ConditionedGaussianProcess¶

class
ConditionedGaussianProcess
(*args)¶ Conditioned Gaussian process.
ConditionedGaussianProcess(krigingResult, mesh)
 Parameters
 krigingResult
KrigingResult
Structure that contains all elements of kriging computations.
 mesh
Mesh
Mesh over which the domain is discretized.
 krigingResult
Notes
ConditionedGaussianProcess helps to create Gaussian random fields, where , with covariance function ( in the stationary case), conditionaly to some observations.
Let be the observations of the Gaussian process. We assume the same Gaussian prior as in the
KrigingAlgorithm
:with a general linear model, a zeromean Gaussian process with a stationary autocorrelation function :
The ConditionedGaussianProcess generates realizations of the conditioned process. It focuses first on the
KrigingAlgorithm
to build such prior. Results are stored in aKrigingResult
structure, which is given as input argument of the class. This last one, combined with the mesh argument, define both the prior and the covariance evaluation on the mesh vertices conditionnaly to the previous observations. It follows that the realizations are randomly generated from the Gaussian distribution .In practice, we do not store the Gaussian distribution as we need only the random realization method. For that purpose, we use the Cholesky method : we compute the Cholesky factor of the covariance matrix such as . It follows that the random realizations are obtained as following : with a centered & reduced random Gaussian realization.
Examples
>>> import openturns as ot >>> ot.RandomGenerator.SetSeed(0) >>> # Kriging use case >>> # Learning data >>> levels = [8.0, 5.0] >>> box = ot.Box(levels) >>> inputSample = box.generate() >>> # Scale each direction >>> inputSample *= 10 >>> # Define model >>> model = ot.SymbolicFunction(['x', 'y'], ['cos(0.5*x) + sin(y)']) >>> outputSample = model(inputSample) >>> # Definition of exponential model >>> inputDimension = 2 >>> covarianceModel = ot.SquaredExponential([1.988, 0.924], [3.153]) >>> # Basis definition >>> basis = ot.ConstantBasisFactory(inputDimension).build() >>> # Kriring algorithm >>> algo = ot.KrigingAlgorithm(inputSample, outputSample, covarianceModel, basis) >>> algo.run() >>> result = algo.getResult() >>> vertices = [[1.0, 0.0], [2.0, 0.0], [2.0, 1.0], [1.0, 1.0], [1.5, 0.5]] >>> simplices = [[0, 1, 4], [1, 2, 4], [2, 3, 4], [3, 0, 4]] >>> mesh2D = ot.Mesh(vertices, simplices) >>> process = ot.ConditionedGaussianProcess(result, mesh2D)
Methods
getClassName
(self)Accessor to the object’s name.
getContinuousRealization
(self)Get a continuous realization.
getCovarianceModel
(self)Get the covariance model.
getDescription
(self)Get the description of the process.
getFuture
(self, \*args)Prediction of the future iterations of the process.
getId
(self)Accessor to the object’s id.
getInputDimension
(self)Get the dimension of the domain .
getMarginal
(self, indices)Get the marginal of the random process.
getMesh
(self)Get the mesh.
getName
(self)Accessor to the object’s name.
getOutputDimension
(self)Get the dimension of the domain .
getRealization
(self)Return a realization of the process.
getSample
(self, size)Get realizations of the process.
getShadowedId
(self)Accessor to the object’s shadowed id.
getTimeGrid
(self)Get the time grid of observation of the process.
getTrend
(self)Get the trend function.
getVisibility
(self)Accessor to the object’s visibility state.
hasName
(self)Test if the object is named.
hasVisibleName
(self)Test if the object has a distinguishable name.
isComposite
(self)Test whether the process is composite or not.
isNormal
(self)Test whether the process is normal or not.
isStationary
(self)Test whether the process is stationary or not.
isTrendStationary
(self)Tell if the process is trend stationary or not.
setDescription
(self, description)Set the description of the process.
setMesh
(self, \*args)Set the mesh.
setName
(self, name)Accessor to the object’s name.
setSamplingMethod
(self, \*args)Set the used method for getRealization.
setShadowedId
(self, id)Accessor to the object’s shadowed id.
setTimeGrid
(self, \*args)Set the time grid of observation of the process.
setVisibility
(self, visible)Accessor to the object’s visibility state.

__init__
(self, *args)¶ Initialize self. See help(type(self)) for accurate signature.

getClassName
(self)¶ Accessor to the object’s name.
 Returns
 class_namestr
The object class name (object.__class__.__name__).

getContinuousRealization
(self)¶ Get a continuous realization.
 Returns
 realization
Function
According to the process, the continuous realizations are built:
either using a dedicated functional model if it exists: e.g. a functional basis process.
or using an interpolation from a discrete realization of the process on : in dimension , a linear interpolation and in dimension , a piecewise constant function (the value at a given position is equal to the value at the nearest vertex of the mesh of the process).
 realization

getCovarianceModel
(self)¶ Get the covariance model.
 Returns
 covarianceModel
CovarianceModel
Temporal covariance model .
 covarianceModel

getDescription
(self)¶ Get the description of the process.
 Returns
 description
Description
Description of the process.
 description

getFuture
(self, *args)¶ Prediction of the future iterations of the process.
 Parameters
 stepNumberint,
Number of future steps.
 sizeint, , optional
Number of futures needed. Default is 1.
 Returns
 prediction
ProcessSample
orTimeSeries
future iterations of the process. If , prediction is a
TimeSeries
. Otherwise, it is aProcessSample
.
 prediction

getId
(self)¶ Accessor to the object’s id.
 Returns
 idint
Internal unique identifier.

getInputDimension
(self)¶ Get the dimension of the domain .
 Returns
 nint
Dimension of the domain : .

getMarginal
(self, indices)¶ Get the marginal of the random process.
 Parameters
 kint or list of ints
Index of the marginal(s) needed.
 Returns
 marginals
Process
Process defined with marginal(s) of the random process.
 marginals

getName
(self)¶ Accessor to the object’s name.
 Returns
 namestr
The name of the object.

getOutputDimension
(self)¶ Get the dimension of the domain .
 Returns
 dint
Dimension of the domain .

getRealization
(self)¶ Return a realization of the process.
 Returns
 realization
Field
A realization of the process.
 realization
Examples
>>> import openturns as ot >>> ot.RandomGenerator.SetSeed(0) >>> # Kriging use case >>> # Learning data >>> levels = [8.0, 5.0] >>> box = ot.Box(levels) >>> inputSample = box.generate() >>> # Scale each direction >>> inputSample *= 10 >>> # Define model >>> model = ot.SymbolicFunction(['x', 'y'], ['cos(0.5*x) + sin(y)']) >>> outputSample = model(inputSample) >>> # Definition of exponential model >>> inputDimension = 2 >>> covarianceModel = ot.SquaredExponential(inputDimension *[0.95]) >>> # Basis definition >>> basis = ot.ConstantBasisFactory(inputDimension).build() >>> # Kriring algorithm >>> algo = ot.KrigingAlgorithm(inputSample, outputSample, covarianceModel, basis) >>> algo.run() >>> result = algo.getResult() >>> vertices = [[1.0, 0.0], [2.0, 0.0], [2.0, 1.0],[1.0, 1.0], [1.5, 0.5]] >>> simplices = [[0, 1, 4], [1, 2, 4], [2, 3, 4], [3, 0, 4]] >>> mesh2D = ot.Mesh(vertices, simplices) >>> process = ot.ConditionedGaussianProcess(result, mesh2D) >>> # Get a realization of the process >>> realization = process.getRealization()

getSample
(self, size)¶ Get realizations of the process.
 Parameters
 nint,
Number of realizations of the process needed.
 Returns
 processSample
ProcessSample
realizations of the random process. A process sample is a collection of fields which share the same mesh .
 processSample

getShadowedId
(self)¶ Accessor to the object’s shadowed id.
 Returns
 idint
Internal unique identifier.

getTimeGrid
(self)¶ Get the time grid of observation of the process.
 Returns
 timeGrid
RegularGrid
Time grid of a process when the mesh associated to the process can be interpreted as a
RegularGrid
. We check if the vertices of the mesh are scalar and are regularly spaced in but we don’t check if the connectivity of the mesh is conform to the one of a regular grid (without any hole and composed of ordered instants).
 timeGrid

getTrend
(self)¶ Get the trend function.
 Returns
 trend
TrendTransform
Trend function.
 trend

getVisibility
(self)¶ Accessor to the object’s visibility state.
 Returns
 visiblebool
Visibility flag.

hasName
(self)¶ Test if the object is named.
 Returns
 hasNamebool
True if the name is not empty.

hasVisibleName
(self)¶ Test if the object has a distinguishable name.
 Returns
 hasVisibleNamebool
True if the name is not empty and not the default one.

isComposite
(self)¶ Test whether the process is composite or not.
 Returns
 isCompositebool
True if the process is composite (built upon a function and a process).

isNormal
(self)¶ Test whether the process is normal or not.
 Returns
 isNormalbool
True if the process is normal.
Notes
A stochastic process is normal if all its finite dimensional joint distributions are normal, which means that for all and , with , there is and such that:
where , and and is the symmetric matrix:
A Gaussian process is entirely defined by its mean function and its covariance function (or correlation function ).

isStationary
(self)¶ Test whether the process is stationary or not.
 Returns
 isStationarybool
True if the process is stationary.
Notes
A process is stationary if its distribution is invariant by translation: , , , we have:

isTrendStationary
(self)¶ Tell if the process is trend stationary or not.
 Returns
 isTrendStationarybool
True if the process is trend stationary.

setDescription
(self, description)¶ Set the description of the process.
 Parameters
 descriptionsequence of str
Description of the process.

setName
(self, name)¶ Accessor to the object’s name.
 Parameters
 namestr
The name of the object.

setSamplingMethod
(self, *args)¶ Set the used method for getRealization.
Available parameters are :
0 : Cholesky factor sampling (default method)
1 : HMatrix method (if HMat available)
2 : Gibbs method (in dimension 1 only)
 Parameters
 samplingMethodint
Fix a method for sampling.

setShadowedId
(self, id)¶ Accessor to the object’s shadowed id.
 Parameters
 idint
Internal unique identifier.

setTimeGrid
(self, *args)¶ Set the time grid of observation of the process.
 Returns
 timeGrid
RegularGrid
Time grid of observation of the process when the mesh associated to the process can be interpreted as a
RegularGrid
. We check if the vertices of the mesh are scalar and are regularly spaced in but we don’t check if the connectivity of the mesh is conform to the one of a regular grid (without any hole and composed of ordered instants).
 timeGrid

setVisibility
(self, visible)¶ Accessor to the object’s visibility state.
 Parameters
 visiblebool
Visibility flag.