WhiteNoise

(Source code, png)

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

White Noise process.

Parameters:
distributionDistribution

Distribution of dimension d of the white noise process.

meshMesh, optional

Mesh in \Rset^n over which the process is discretized. By default, the mesh is reduced to one point in \Rset which coordinate is equal to 0.

Notes

A second order white noise \varepsilon: \Omega \times \cD \rightarrow \Rset^d is a stochastic process of dimension d such that the covariance function C(\vect{s},\vect{t})=\delta(\vect{t}-\vect{s})C(\vect{s},\vect{s}) where C(\vect{s},\vect{s}) is the covariance matrix of the process at vertex \vect{s} and \delta the Kroenecker function.

A process \varepsilon is a white noise if all finite family of locations (\vect{t}_i)_{i=1, \dots, n} \in \cD, (\varepsilon_{\vect{t}_i})_{i=1, \dots, n} is independent and identically distributed.

Examples

Create a normal normal white noise of dimension 1:

>>> import openturns as ot
>>> myDist = ot.Normal()
>>> myMesh = ot.IntervalMesher([10]*2).build(ot.Interval([0.0]*2, [1.0]*2))
>>> myWN = ot.WhiteNoise(myDist, myMesh)

Get a realization:

>>> myReal =myWN.getRealization()

Methods

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()

Accessor to the distribution.

getFuture(*args)

Prediction of the N future iterations of the process.

getInputDimension()

Get the dimension of the domain \cD.

getMarginal(indices)

Accessor to the marginal 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.

getTimeGrid()

Get the time grid of observation of the process.

getTrend()

Accessor to the trend.

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.

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.

setTimeGrid(timeGrid)

Set the time grid of observation of the process.

__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()

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()

Accessor to the distribution.

Returns:
distributionDistribution

The distribution of dimension d of the white noise.

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)

Accessor to the marginal process.

Parameters:
Nint

The index of the marginal to be extracted.

indicesIndices, optional

The list of the indexes of the marginal to be extracted.

Returns:
wnWhiteNoise

The marginal white noise.

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.

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.

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}})

setDescription(description)

Set the description of the process.

Parameters:
descriptionsequence of str

Description of the process.

setDistribution(distribution)

Accessor to the distribution.

Parameters:
distributionDistribution

The distribution of dimension d of the white noise.

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.

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

Estimate a multivariate ARMA process

Estimate a multivariate ARMA process

Estimate a scalar ARMA process

Estimate a scalar ARMA process

Aggregate processes

Aggregate processes

Create a white noise process

Create a white noise process

Manipulate a time series

Manipulate a time series

Create and manipulate an ARMA process

Create and manipulate an ARMA process