# GaussProductExperiment¶

class GaussProductExperiment(*args)

Gauss product experiment.

Available constructors:

GaussProductExperiment(marginalSizes)

GaussProductExperiment(distribution)

GaussProductExperiment(distribution, marginalSizes)

Parameters:
marginalSizessequence of positive int

Numbers of points for each direction. Then, the total number of points generated is . By default, the value of is equal to . The default marginal size is defined in the GaussProductExperiment-DefaultMarginalSize key of the ResourceMap.

distribution

Distribution of dimension with an independent copula.

Notes

The Gauss product experiment is a tensor product experiment which uses Gauss points in each direction. Using the notations of the TensorProductExperiment documentation, we have and for every index .

For each marginal, the algorithm computes the family of orthogonal polynomials depending on the marginal distribution using the StandardDistributionPolynomialFactory class. The input distribution must have an independent copula.

Examples

>>> import openturns as ot
>>> marginal_1 = ot.Exponential()
>>> marginal_2 = ot.Triangular(-1.0, -0.5, 1.0)
>>> distribution = ot.ComposedDistribution([marginal_1, marginal_2])
>>> marginalSizes = [3, 2]
>>> experiment = ot.GaussProductExperiment(distribution, marginalSizes)
>>> nodes, weights = experiment.generateWithWeights()
>>> print(nodes)
[ X0        X1        ]
0 : [  0.415775 -0.511215 ]
1 : [  2.29428  -0.511215 ]
2 : [  6.28995  -0.511215 ]
3 : [  0.415775  0.357369 ]
4 : [  2.29428   0.357369 ]
5 : [  6.28995   0.357369 ]
>>> print(weights)
[0.429018,0.168036,0.00626806,0.282075,0.110482,0.00412119]

In the following example [morokoff1995], we integrate a dimension 5 integrand with marginal probability density functions. We use 7 nodes for each marginal, leading to a total of nodes for the tensor product Gauss quadrature.

>>> import openturns as ot
>>> def g_function_py(x):
...     value = (1.0 + 1.0 / dimension) ** dimension
...     for i in range(dimension):
...         value *= x[i] ** (1.0 / dimension)
...     return [value]
>>>
>>> dimension = 5
>>> g_function = ot.PythonFunction(dimension, 1, g_function_py)
>>> interval = ot.Interval([0.0] * dimension, [1.0] * dimension)
>>> integral = 1.0
>>> print('Exact integral = ', integral)
Exact integral =  1.0
>>> marginal_levels = [7] * dimension
>>> distribution = ot.ComposedDistribution([ot.Uniform(0.0, 1.0)] * dimension)
>>> experiment = ot.GaussProductExperiment(distribution, marginal_levels)
>>> nodes, weights = experiment.generateWithWeights()
>>> number_of_nodes = nodes.getSize()
>>> print('Number of nodes = ', number_of_nodes)
Number of nodes =  16807
>>> function_values = g_function(nodes).asPoint()
>>> approximate_integral = function_values.dot(weights)
>>> print('Approximate integral = ', approximate_integral)
Approximate integral =  1.0040...

Methods

 generate() Generate points according to the type of the experiment. Generate points and their associated weight according to the type of the experiment. Accessor to the object's name. Accessor to the distribution. getId() Accessor to the object's id. Get the marginal sizes. getName() Accessor to the object's name. Accessor to the object's shadowed id. getSize() Accessor to the size of the generated sample. Accessor to the object's visibility state. hasName() Test if the object is named. Ask whether the experiment has uniform weights. Test if the object has a distinguishable name. isRandom() Accessor to the randomness of quadrature. setDistribution(distribution) Accessor to the distribution. setMarginalSizes(marginalSizes) Set the marginal sizes. setName(name) Accessor to the object's name. setShadowedId(id) Accessor to the object's shadowed id. setSize(size) Accessor to the size of the generated sample. setVisibility(visible) Accessor to the object's visibility state.
__init__(*args)
generate()

Generate points according to the type of the experiment.

Returns:
sampleSample

Points which constitute the design of experiments with . The sampling method is defined by the nature of the weighted experiment.

Examples

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> myExperiment = ot.MonteCarloExperiment(ot.Normal(2), 5)
>>> sample = myExperiment.generate()
>>> print(sample)
[ X0        X1        ]
0 : [  0.608202 -1.26617  ]
1 : [ -0.438266  1.20548  ]
2 : [ -2.18139   0.350042 ]
3 : [ -0.355007  1.43725  ]
4 : [  0.810668  0.793156 ]
generateWithWeights()

Generate points and their associated weight according to the type of the experiment.

Returns:
sampleSample

The points which constitute the design of experiments. The sampling method is defined by the nature of the experiment.

weightsPoint of size

Weights associated with the points. By default, all the weights are equal to .

Examples

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> myExperiment = ot.MonteCarloExperiment(ot.Normal(2), 5)
>>> sample, weights = myExperiment.generateWithWeights()
>>> print(sample)
[ X0        X1        ]
0 : [  0.608202 -1.26617  ]
1 : [ -0.438266  1.20548  ]
2 : [ -2.18139   0.350042 ]
3 : [ -0.355007  1.43725  ]
4 : [  0.810668  0.793156 ]
>>> print(weights)
[0.2,0.2,0.2,0.2,0.2]
getClassName()

Accessor to the object’s name.

Returns:
class_namestr

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

getDistribution()

Accessor to the distribution.

Returns:
distributionDistribution

Distribution used to generate the set of input data.

getId()

Accessor to the object’s id.

Returns:
idint

Internal unique identifier.

getMarginalSizes()

Get the marginal sizes.

Returns:
marginalSizesIndices

Numbers of points for each direction.

getName()

Accessor to the object’s name.

Returns:
namestr

The name of the object.

Accessor to the object’s shadowed id.

Returns:
idint

Internal unique identifier.

getSize()

Accessor to the size of the generated sample.

Returns:
sizepositive int

Number of points constituting the design of experiments.

getVisibility()

Accessor to the object’s visibility state.

Returns:
visiblebool

Visibility flag.

hasName()

Test if the object is named.

Returns:
hasNamebool

True if the name is not empty.

hasUniformWeights()

Ask whether the experiment has uniform weights.

Returns:
hasUniformWeightsbool

Whether the experiment has uniform weights.

hasVisibleName()

Test if the object has a distinguishable name.

Returns:
hasVisibleNamebool

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

isRandom()

Accessor to the randomness of quadrature.

Parameters:
isRandombool

Is true if the design of experiments is random. Otherwise, the design of experiment is assumed to be deterministic.

setDistribution(distribution)

Accessor to the distribution.

Parameters:
distributionDistribution

Distribution used to generate the set of input data.

setMarginalSizes(marginalSizes)

Set the marginal sizes.

Parameters:
marginalSizessequence of positive int

Numbers of points for each direction.

setName(name)

Accessor to the object’s name.

Parameters:
namestr

The name of the object.

Accessor to the object’s shadowed id.

Parameters:
idint

Internal unique identifier.

setSize(size)

Accessor to the size of the generated sample.

Parameters:
sizepositive int

Number of points constituting the design of experiments. Only available in dimension 1.

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:
visiblebool

Visibility flag.

## Examples using the class¶

Create a polynomial chaos metamodel by integration on the cantilever beam

Create a polynomial chaos metamodel by integration on the cantilever beam

Create a sparse chaos by integration

Create a sparse chaos by integration

Create a Gauss product design

Create a Gauss product design

Plot Smolyak multi-indices

Plot Smolyak multi-indices