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

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 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.

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.

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(size)

Compute a sample of realizations of the conditional Gaussian process (conditional on the learning set).

getThreshold()

Accessor to the threshold of the Event.

hasName()

Test if the object is named.

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.

__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 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.

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.

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.

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}).

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.

See also

getSample
getSample(size)

Compute a sample of realizations of the conditional Gaussian process (conditional on the learning set).

The realization predicts the value on the given input points.

Returns:
realizationsSample

2-d float sequence of values of the Gaussian process.

See also

getRealization
getThreshold()

Accessor to the threshold of the Event.

Returns:
thresholdfloat

Threshold of the RandomVector.

hasName()

Test if the object is named.

Returns:
hasNamebool

True if the name is not empty.

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.

Examples using the class

Advanced Kriging

Advanced Kriging