ImportanceSamplingExperiment¶

(Source code, png)

class ImportanceSamplingExperiment(*args)¶

Importance Sampling experiment.

Available constructors:

ImportanceSamplingExperiment(importanceDistribution)

ImportanceSamplingExperiment(importanceDistribution, size)

ImportanceSamplingExperiment(initialDistribution, importanceDistribution, size)

Parameters:

initialDistributionDistribution: Distribution $\mu$ which is the initial distribution used to generate the set of input data.
sizepositive int: Number of points that will be generated in the experiment.
importanceDistributionDistribution: Distribution $p$ according to which the points of the experiments will be generated with the Importance Sampling technique.

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.
`getImportanceDistribution`()	Accessor to the importance distribution.
`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.
`setName`(name)	Accessor to the object's name.
`setSize`(size)	Accessor to the size of the generated sample.

See also

WeightedExperiment

Notes

ImportanceSamplingExperiment is a random weighted design of experiments. We can use it to generate a sample $(\inputReal_i)_{1 \leq i \leq \sampleSize}$ based on independent observations from the distribution $\inputMeasure$ (see [hammersley1961] page 57, [lemieux2009] page 11). Importance sampling is a variance reduction method i.e. it aims at reducing the variance of the estimator of the weighted integral. The sample is generated from the importance distribution $p$ and each realization is weighted by $w_i = \frac{\inputProbabilityDensityFunction(\inputReal_i)}{p(\inputReal_i)}$ where $\inputProbabilityDensityFunction$ is the probability density function of the input random vector.

There is no general method in the library to provide the importance distribution, which must be specified by the user. In the specific case of rare event estimation, [morio2015] page 53 provides different methods to compute the instrumental distribution, including simple changes of measure and exponential twisting. The PostAnalyticalImportanceSampling class combines the FORM class and importance sampling to estimate a probability.

The ImportanceSamplingExperiment class is nonadaptive, i.e. the parameters of the instrumental distribution are set once for all. See NAIS for an adaptive importance sampling algorithm and CrossEntropyImportanceSampling for another algorithm.

Examples

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> distribution = ot.JointDistribution([ot.Uniform(0, 1)] * 2)
>>> importanceDistribution = ot.JointDistribution([ot.Uniform(0, 1)] * 2)
>>> experiment = ot.ImportanceSamplingExperiment(distribution, importanceDistribution, 5)
>>> print(experiment.generate())
    [ X0        X1        ]
0 : [ 0.629877  0.882805  ]
1 : [ 0.135276  0.0325028 ]
2 : [ 0.347057  0.969423  ]
3 : [ 0.92068   0.50304   ]
4 : [ 0.0632061 0.292757  ]

__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]

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 of the input random vector.

getImportanceDistribution()¶

Accessor to the importance distribution.

Returns:

importanceDistributionDistribution: Distribution $p$ according to which the points of the design of experiments will be generated with the Importance Sampling technique.

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getSize()¶

Accessor to the size of the generated sample.

Returns:

sizepositive int: Number $\sampleSize$ of points constituting the design of experiments.

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.

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 of the input random vector.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setSize(size)¶

Accessor to the size of the generated sample.

Parameters:

sizepositive int: Number $\sampleSize$ of points constituting the design of experiments.

Examples using the class¶

Use the Importance Sampling algorithm

Axial stressed beam : comparing different methods to estimate a probability

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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ImportanceSamplingExperiment¶

Examples using the class¶