SpectralGaussianProcess¶

(Source code, png)

class SpectralGaussianProcess(*args)¶

Spectral Gaussian process.

Available constructors:

SpectralGaussianProcess(spectralModel, timeGrid)

SpectralGaussianProcess(spectralModel, maxFreq, N)

Parameters:

timeGridRegularGrid: The time grid associated to the process. The algorithm is only implemented when the mesh is a regular grid.
spectralModelSpectralModel
maxFreqfloat: Equal to the maximal frequency minus $\Delta f$ .
Nfloat: The number of points in the frequency grid, which is equal to the number of time stamps of the time grid.

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.
`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.
`getInputDimension`()	Get the dimension of the domain $\cD$ .
`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.
`getOutputDimension`()	Get the dimension of the domain $\cD$ .
`getRealization`()	Get a realization of the process.
`getSample`(size)	Get $n$ realizations of the process.
`getSpectralModel`()	Get the spectral model.
`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.
`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.
`setTimeGrid`(timeGrid)	Set the time grid of observation of the process.

AdaptGrid

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)

__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.

getFFTAlgorithm()¶

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

Returns:

fftAlgoFFT: 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:

freqGridRegularGrid: The frequency grid used to discretize the spectral model.

getFrequencyStep()¶

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

Returns:

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

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(*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.

getMaximalFrequency()¶

Get the maximal frequency used in the computation.

Returns:

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

getMesh()¶

Get the mesh.

Returns:

meshMesh: Mesh over which the domain $\cD$ is discretized.

getNFrequency()¶

Get the number of points in the frequency grid.

Returns:

freqGridRegularGrid: 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:

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 $\cM \in \Rset^n$ .

getSpectralModel()¶

Get the spectral model.

Returns:

specModSpectralModel: The spectral model defining the process.

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.

setFFTAlgorithm(fft)¶

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

Parameters:

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

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 spectral density function

Create a Gaussian process

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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SpectralGaussianProcess¶

Examples using the class¶