RandomWalk

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

Random walk process.

Parameters:

origin : NumericalPoint

Origin of the random walk.

distribution : Distribution

Distribution of dimension equal to the dimension of origin.

timeGrid : TimeGrid, optional

The time grid of the process. By default, the time grid is reduced to one time stamp equal to 0.

Notes

A random walk is a process X: \Omega \times \cD \rightarrow \Rset^d where \cD=\Rset discretized on the time grid (t_i)_{i \geq 0} such that:

\forall n>0,\: X_{t_n}  =  X_{t_{n-1}} + \varepsilon_{t_n}

where \vect{x}_0 \in \Rset^d and \varepsilon is a white noise of dimension d.

Examples

Create a random walk:

>>> import openturns as ot
>>> myTimeGrid = ot.RegularGrid(0, 0.1, 10)
>>> myDist = ot.ComposedDistribution([ot.Normal(), ot.Exponential(0.2)], ot.ClaytonCopula(0.5))
>>> myOrigin = ot.NumericalPoint(myDist.getMean())
>>> myRandomWalk = ot.RandomWalk(myOrigin, myDist, myTimeGrid)

Get a realization:

>>> myReal = myRandomWalk.getRealization()

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.
getDistribution() Accessor to the distribution.
getFuture(*args) Prediction of the N future iterations of the process.
getId() Accessor to the object’s id.
getMarginal(*args) Get the k^{th} marginal of the random process.
getMesh() Get the mesh.
getName() Accessor to the object’s name.
getOrigin() Accessor to the origin.
getRealization() Get a realization of the process.
getSample(size) Get n realizations of the process.
getShadowedId() Accessor to the object’s shadowed id.
getSpatialDimension() Get the dimension of the domain \cD.
getTimeGrid() Get the time grid of observation of the process.
getTrend()
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.
setDescription(description) Set the description of the process.
setDistribution(distribution) Accessor to the distribution.
setMesh(mesh) Set the mesh.
setName(name) Accessor to the object’s name.
setOrigin(origin) Accessor to the origin.
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)
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.

getDistribution()

Accessor to the distribution.

Returns:

distribution : Distribution

The distribution of dimension d of the white noise.

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.

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.

getOrigin()

Accessor to the origin.

Returns:

origin : NumericalPoint

The origin of the random walk.

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.

getShadowedId()

Accessor to the object’s shadowed id.

Returns:

id : int

Internal unique identifier.

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).

getVisibility()

Accessor to the object’s visibility state.

Returns:

visible : bool

Visibility flag.

hasName()

Test if the object is named.

Returns:

hasName : bool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns:

hasVisibleName : bool

True if the name is not empty and not the default one.

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.

setDistribution(distribution)

Accessor to the distribution.

Parameters:

distribution : Distribution

The distribution of dimension d of the white noise.

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.

setOrigin(origin)

Accessor to the origin.

Parameters:

origin : NumericalPoint

The origin of the random walk.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters:

id : int

Internal unique identifier.

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).

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:

visible : bool

Visibility flag.