GaussProductExperiment¶

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class GaussProductExperiment(*args)¶

Gauss product experiment.

Available constructors:

GaussProductExperiment(marginalSizes)

GaussProductExperiment(distribution)

GaussProductExperiment(distribution, marginalSizes)

Parameters

marginalSizessequence of positive int: Numbers of points $s_j$ for each direction. Then, the total number of points generated is $\textrm{card}(I)=\prod_{j = 1}^{d_x} s_j$ . By default, the value of $s_j$ is equal to $5$ . The default marginal size is defined in the GaussProductExperiment-DefaultMarginalSize key of the ResourceMap.
distribution :: Distribution $\mu$ of dimension $d_x$ with an independent copula.

See also

WeightedExperiment, TensorProductExperiment

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 $n_e = d_x$ and $d_j=1$ for every index $j = 1, ..., n_e$ .

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 $\mathcal{U}(0, 1)$ marginal probability density functions. We use 7 nodes for each marginal, leading to a total of $7^5 = 16807$ 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.
`generateWithWeights`()	Generate points and their associated weight according to the type of the experiment.
`getClassName`()	Accessor to the object's name.
`getDistribution`()	Accessor to the distribution.
`getId`()	Accessor to the object's id.
`getMarginalSizes`()	Get the marginal sizes.
`getName`()	Accessor to the object's name.
`getShadowedId`()	Accessor to the object's shadowed id.
`getSize`()	Accessor to the size of the generated sample.
`getVisibility`()	Accessor to the object's visibility state.
`hasName`()	Test if the object is named.
`hasUniformWeights`()	Ask whether the experiment has uniform weights.
`hasVisibleName`()	Test if the object has a distinguishable name.
`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 $(\Xi_i)_{i \in I}$ which constitute the design of experiments with $card I = size$ . 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 $cardI$: Weights $(\omega_i)_{i \in I}$ associated with the points. By default, all the weights are equal to $1/cardI$ .

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 $s_j$ for each direction.

getName()¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getShadowedId()¶

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 $cardI$ 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.

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 $s_j$ for each direction.

setName(name)¶

Accessor to the object’s name.

Parameters

namestr: The name of the object.

setShadowedId(id)¶

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 $cardI$ of points constituting the design of experiments.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters

visiblebool: Visibility flag.

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