SpectralGaussianProcess

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

Spectral Gaussian process.

Available constructors:

SpectralGaussianProcess(secondOrderModel, timeGrid)

SpectralGaussianProcess(spectralModel, timeGrid)

SpectralGaussianProcess(secondOrderModel, fmax, N)

SpectralGaussianProcess(spectralModel, maxFreq, N)

Parameters:

secondOrderModel : SecondOrderModel

timeGrid : RegularGrid

The time grid associated to the process. OpenTURNS has only implemented the algorithm when the mesh is a regular grid.

spectralModel : SpectralModel

maxFreq : float

Equal to the maximal frequency minus \Delta f.

N : float

The number of points in the frequency grid, which is equal to the number of time stamps of the time grid.

Notes

  • In the first usage, we fix the time grid and the second order model (spectral density model) which implements the process. The frequency discretization is deduced from the time discretization by the formulas f_{max} = \frac{1}{\Delta t}, \quad \Delta f = \frac{1}{t_{max}}, N = \frac{f_{max}}{\Delta f}= \frac{t_{max}}{\Delta t}
  • In the second usage, the process is fixed in the frequency domain. fmax value and N induce the time grid. Care: the maximal frequency used in the computation is not fmax but (1-1/N)fmax = fmax - \Delta f.
  • In the third usage, the spectral model is given and the other arguments are the same as the first usage.
  • In the fourth usage, the spectral model is given and the other arguments are the same as the second usage.

The first call of getRealization() might be time consuming because it computes N hermitian matrices of size d \times \ d, where d is the dimension of the spectral model. These matrices are factorized and stored in order to be used for each call of the getRealization method.

Examples

Create a SpectralGaussianProcess from a spectral model and a time grid:

>>> import openturns as ot
>>> amplitude = [1.0, 2.0]
>>> scale = [4.0, 5.0]
>>> spatialCorrelation = ot.CorrelationMatrix(2)
>>> spatialCorrelation[0,1] = 0.8
>>> myTimeGrid =  ot.RegularGrid(0.0, 0.1, 20)
>>> mySpectralModel = ot.CauchyModel(scale, amplitude, spatialCorrelation)
>>> mySpectNormProc1 = ot.SpectralGaussianProcess(mySpectralModel, myTimeGrid)

Create a SpectralGaussianProcess from a second order model and a time grid:

>>> import openturns as ot
>>> amplitude = [1.0]
>>> scale = [4.0, 5.0]
>>> myTimeGrid =  ot.RegularGrid(0.0, 0.1, 20)
>>> mySecondOrderModel = ot.ExponentialCauchy(scale, amplitude)
>>> mySpectNormProc2 = ot.SpectralGaussianProcess(mySecondOrderModel, myTimeGrid)

Methods

AdaptGrid
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.
getDimension() Get the dimension of the domain \cD.
getFFTAlgorithm() Get the FFT algorithm used to generate realizations of the spectral Gaussian process.
getFrequencyGrid() Get the frequency grid used to discretize the spectral model.
getFrequencyStep() Get the frequency step \Delta f used to discretize the spectral model.
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.
getMaximalFrequency() Get the maximal frequency used in the computation.
getMesh() Get the mesh.
getNFrequency() Get the number of points in the frequency grid.
getName() Accessor to the object’s name.
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.
getSpectralModel() Get the spectral model.
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.
setDescription(description) Set the description of the process.
setFFTAlgorithm(fft) Set the FFT algorithm used to generate realizations of the spectral Gaussian process.
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)

x.__init__(…) initializes x; see help(type(x)) for signature

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 : Function

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_model : CovarianceModel

Covariance model, if any.

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.

getFFTAlgorithm()

Get the FFT algorithm used to generate realizations of the spectral Gaussian process.

Returns:

fftAlgo : FFT

FFT algorithm used to generate realizations of the spectral Gaussian process. By default, it is the KissFFT algorithm.

getFrequencyGrid()

Get the frequency grid used to discretize the spectral model.

Returns:

freqGrid : RegularGrid

The frequency grid used to discretize the spectral model.

getFrequencyStep()

Get the frequency step \Delta f used to discretize the spectral model.

Returns:

freqStep : float

The frequency step \Delta f used to discretize the spectral model.

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.

getMaximalFrequency()

Get the maximal frequency used in the computation.

Returns:

freqMax : float

The maximal frequency used in the computation: (1-1/N)fmax = fmax - \Delta f.

getMesh()

Get the mesh.

Returns:

mesh : Mesh

Mesh over which the domain \cD is discretized.

getNFrequency()

Get the number of points in the frequency grid.

Returns:

freqGrid : RegularGrid

The number N of points in the frequency grid, which is equal to the number of time stamps of the time grid.

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.

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.

getSpectralModel()

Get the spectral model.

Returns:

specMod : SpectralModel

The spectral model defining the process.

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).

getTrend()

Accessor to the trend.

Returns:

trend : TrendTransform

Trend, if any.

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 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:

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.

setFFTAlgorithm(fft)

Set the FFT algorithm used to generate realizations of the spectral Gaussian process.

Parameters:

fftAlgo : FFT

FFT algorithm that will be used to generate realizations of the spectral Gaussian process. OpenTURNS provides the KissFFT algorithm. More efficient implementations are provided by the openturns-fftw module.

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.

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.