ConditionedGaussianProcess¶

class ConditionedGaussianProcess(*args)¶

Conditioned Gaussian process.

Warning

This class is experimental and likely to be modified in future releases. To use it, import the openturns.experimental submodule.

Parameters:

gprResultGaussianProcessRegressionResult: Structure that contains all the elements of Gaussian Process Regression computations.
meshMesh: Mesh $\cM$ over which the domain $\cD$ is discretized.

Methods

`getClassName`()	Accessor to the object's name.
`getContinuousRealization`()	Get a continuous realization.
`getCovarianceModel`()	Get the covariance model.
`getDescription`()	Get the description of the process.
`getFuture`(*args)	Prediction of the $N$ future iterations of the process.
`getInputDimension`()	Get the dimension of the domain $\cD$ .
`getMarginal`(indices)	Get the $k^{th}$ marginal of the random process.
`getMesh`()	Get the mesh.
`getName`()	Accessor to the object's name.
`getOutputDimension`()	Get the dimension of the domain $\cD$ .
`getRealization`()	Return a realization of the process.
`getSample`(size)	Get $n$ realizations of the process.
`getSamplingMethod`()	Get the used method for getRealization.
`getTimeGrid`()	Get the time grid of observation of the process.
`getTrend`()	Get the trend function.
`hasName`()	Test if the object is named.
`isComposite`()	Test whether the process is composite or not.
`isNormal`()	Test whether the process is normal or not.
`isStationary`()	Test whether the process is stationary or not.
`isTrendStationary`()	Tell if the process is trend stationary or not.
`setDescription`(description)	Set the description of the process.
`setMesh`(mesh)	Set the mesh.
`setName`(name)	Accessor to the object's name.
`setSamplingMethod`(*args)	Set the used method for getRealization.
`setTimeGrid`(timeGrid)	Set the time grid of observation of the process.

Notes

This class helps to generate fields from the conditioned gaussian process resulting from a Gaussian process regression algorithm.

Refer to the documentation of GaussianProcessRegression to get details on the notations.

Examples

We consider the model $g: \Rset^2 \rightarrow \Rset$ defined by $g(x,y) = cos(x/2) + sin(y)$ .

>>> import openturns as ot
>>> import openturns.experimental as otexp
>>> ot.RandomGenerator.SetSeed(0)
>>> model = ot.SymbolicFunction(['x', 'y'], ['cos(0.5*x) + sin(y)'])

Then we define the train sample as a box with 8 levels on the x-axis and 5 levels on the y-axis.

>>> levels = [8.0, 5.0]
>>> box = ot.Box(levels)
>>> x_train = box.generate()
>>> x_train *= 10
>>> y_train = model(x_train)

We define the covariance model as a SquaredExponential model:

>>> dim = 2
>>> covarianceModel = ot.SquaredExponential([1.0, 1.0], [1.0])

We define the functions basis to estimate the trend: this basis only contains constant functions:

>>> basis = ot.ConstantBasisFactory(dim).build()

We estimate the Gaussian process regression, using first the class GaussianProcessFitter, then using the class GaussianProcessRegression:

>>> fitter_algo = otexp.GaussianProcessFitter(x_train, y_train, covarianceModel, basis)
>>> fitter_algo.run()
>>> fitter_result = fitter_algo.getResult()
>>> gpr_algo = otexp.GaussianProcessRegression(fitter_result)
>>> gpr_algo.run()
>>> gpr_result = gpr_algo.getResult()

Now, we define the mesh on which the gconditionned gaussian process is generated:

>>> 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 = otexp.ConditionedGaussianProcess(gpr_result, mesh2D)

We get a realization of the conditioned Gaussian process:

>>> realization = process.getRealization()

__init__(*args)¶

getClassName()¶

Accessor to the object’s name.

Returns:

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

getContinuousRealization()¶

Get a continuous realization.

Returns:

realizationFunction

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 $\cM$ : in dimension $d=1$ , a linear interpolation and in dimension $d \geq 2$ , 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).

getCovarianceModel()¶

Get the covariance model.

Returns:

covarianceModelCovarianceModel: Temporal covariance model $C$ .

getDescription()¶

Get the description of the process.

Returns:

descriptionDescription: Description of the process.

getFuture(*args)¶

Prediction of the $N$ future iterations of the process.

Parameters:

stepNumberint, $N \geq 0$: Number of future steps.
sizeint, $size \geq 0$ , optional: Number of futures needed. Default is 1.

Returns:

predictionProcessSample or TimeSeries: $N$ future iterations of the process. If $size = 1$ , prediction is a TimeSeries. Otherwise, it is a ProcessSample.

getInputDimension()¶

Get the dimension of the domain $\cD$ .

Returns:

nint: Dimension of the domain $\cD$ : $n$ .

getMarginal(indices)¶

Get the $k^{th}$ marginal of the random process.

Parameters:

kint or list of ints $0 \leq k < d$: Index of the marginal(s) needed.

Returns:

marginalsProcess: Process defined with marginal(s) of the random process.

getMesh()¶

Get the mesh.

Returns:

meshMesh: Mesh over which the domain $\cD$ is discretized.

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getOutputDimension()¶

Get the dimension of the domain $\cD$ .

Returns:

dint: Dimension of the domain $\cD$ .

getRealization()¶

Return a realization of the process.

Returns:

realizationField: A realization of the process.

getSample(size)¶

Get $n$ realizations of the process.

Parameters:

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

Returns:

processSampleProcessSample: $n$ realizations of the random process. A process sample is a collection of fields which share the same mesh $\cM \in \Rset^n$ .

getSamplingMethod()¶

Get the used method for getRealization.

Returns:

samplingMethodint: Used method for sampling.

Notes

Available parameters are :

0 (GaussianProcess.CHOLESKY) : Cholesky factor sampling (default method)

1 (GaussianProcess.HMAT) : H-Matrix method (if H-Mat available)

2 (GaussianProcess.GALLIGAOGIBBS) : Gibbs method (in dimension 1 only)

getTimeGrid()¶

Get the time grid of observation of the process.

Returns:

timeGridRegularGrid: 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 $\Rset$ 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).

getTrend()¶

Get the trend function.

Returns:

trendTrendTransform: Trend function.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

isComposite()¶

Test whether the process is composite or not.

Returns:

isCompositebool: True if the process is composite (built upon a function and a process).

isNormal()¶

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 $k \in \Nset$ and $I_k \in \Nset^*$ , with $cardI_k=k$ , there is $\vect{m}_1, \dots, \vect{m}_k \in \Rset^d$ and $\mat{C}_{1,\dots,k}\in\mathcal{M}_{kd,kd}(\Rset)$ such that:

$\Expect{\exp\left\{i\Tr{\vect{X}}_{I_k} \vect{U}_{k} \right\}} = \exp{\left\{i\Tr{\vect{U}}_{k}\vect{M}_{k}-\frac{1}{2}\Tr{\vect{U}}_{k}\mat{C}_{1,\dots,k}\vect{U}_{k}\right\}}$

where $\Tr{\vect{X}}_{I_k} = (\Tr{X}_{\vect{t}_1}, \hdots, \Tr{X}_{\vect{t}_k})$ , $\\Tr{vect{U}}_{k} = (\Tr{\vect{u}}_{1}, \hdots, \Tr{\vect{u}}_{k})$ and $\Tr{\vect{M}}_{k} = (\Tr{\vect{m}}_{1}, \hdots, \Tr{\vect{m}}_{k})$ and $\mat{C}_{1,\dots,k}$ is the symmetric matrix:

$\mat{C}_{1,\dots,k} = \left( \begin{array}{cccc} C(\vect{t}_1, \vect{t}_1) &C(\vect{t}_1, \vect{t}_2) & \hdots & C(\vect{t}_1, \vect{t}_{k}) \\ \hdots & C(\vect{t}_2, \vect{t}_2) & \hdots & C(\vect{t}_2, \vect{t}_{k}) \\ \hdots & \hdots & \hdots & \hdots \\ \hdots & \hdots & \hdots & C(\vect{t}_{k}, \vect{t}_{k}) \end{array} \right)$

A Gaussian process is entirely defined by its mean function $m$ and its covariance function $C$ (or correlation function $R$ ).

isStationary()¶

Test whether the process is stationary or not.

Returns:

isStationarybool: True if the process is stationary.

Notes

A process $X$ is stationary if its distribution is invariant by translation: $\forall k \in \Nset$ , $\forall (\vect{t}_1, \dots, \vect{t}_k) \in \cD$ , $\forall \vect{h}\in \Rset^n$ , we have:

$(X_{\vect{t}_1}, \dots, X_{\vect{t}_k}) \stackrel{\mathcal{D}}{=} (X_{\vect{t}_1+\vect{h}}, \dots, X_{\vect{t}_k+\vect{h}})$

isTrendStationary()¶

Tell if the process is trend stationary or not.

Returns:

isTrendStationarybool: True if the process is trend stationary.

setDescription(description)¶

Set the description of the process.

Parameters:

descriptionsequence of str: Description of the process.

setMesh(mesh)¶

Set the mesh.

Parameters:

meshMesh: Mesh over which the domain $\cD$ is discretized.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setSamplingMethod(*args)¶

Set the used method for getRealization.

Available parameters are :

0 (GaussianProcess.CHOLESKY) : Cholesky factor sampling (default method)

1 (GaussianProcess.HMAT) : H-Matrix method (if H-Mat available)

2 (GaussianProcess.GALLIGAOGIBBS) : Gibbs method (in dimension 1 only)

Parameters:

samplingMethodint: Fix a method for sampling.

setTimeGrid(timeGrid)¶

Set the time grid of observation of the process.

Returns:

timeGridRegularGrid: 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 $\Rset$ 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).

Examples using the class¶

Advanced Gaussian process regression

Gaussian Process Regression : generate trajectories from the metamodel

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ConditionedGaussianProcess¶

Examples using the class¶