GaussProductExperiment¶

(Source code, png)

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.JointDistribution([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.JointDistribution([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.
`getMarginalSizes`()	Get the marginal sizes.
`getName`()	Accessor to the object's name.
`getSize`()	Accessor to the size of the generated sample.
`hasName`()	Test if the object is named.
`hasUniformWeights`()	Ask whether the experiment has uniform weights.
`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.
`setSize`(size)	Accessor to the size of the generated sample.

__init__(*args)¶

generate()¶

Generate points according to the type of the experiment.

Returns:

sampleSample: Points $(\inputReal_i)_{i = 1, ..., \sampleSize}$ of the design of experiments. The sampling method is defined by the type 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 of the design of experiments. The sampling method is defined by the nature of the experiment.
weightsPoint of size $\sampleSize$: Weights $(w_i)_{i = 1, ..., \sampleSize}$ associated with the points. By default, all the weights are equal to $\frac{1}{\sampleSize}$ .

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]