TensorProductExperiment¶

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class TensorProductExperiment(*args)¶

Tensor product experiment.

Parameters:

experimentslist of WeightedExperiment: List of $n$ marginal experiments of the tensor product experiment. Each marginal experiment can have arbitrary dimension.

See also

WeightedExperiment, GaussProductExperiment

Notes

The tensor product design of experiments (DOE) is based on a collection of marginal multidimensional elementary designs of experiments. It is anisotropic in the sense that each marginal DOE does not necessarily have the same size. Using more points in one component may be useful if we want to get more accurate results in that particular direction.

Furthermore, each marginal DOE does not necessarily have a dimension equal to 1. In this sense, this is a generalization of a classical tensor product DOE.

Let $\mathcal{X} \subset \mathbb{R}^{d_x}$ be the integration domain and let $g : \mathcal{X} \rightarrow \mathbb{R}^{d_y}$ be an integrable function. Let $f : \mathcal{X} \rightarrow \mathbb{R}$ be a probability density function. The tensor product experiment produces an approximation of the integral :

$\int_{\mathcal{X}} g(\boldsymbol{x}) f(\boldsymbol{x}) d\boldsymbol{x} \approx \sum_{i = 1}^{s_t} w_i g\left(\boldsymbol{x}_i\right)$

where $s_t \in \mathbb{N}$ is the size of the tensor product design of experiments, $w_1, ..., w_{s_t} \in \mathbb{R}$ are the weights and $\boldsymbol{x}_1, ..., \boldsymbol{x}_{s_t} \in \mathbb{R}^{d_x}$ are the nodes.

Let $n_e \in \mathbb{N}$ be the number of marginal DOEs. For each marginal DOE index $j = 1, ..., n_e$ , let $s_j \in \mathbb{N}$ be its size, representing the number of nodes in the marginal DOE, and let $d_j \in \mathbb{N}$ be the dimension of the $j$ -th marginal DOE. Let $\left\{ Q_{s_j}^{(d_j)} \right\}_{j = 1, ..., n_e}$ be the collection of marginal DOEs. The dimension of the tensor product experiment is equal to the sum of the dimensions of the marginal DOEs:

$d_x = \sum_{j = 1}^{n_e} d_j.$

The size of the tensor product experiment is equal to the product of the marginal DOE sizes:

$s_t = \prod_{j = 1}^{n_e} s_j.$

Let $\boldsymbol{k} = \left(k_1, ..., k_{n_e}\right) \in (\mathbb{N}^\star)^{n_e}$ be a multi-index where $\mathbb{N}^\star = \{1, 2, ..., \}$ is the set of positive integers. For any marginal DOE $j = 1,..., n_e$ , let $\left\{w_{s_j, k_j} \right\}_{k_j = 1, ..., s_j}$ be its weights and let $\left\{ \boldsymbol{x}_{s_j, k_j} \in \mathbb{R}^{d_j} \right\}_{k_j = 1, ..., s_j}$ be its nodes.

The tensor product quadrature is ([gerstner1998] page 214):

$Q^{(d_x)}_{s_t} = Q_{s_1}^{(d_1)} \otimes \cdots \otimes Q_{s_{n_e}}^{(d_{n_e})}$

where:

$Q_{s_1}^{(d_1)} \otimes \cdots \otimes Q_{s_{n_e}}^{(d_{n_e})} (g) = \sum_{k_1 = 1}^{s_1} \cdots \sum_{k_{n_e} = 1}^{s_{n_e}} w_{s_1, k_1} \cdots w_{s_{n_e}, k_{n_e}} g\left(\boldsymbol{x}_{s_1, k_1}, \ldots, \boldsymbol{x}_{s_{n_e}, k_{n_e}}\right).$

For any multi-index $\boldsymbol{k} = (k_1, ..., k_{n_e}) \in (\mathbb{N}^\star)^{n_e}$ , we write

$\boldsymbol{k} \leq \left(s_1, ..., s_{n_e}\right)$

if and only if $k_j \leq s_j$ for $j = 1, ..., n_e$ . This means that each component of the multi-index is less than or equal to its corresponding marginal size. The set of multi-indices such that $\boldsymbol{k} \leq \left(s_1, ..., s_{n_e}\right)$ is produced using all possible combinations of the indices with the Tuples class.

The tensor product experiment is:

$Q_{s_1}^{(d_1)} \otimes \cdots \otimes Q_{s_{n_e}}^{(d_{n_e})} (g) = \sum_{\boldsymbol{k} \leq \left(s_1, ..., s_{n_e}\right)} w_{\boldsymbol{k}} g\left(\boldsymbol{x}_{\boldsymbol{k}}\right),$

where each weight is equal to the product of the marginal elementary weights:

$w_{\boldsymbol{k}} = \prod_{j = 1}^{n_e} w_{s_j, k_j} \in \mathbb{R}$

and each node is equal to the agregation of the marginal elementary nodes:

$\boldsymbol{x}_{\boldsymbol{k}} = \left(\boldsymbol{x}_{s_1, k_1}, \ldots, \boldsymbol{x}_{s_{n_e}, k_{n_e}}\right) \in \mathbb{R}^{d_x}.$

Examples

In the following example, we tensorize two Gauss-Legendre quadratures, using 3 nodes in the first dimension and 5 nodes in the second.

>>> import openturns as ot
>>> experiment1 = ot.GaussProductExperiment(ot.Uniform(0.0, 1.0), [3])
>>> experiment2 = ot.GaussProductExperiment(ot.Uniform(0.0, 1.0), [5])
>>> collection = [experiment1, experiment2]
>>> multivariate_experiment = ot.TensorProductExperiment(collection)
>>> nodes, weights = multivariate_experiment.generateWithWeights()

Marginal DOEs do not necessarily have dimension 1. In the following example, we tensorize two DOEs, where the first one has dimension 2, and the second one has dimension 3.

>>> # Experiment 1 : Uniform * 2 with 3 and 2 nodes.
>>> marginal_sizes_1 = [3, 2]
>>> dimension_1 = len(marginal_sizes_1)
>>> distribution_1 = ot.ComposedDistribution([ot.Uniform()] * dimension_1)
>>> experiment_1 = ot.GaussProductExperiment(distribution_1, marginal_sizes_1)
>>> # Experiment 2 : Normal * 3 with 2, 2 and 1 nodes.
>>> marginal_sizes_2 = [2, 2, 1]
>>> dimension_2 = len(marginal_sizes_2)
>>> distribution_2 = ot.ComposedDistribution([ot.Normal()] * dimension_2)
>>> experiment_2 = ot.GaussProductExperiment(distribution_2, marginal_sizes_2)
>>> # Tensor product
>>> collection = [experiment_1, experiment_2]
>>> multivariate_experiment = ot.TensorProductExperiment(collection)
>>> nodes, weights = multivariate_experiment.generateWithWeights()

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.
`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.
`getWeightedExperimentCollection`()	Get the marginals of the experiment.
`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.
`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.
`setWeightedExperimentCollection`(coll)	Set the marginals of the experiment.

__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.

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.

getWeightedExperimentCollection()¶

Get the marginals of the experiment.

Returns:

colllist of WeightedExperiment: List of the marginals of the experiment.

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.

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.

setWeightedExperimentCollection(coll)¶

Set the marginals of the experiment.

Parameters:

colllist of WeightedExperiment: List of the marginals of the experiment.

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