FunctionalChaosRandomVector¶

class FunctionalChaosRandomVector(*args)¶

Functional chaos random vector.

Allows one to simulate a variable through a chaos decomposition, and retrieve its mean and covariance analytically from the chaos coefficients.

Parameters:

functionalChaosResultFunctionalChaosResult: A result from a functional chaos decomposition.

See also

FunctionalChaosAlgorithm, FunctionalChaosResult

Methods

`asComposedEvent`()	If the random vector can be viewed as the composition of several `ThresholdEvent` objects, this method builds and returns the composition.
`getAntecedent`()	Accessor to the antecedent RandomVector in case of a composite RandomVector.
`getClassName`()	Accessor to the object's name.
`getCovariance`()	Accessor to the covariance of the functional chaos expansion.
`getDescription`()	Accessor to the description of the RandomVector.
`getDimension`()	Accessor to the dimension of the RandomVector.
`getDistribution`()	Accessor to the distribution of the RandomVector.
`getDomain`()	Accessor to the domain of the Event.
`getFrozenRealization`(fixedPoint)	Compute realizations of the RandomVector.
`getFrozenSample`(fixedSample)	Compute realizations of the RandomVector.
`getFunction`()	Accessor to the Function in case of a composite RandomVector.
`getFunctionalChaosResult`()	Accessor to the functional chaos result.
`getId`()	Accessor to the object's id.
`getMarginal`(*args)	Get the random vector corresponding to the $i^{th}$ marginal component(s).
`getMean`()	Accessor to the mean of the functional chaos expansion.
`getName`()	Accessor to the object's name.
`getOperator`()	Accessor to the comparaison operator of the Event.
`getParameter`()	Accessor to the parameter of the distribution.
`getParameterDescription`()	Accessor to the parameter description of the distribution.
`getProcess`()	Get the stochastic process.
`getRealization`()	Compute one realization of the RandomVector.
`getSample`(size)	Compute realizations of the RandomVector.
`getShadowedId`()	Accessor to the object's shadowed id.
`getThreshold`()	Accessor to the threshold of the Event.
`getVisibility`()	Accessor to the object's visibility state.
`hasName`()	Test if the object is named.
`hasVisibleName`()	Test if the object has a distinguishable name.
`isComposite`()	Accessor to know if the RandomVector is a composite one.
`isEvent`()	Whether the random vector is an event.
`setDescription`(description)	Accessor to the description of the RandomVector.
`setName`(name)	Accessor to the object's name.
`setParameter`(parameters)	Accessor to the parameter of the distribution.
`setShadowedId`(id)	Accessor to the object's shadowed id.
`setVisibility`(visible)	Accessor to the object's visibility state.

__init__(*args)¶

asComposedEvent()¶

If the random vector can be viewed as the composition of several ThresholdEvent objects, this method builds and returns the composition. Otherwise throws.

Returns:

composedRandomVector: Composed event.

getAntecedent()¶

Accessor to the antecedent RandomVector in case of a composite RandomVector.

Returns:

antecedentRandomVector: Antecedent RandomVector $\vect{X}$ in case of a CompositeRandomVector such as: $\vect{Y}=f(\vect{X})$ .

getClassName()¶

Accessor to the object’s name.

Returns:

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

getCovariance()¶

Accessor to the covariance of the functional chaos expansion.

Let $n_X \in \Nset$ be the dimension of the input random vector, let $n_Y \in \Nset$ be the dimension of the output random vector. and let $P \in \Nset$ be the number of coefficients in the basis. We consider the following functional chaos expansion:

$\widetilde{\vect{Y}} = \sum_{k = 0}^P \vect{a}_k \psi_k(\vect{Z})$

where $\widetilde{\vect{Y}} \in \Rset^{n_Y}$ is the approximation of the output random variable $\vect{Y}$ by the expansion, $\left\{\vect{a}_k \in \Rset^{n_Y}\right\}_{k = 0, ..., P}$ are the coefficients, $\left\{\psi_k: \Rset^{n_X} \rightarrow \Rset\right\}_{k = 0, ..., P}$ are the orthonormal functions in the basis, and $\vect{Z} \in \Rset^{n_X}$ is the standardized random input vector. The previous equation can be equivalently written as follows:

$\widetilde{Y}_i = \sum_{k = 0}^P a_{ki} \psi_k(\vect{Z})$

for $i = 1, ..., n_Y$ where $a_{ki} \in \Rset$ is the $i$ -th component of the $k$ -th coefficient in the expansion:

$\vect{a}_k = \begin{pmatrix}a_{k, 1} \\ \vdots\\ a_{k, n_Y} \end{pmatrix}.$

The covariance matrix of the functional chaos expansion is the matrix $\matcov \in \Rset^{n_Y \times n_Y}$ , where each component is:

$c_{ij} = \Cov{\widetilde{Y}_i, \tilde{Y}_j}$

for $i,j = 1, ..., n_Y$ . The covariance can be computed using the coefficients of the expansion:

$\Cov{\widetilde{Y}_i, \tilde{Y}_j} = \sum_{k = 1}^P a_{ki} a_{jk}$

for $i,j = 1, ..., n_Y$ . This covariance involves all the coefficients, except the first one. The diagonal of the covariance matrix is the marginal variance:

$\Var{\widetilde{Y}_i} = \sum_{k = 1}^P a_{ki}^2$

for $i,j = 1, ..., n_Y$ .

Returns:

covarianceCovarianceMatrix, dimension $n_Y \times n_Y$: The covariance of the functional chaos expansion.

Examples

>>> from openturns.usecases import ishigami_function
>>> import openturns as ot
>>> import math
>>> im = ishigami_function.IshigamiModel()
>>> sampleSize = 1000
>>> inputTrain = im.distributionX.getSample(sampleSize)
>>> outputTrain = im.model(inputTrain)
>>> multivariateBasis = ot.OrthogonalProductPolynomialFactory([im.X1, im.X2, im.X3])
>>> selectionAlgorithm = ot.LeastSquaresMetaModelSelectionFactory()
>>> projectionStrategy = ot.LeastSquaresStrategy(selectionAlgorithm)
>>> totalDegree = 10
>>> enumerateFunction = multivariateBasis.getEnumerateFunction()
>>> basisSize = enumerateFunction.getBasisSizeFromTotalDegree(totalDegree)
>>> adaptiveStrategy = ot.FixedStrategy(multivariateBasis, basisSize)
>>> chaosAlgo = ot.FunctionalChaosAlgorithm(
...     inputTrain, outputTrain, im.distributionX, adaptiveStrategy, projectionStrategy
... )
>>> chaosAlgo.run()
>>> chaosResult = chaosAlgo.getResult()
>>> randomVector = ot.FunctionalChaosRandomVector(chaosResult)
>>> covarianceMatrix = randomVector.getCovariance()
>>> covarianceMatrix = randomVector.getCovariance()
>>> print('covarianceMatrix=', covarianceMatrix[0, 0])
covarianceMatrix= 13.8...
>>> outputDimension = outputTrain.getDimension()
>>> stdDev = ot.Point([math.sqrt(covarianceMatrix[i, i]) for i in range(outputDimension)])
>>> print('stdDev=', stdDev[0])
stdDev= 3.72...

getDescription()¶

Accessor to the description of the RandomVector.

Returns:

descriptionDescription: Describes the components of the RandomVector.

getDimension()¶

Accessor to the dimension of the RandomVector.

Returns:

dimensionpositive int: Dimension of the RandomVector.

getDistribution()¶

Accessor to the distribution of the RandomVector.

Returns:

distributionDistribution: Distribution of the considered UsualRandomVector.

Examples

>>> import openturns as ot
>>> distribution = ot.Normal([0.0, 0.0], [1.0, 1.0], ot.CorrelationMatrix(2))
>>> randomVector = ot.RandomVector(distribution)
>>> ot.RandomGenerator.SetSeed(0)
>>> print(randomVector.getDistribution())
Normal(mu = [0,0], sigma = [1,1], R = [[ 1 0 ]
 [ 0 1 ]])

getDomain()¶

Accessor to the domain of the Event.

Returns:

domainDomain: Describes the domain of an event.

getFrozenRealization(fixedPoint)¶

Compute realizations of the RandomVector.

In the case of a CompositeRandomVector or an event of some kind, this method returns the value taken by the random vector if the root cause takes the value given as argument.

Parameters:

fixedPointPoint: Point chosen as the root cause of the random vector.

Returns:

realizationPoint: The realization corresponding to the chosen root cause.

See also

getRealization
getFrozenSample

Examples

>>> import openturns as ot
>>> distribution = ot.Normal()
>>> randomVector = ot.RandomVector(distribution)
>>> f = ot.SymbolicFunction('x', 'x')
>>> compositeRandomVector = ot.CompositeRandomVector(f, randomVector)
>>> event = ot.ThresholdEvent(compositeRandomVector, ot.Less(), 0.0)
>>> print(event.getFrozenRealization([0.2]))
[0]
>>> print(event.getFrozenRealization([-0.1]))
[1]

getFrozenSample(fixedSample)¶

Compute realizations of the RandomVector.

In the case of a CompositeRandomVector or an event of some kind, this method returns the different values taken by the random vector when the root cause takes the values given as argument.

Parameters:

fixedSampleSample: Sample of root causes of the random vector.

Returns:

sampleSample: Sample of the realizations corresponding to the chosen root causes.

See also

getSample
getFrozenRealization

Examples

>>> import openturns as ot
>>> distribution = ot.Normal()
>>> randomVector = ot.RandomVector(distribution)
>>> f = ot.SymbolicFunction('x', 'x')
>>> compositeRandomVector = ot.CompositeRandomVector(f, randomVector)
>>> event = ot.ThresholdEvent(compositeRandomVector, ot.Less(), 0.0)
>>> print(event.getFrozenSample([[0.2], [-0.1]]))
    [ y0 ]
0 : [ 0  ]
1 : [ 1  ]

getFunction()¶

Accessor to the Function in case of a composite RandomVector.

Returns:

functionFunction: Function used to define a CompositeRandomVector as the image through this function of the antecedent $\vect{X}$ : $\vect{Y}=f(\vect{X})$ .

getFunctionalChaosResult()¶

Accessor to the functional chaos result.

Returns:

functionalChaosResultFunctionalChaosResult: The result from a functional chaos decomposition.

getId()¶

Accessor to the object’s id.

Returns:

idint: Internal unique identifier.

getMarginal(*args)¶

Get the random vector corresponding to the $i^{th}$ marginal component(s).

Parameters:

iint or list of ints, $0\leq i < dim$: Indicates the component(s) concerned. $dim$ is the dimension of the RandomVector.

Returns:

vectorRandomVector: RandomVector restricted to the concerned components.

Notes

Let’s note $\vect{Y}=\Tr{(Y_1,\dots,Y_n)}$ a random vector and $I \in [1,n]$ a set of indices. If $\vect{Y}$ is a UsualRandomVector, the subvector is defined by $\tilde{\vect{Y}}=\Tr{(Y_i)}_{i \in I}$ . If $\vect{Y}$ is a CompositeRandomVector, defined by $\vect{Y}=f(\vect{X})$ with $f=(f_1,\dots,f_n)$ , $f_i$ some scalar functions, the subvector is $\tilde{\vect{Y}}=(f_i(\vect{X}))_{i \in I}$ .

Examples

>>> import openturns as ot
>>> distribution = ot.Normal([0.0, 0.0], [1.0, 1.0], ot.CorrelationMatrix(2))
>>> randomVector = ot.RandomVector(distribution)
>>> ot.RandomGenerator.SetSeed(0)
>>> print(randomVector.getMarginal(1).getRealization())
[0.608202]
>>> print(randomVector.getMarginal(1).getDistribution())
Normal(mu = 0, sigma = 1)

getMean()¶

Accessor to the mean of the functional chaos expansion.

Let $n_X \in \Nset$ be the dimension of the input random vector, let $n_Y \in \Nset$ be the dimension of the output random vector. and let $P \in \Nset$ be the number of coefficients in the basis. We consider the following functional chaos expansion:

$\widetilde{\vect{Y}} = \sum_{k = 0}^P \vect{a}_k \psi_k(\vect{Z})$

where $\widetilde{\vect{Y}} \in \Rset^{n_Y}$ is the approximation of the output random variable $\vect{Y}$ by the expansion, $\left\{\vect{a}_k \in \Rset^{n_Y}\right\}_{k = 0, ..., P}$ are the coefficients, $\left\{\psi_k: \Rset^{n_X} \rightarrow \Rset\right\}_{k = 0, ..., P}$ are the orthonormal functions in the basis, and $\vect{Z} \in \Rset^{n_X}$ is the standardized random input vector. The previous equation can be equivalently written as follows:

$\widetilde{Y}_i = \sum_{k = 0}^P a_{ki} \psi_k(\vect{Z})$

for $i = 1, ..., n_Y$ where $a_{ki} \in \Rset$ is the $i$ -th component of the $k$ -th coefficient in the expansion:

$\vect{a}_k = \begin{pmatrix}a_{k, 1} \\ \vdots\\ a_{k, n_Y} \end{pmatrix}.$

The mean of the functional chaos expansion is the first coefficient in the expansion:

$\Expect{\widetilde{Y}_i} = a_{i,0}$

for $i = 1, ..., n_Y$ .

Returns:

meanPoint, dimension $n_Y$: The mean of the functional chaos expansion.

Examples

>>> from openturns.usecases import ishigami_function
>>> import openturns as ot
>>> im = ishigami_function.IshigamiModel()
>>> sampleSize = 1000
>>> inputTrain = im.distributionX.getSample(sampleSize)
>>> outputTrain = im.model(inputTrain)
>>> multivariateBasis = ot.OrthogonalProductPolynomialFactory([im.X1, im.X2, im.X3])
>>> selectionAlgorithm = ot.LeastSquaresMetaModelSelectionFactory()
>>> projectionStrategy = ot.LeastSquaresStrategy(selectionAlgorithm)
>>> totalDegree = 10
>>> enumerateFunction = multivariateBasis.getEnumerateFunction()
>>> basisSize = enumerateFunction.getBasisSizeFromTotalDegree(totalDegree)
>>> adaptiveStrategy = ot.FixedStrategy(multivariateBasis, basisSize)
>>> chaosAlgo = ot.FunctionalChaosAlgorithm(
...     inputTrain, outputTrain, im.distributionX, adaptiveStrategy, projectionStrategy
... )
>>> chaosAlgo.run()
>>> chaosResult = chaosAlgo.getResult()
>>> randomVector = ot.FunctionalChaosRandomVector(chaosResult)
>>> mean = randomVector.getMean()
>>> print('mean=', mean[0])
mean= 3.50...

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getOperator()¶

Accessor to the comparaison operator of the Event.

Returns:

operatorComparisonOperator: Comparaison operator used to define the RandomVector.

getParameter()¶

Accessor to the parameter of the distribution.

Returns:

parameterPoint: Parameter values.

getParameterDescription()¶

Accessor to the parameter description of the distribution.

Returns:

descriptionDescription: Parameter names.

getProcess()¶

Get the stochastic process.

Returns:

processProcess: Stochastic process used to define the RandomVector.

getRealization()¶

Compute one realization of the RandomVector.

Returns:

aRealizationPoint: Sequence of values randomly determined from the RandomVector definition. In the case of an event: one realization of the event (considered as a Bernoulli variable) which is a boolean value (1 for the realization of the event and 0 else).

See also

getSample

Examples

>>> import openturns as ot
>>> distribution = ot.Normal([0.0, 0.0], [1.0, 1.0], ot.CorrelationMatrix(2))
>>> randomVector = ot.RandomVector(distribution)
>>> ot.RandomGenerator.SetSeed(0)
>>> print(randomVector.getRealization())
[0.608202,-1.26617]
>>> print(randomVector.getRealization())
[-0.438266,1.20548]

getSample(size)¶

Compute realizations of the RandomVector.

Parameters:

nint, $n \geq 0$: Number of realizations needed.

Returns:

realizationsSample: n sequences of values randomly determined from the RandomVector definition. In the case of an event: n realizations of the event (considered as a Bernoulli variable) which are boolean values (1 for the realization of the event and 0 else).

See also

getRealization

Examples

>>> import openturns as ot
>>> distribution = ot.Normal([0.0, 0.0], [1.0, 1.0], ot.CorrelationMatrix(2))
>>> randomVector = ot.RandomVector(distribution)
>>> ot.RandomGenerator.SetSeed(0)
>>> print(randomVector.getSample(3))
    [ X0        X1        ]
0 : [  0.608202 -1.26617  ]
1 : [ -0.438266  1.20548  ]
2 : [ -2.18139   0.350042 ]

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns:

idint: Internal unique identifier.

getThreshold()¶

Accessor to the threshold of the Event.

Returns:

thresholdfloat: Threshold of the RandomVector.

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.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns:

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

isComposite()¶

Accessor to know if the RandomVector is a composite one.

Returns:

isCompositebool: Indicates if the RandomVector is of type Composite or not.

isEvent()¶

Whether the random vector is an event.

Returns:

isEventbool: Whether it takes it values in {0, 1}.

setDescription(description)¶

Accessor to the description of the RandomVector.

Parameters:

descriptionstr or sequence of str: Describes the components of the RandomVector.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setParameter(parameters)¶

Accessor to the parameter of the distribution.

Parameters:

parametersequence of float: Parameter values.

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters:

idint: Internal unique identifier.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters:

visiblebool: Visibility flag.

Examples using the class¶

Create a polynomial chaos for the Ishigami function: a quick start guide to polynomial chaos

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FunctionalChaosRandomVector¶

Examples using the class¶