Process

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../../_images/openturns-Process-1.png
class Process(*args)

Base class for stochastic processes.

Notes

The Process class enables to model a stochastic process.

A multivariate stochastic process X of dimension d is defined by:

X: \Omega \times\cD \mapsto \Rset^d

where \omega \in \Omega is an event, \cD is a domain of \Rset^n discretized on the mesh \cM, \vect{t}\in \cD is a multivariate index and X(\omega, \vect{t}) \in \Rset^d.

A realization of the process X, for a given \omega \in \Omega is X(\omega): \cD  \mapsto \Rset^d defined by:

X(\omega)(\vect{t}) = X(\omega, \vect{t})

X_{\vect{t}}: \Omega \rightarrow \Rset^d is the random variable at index \vect{t} \in \cD defined by:

X_{\vect{t}}(\omega) = X(\omega, \vect{t})

A Process object can be created only through its derived classes:

SpectralGaussianProcess, GaussianProcess, CompositeProcess, ARMA, RandomWalk, FunctionalBasisProcess and WhiteNoise.

Methods

getClassName(self)

Accessor to the object’s name.

getContinuousRealization(self)

Get a continuous realization.

getCovarianceModel(self)

Accessor to the covariance model.

getDescription(self)

Get the description of the process.

getFuture(self, \*args)

Prediction of the N future iterations of the process.

getId(self)

Accessor to the object’s id.

getImplementation(self, \*args)

Accessor to the underlying implementation.

getInputDimension(self)

Get the dimension of the domain \cD.

getMarginal(self, \*args)

Get the k^{th} 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 \cD.

getRealization(self)

Get a realization of the process.

getSample(self, size)

Get n realizations of the process.

getTimeGrid(self)

Get the time grid of observation of the process.

getTrend(self)

Accessor to the trend.

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.

setDescription(self, description)

Set the description of the process.

setMesh(self, mesh)

Set the mesh.

setName(self, name)

Accessor to the object’s name.

setTimeGrid(self, timeGrid)

Set the time grid of observation of the process.

__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
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(self)

Accessor to the covariance model.

Returns
cov_modelCovarianceModel

Covariance model, if any.

getDescription(self)

Get the description of the process.

Returns
descriptionDescription

Description of the process.

getFuture(self, *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(self)

Accessor to the object’s id.

Returns
idint

Internal unique identifier.

getImplementation(self, *args)

Accessor to the underlying implementation.

Returns
implImplementation

The implementation class.

getInputDimension(self)

Get the dimension of the domain \cD.

Returns
nint

Dimension of the domain \cD: n.

getMarginal(self, *args)

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(self)

Get the mesh.

Returns
meshMesh

Mesh over which the domain \cD is discretized.

getName(self)

Accessor to the object’s name.

Returns
namestr

The name of the object.

getOutputDimension(self)

Get the dimension of the domain \cD.

Returns
dint

Dimension of the domain \cD.

getRealization(self)

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(self, 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.

getTimeGrid(self)

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(self)

Accessor to the trend.

Returns
trendTrendTransform

Trend, if any.

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 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(self)

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}})

setDescription(self, description)

Set the description of the process.

Parameters
descriptionsequence of str

Description of the process.

setMesh(self, mesh)

Set the mesh.

Parameters
meshMesh

Mesh over which the domain \cD is discretized.

setName(self, name)

Accessor to the object’s name.

Parameters
namestr

The name of the object.

setTimeGrid(self, 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).