KrigingRandomVector¶

class KrigingRandomVector(*args)¶

KrigingRandom vector, a conditioned Gaussian process.

Parameters

krigingResultKrigingResult: Structure that contains elements of computation of a kriging algorithm
points1-d or 2-d sequence of float: Sequence of values defining a Point or a Sample.

Notes

KrigingRandomVector helps to create Gaussian random vector, $Y: \Rset^n \mapsto \Rset^d$ , with stationary covariance function $\cC^{stat}: \Rset^n \mapsto \cM_{d \times d}(\Rset)$ , conditionally to some observations.

Let $Y(x=x_1)=y_1,\cdots,Y(x=x_n)=y_n$ be the observations of the Gaussian process. We assume the same Gaussian prior as in the KrigingAlgorithm:

$Y(\vect{x}) = \Tr{\vect{f}(\vect{x})} \vect{\beta} + Z(\vect{x})$

with $\Tr{\vect{f}(\vect{x})} \vect{\beta}$ a general linear model, $Z(\vect{x})$ a zero-mean Gaussian process with a stationary autocorrelation function $\cC^{stat}$ :

$\mathbb{E}[Z(\vect{x}), Z(\vect{\tilde{x}})] = \sigma^2 \cC^{stat}_{\theta}(\vect{x} - \vect{\tilde{x}})$

The objective is to generate realizations of the random vector $Y$ , on new points $\vect{\tilde{x}}$ , conditionally to these observations. For that purpose, KrigingAlgorithm build such a prior and stores results in a KrigingResult structure on a first step. This structure is given as input argument.

Then, in a second step, both the prior and the covariance on input points $\vect{\tilde{x}}$ , conditionally to the previous observations, are evaluated (respectively $Y(\vect{\tilde{x}})$ and $\cC^{stat}_{\theta}(\vect{\tilde{x}})$ ).

Finally realizations are randomly generated by the Gaussian distribution $\cN ( Y(\vect{\tilde{x}}), \cC^{stat}_{\theta}(\vect{\tilde{x}}) )$

KrigingRandomVector class inherits from UsualRandomVector. Thus it stores the previous distribution and returns elements thanks to that distribution (realization, mean, covariance, sample…)

Examples

Create the model $\cM: \Rset \mapsto \Rset$ and the samples:

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x'],  ['x * sin(x)'])
>>> sampleX = [[1.0], [2.0], [3.0], [4.0], [5.0], [6.0], [7.0], [8.0]]
>>> sampleY = f(sampleX)

Create the algorithm:

>>> basis = ot.Basis([ot.SymbolicFunction(['x'], ['x']), ot.SymbolicFunction(['x'], ['x^2'])])
>>> covarianceModel = ot.SquaredExponential([1.0])
>>> covarianceModel.setActiveParameter([])
>>> algo = ot.KrigingAlgorithm(sampleX, sampleY, covarianceModel, basis)
>>> algo.run()

Get the results:

>>> result = algo.getResult()
>>> rvector = ot.KrigingRandomVector(result, [[0.0]])

Get a sample of the random vector:

>>> sample = rvector.getSample(5)

Methods

`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 RandomVector.
`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.
`getFunction`()	Accessor to the Function in case of a composite RandomVector.
`getId`()	Accessor to the object's id.
`getKrigingResult`()	Return the kriging result structure.
`getMarginal`(*args)	Get the random vector corresponding to the $i^{th}$ marginal component(s).
`getMean`()	Accessor to the mean of the RandomVector.
`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 a realization of the conditional Gaussian process (conditional on the learning set).
`getSample`(*args)	Compute a sample of realizations of the conditional Gaussian process (conditional on the learning set).
`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)¶

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 RandomVector.

Returns

covarianceCovarianceMatrix: Covariance of the considered UsualRandomVector.

Examples

>>> import openturns as ot
>>> distribution = ot.Normal([0.0, 0.5], [1.0, 1.5], ot.CorrelationMatrix(2))
>>> randomVector = ot.RandomVector(distribution)
>>> ot.RandomGenerator.SetSeed(0)
>>> print(randomVector.getCovariance())
[[ 1    0    ]
 [ 0    2.25 ]]

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.

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 $\vect{X}$ : $\vect{Y}=f(\vect{X})$ .

getId()¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

getKrigingResult()¶

Return the kriging result structure.

Returns

krigResultKrigingResult: The structure containing the elements of a KrigingAlgorithm.

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 RandomVector.

Returns

meanPoint: Mean of the considered UsualRandomVector.

Examples

>>> import openturns as ot
>>> distribution = ot.Normal([0.0, 0.5], [1.0, 1.5], ot.CorrelationMatrix(2))
>>> randomVector = ot.RandomVector(distribution)
>>> ot.RandomGenerator.SetSeed(0)
>>> print(randomVector.getMean())
[0,0.5]

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 a realization of the conditional Gaussian process (conditional on the learning set).

The realization predicts the value on the given input points.

Returns

realizationPoint: Sequence of values of the Gaussian process.

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