FunctionalBasisProcess

(Source code, png)

../../_images/FunctionalBasisProcess.png
class FunctionalBasisProcess(*args)

Functional basis process.

Parameters:
distributionDistribution

The distribution of the random vector \vect{A}=(A_1,\dots, A_K).

basissequence of Function

Collection of deterministic functions.

meshMesh

Mesh \cM over which the domain \cD is discretized.

Notes

A functional basis process X: \Omega \times\cD \mapsto \Rset^d where \cD \in \Rset^n, writes:

X(\omega,\vect{t})=\sum_{i=1}^K A_i(\omega)\phi_i(\vect{t}) \quad  \forall \omega \in \Omega and \forall \vect{t} \in \cD

with \phi_i: \Rset^n \rightarrow \Rset^d for 1 \leq i \leq K and \vect{A}=(A_1,\dots, A_K) a random vector of dimension K.

Examples

Create the coefficients distribution:

>>> import openturns as ot
>>> coefDist = ot.Normal([2]*2, [5]*2, ot.CorrelationMatrix(2))

Create a basis of functions:

>>> phi_1 = ot.SymbolicFunction(['t'], ['sin(t)'])
>>> phi_2 = ot.SymbolicFunction(['t'], ['cos(t)*cos(t)'])
>>> myBasis = ot.Basis([phi_1, phi_2])

Create a mesh:

>>> myMesh = ot.RegularGrid(0.0, 0.1, 10)

Create the functional basis process:

>>> myFBProcess = ot.FunctionalBasisProcess(coefDist, myBasis, myMesh)

Methods

getBasis()

Get the basis of deterministic functions.

getClassName()

Accessor to the object's name.

getContinuousRealization()

Get a continuous realization.

getCovarianceModel()

Accessor to the covariance model.

getDescription()

Get the description of the process.

getDistribution()

Get the coefficients distribution.

getFuture(*args)

Prediction of the N future iterations of the process.

getId()

Accessor to the object's id.

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()

Get a realization of the process.

getSample(size)

Get n realizations of the process.

getShadowedId()

Accessor to the object's shadowed id.

getTimeGrid()

Get the time grid of observation of the process.

getTrend()

Accessor to the trend.

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()

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.

setBasis(basis)

Set the basis of deterministic functions.

setDescription(description)

Set the description of the process.

setDistribution(distribution)

Set the coefficients distribution.

setMesh(mesh)

Set the mesh.

setName(name)

Accessor to the object's name.

setShadowedId(id)

Accessor to the object's shadowed id.

setTimeGrid(timeGrid)

Set the time grid of observation of the process.

setVisibility(visible)

Accessor to the object's visibility state.

__init__(*args)
getBasis()

Get the basis of deterministic functions.

Returns:
basiscollection of Function

Collection of functions (\phi_i)_{1 \leq i \leq K}.

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()

Accessor to the covariance model.

Returns:
cov_modelCovarianceModel

Covariance model, if any.

getDescription()

Get the description of the process.

Returns:
descriptionDescription

Description of the process.

getDistribution()

Get the coefficients distribution.

Returns:
distributionDistribution

The distribution of the random vector \vect{A}=(A_1,\dots, A_K) of dimension K.

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.

getId()

Accessor to the object’s id.

Returns:
idint

Internal unique identifier.

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()

Get a realization of the process.

Returns:
realizationField

Contains a mesh over which the process is discretized and the values of the process at the vertices of the mesh.

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.

getShadowedId()

Accessor to the object’s shadowed id.

Returns:
idint

Internal unique identifier.

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()

Accessor to the trend.

Returns:
trendTrendTransform

Trend, if any.

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()

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}})

setBasis(basis)

Set the basis of deterministic functions.

Parameters:
basissequence of Function

Collection of functions (\phi_i)_{1 \leq i \leq K}.

setDescription(description)

Set the description of the process.

Parameters:
descriptionsequence of str

Description of the process.

setDistribution(distribution)

Set the coefficients distribution.

Parameters:
distributionDistribution

The distribution of the random vector \vect{A}=(A_1,\dots, A_K) of dimension K.

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.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters:
idint

Internal unique identifier.

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).

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:
visiblebool

Visibility flag.

Examples using the class

Create a functional basis process

Create a functional basis process

Create a process from random vectors and processes

Create a process from random vectors and processes

Estimate Sobol indices on a field to point function

Estimate Sobol indices on a field to point function