Parameters

namestr: The name of the object.

setQ(q)¶

Accessor to the norm.

Parameters

qfloat: q-quasi norm parameter.

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters

idint: Internal unique identifier.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters

visiblebool: Visibility flag.

setWeight(weight)¶

Accessor to the weights.

Parameters

wsequence of float: Weights of the indices in each dimension.

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