Process

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() Accessor to the object’s name.
getContinuousRealization() Get a continuous realization.
getCovarianceModel()
getDescription() Get the description of the process.
getDimension() Get the dimension of the domain \cD.
getFuture(*args) Prediction of the N future iterations of the process.
getId() Accessor to the object’s id.
getImplementation(*args) Accessor to the underlying implementation.
getMarginal(*args) Get the k^{th} marginal of the random process.
getMesh() Get the mesh.
getName() Accessor to the object’s name.
getRealization() Get a realization of the process.
getSample(size) Get n realizations of the process.
getSpatialDimension() Get the dimension of the domain \cD.
getTimeGrid() Get the time grid of observation of the process.
getTrend()
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.
setDescription(description) Set the description of the process.
setMesh(mesh) Set the mesh.
setName(name) Accessor to the object’s name.
setTimeGrid(timeGrid) Set the time grid of observation of the process.
__init__(*args)
getClassName()

Accessor to the object’s name.

Returns:

class_name : str

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

getContinuousRealization()

Get a continuous realization.

Returns:

realization : NumericalMathFunction

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).
getDescription()

Get the description of the process.

Returns:

description : Description

Description of the process.

getDimension()

Get the dimension of the domain \cD.

Returns:

d : int

Dimension of the domain \cD.

getFuture(*args)

Prediction of the N future iterations of the process.

Parameters:

stepNumber : int, N \geq 0

Number of future steps.

size : int, size \geq 0, optional

Number of futures needed. Default is 1.

Returns:

prediction : ProcessSample 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:

id : int

Internal unique identifier.

getImplementation(*args)

Accessor to the underlying implementation.

Returns:

impl : Implementation

The implementation class.

getMarginal(*args)

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

Parameters:

k : int or list of ints 0 \leq k < d

Index of the marginal(s) needed.

Returns:

marginals : Process

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

getMesh()

Get the mesh.

Returns:

mesh : Mesh

Mesh over which the domain \cD is discretized.

getName()

Accessor to the object’s name.

Returns:

name : str

The name of the object.

getRealization()

Get a realization of the process.

Returns:

realization : Field

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:

n : int, n \geq 0

Number of realizations of the process needed.

Returns:

processSample : ProcessSample

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

getSpatialDimension()

Get the dimension of the domain \cD.

Returns:

n : int

Dimension of the domain \cD: n.

getTimeGrid()

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 \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).

isComposite()

Test whether the process is composite or not.

Returns:

isComposite : bool

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

isNormal()

Test whether the process is normal or not.

Returns:

isNormal : bool

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 normal 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:

isStationary : bool

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(description)

Set the description of the process.

Parameters:

description : sequence of str

Description of the process.

setMesh(mesh)

Set the mesh.

Parameters:

mesh : Mesh

Mesh over which the domain \cD is discretized.

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setTimeGrid(timeGrid)

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 \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).