WelchFactory¶

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class WelchFactory(*args)¶

Welch estimator of the spectral model of a stationary process.

Refer to Estimation of a spectral density function.

Parameters:

windowFilteringWindows

The filtering window model.

By default, the filtering window model is the Hann model.

blockNumberint

Number of blocks.

By default, blockNumber=1.

overlapfloat, $0 \leq overlap \leq 0.5$ .

Overlap rate parameter of the segments of the time series.

By default, overlap=0.5.

Notes

Let $X: \Omega \times \cD \rightarrow \Rset^d$ be a multivariate second order stationary process, with zero mean, where $\cD \in \Rset^n$ . We only treat here the case where the domain is of dimension 1: $\cD \in \Rset$ (n=1).

If we note $C(\vect{s}, \vect{t})=\Expect{(X_{\vect{s}}-m(\vect{s}))\Tr{(X_{\vect{t}}-m(\vect{t}))}}$ its covariance function, then for all $(i,j), C^{stat}_{i,j} : \Rset^n \rightarrow \Rset^n$ is $\cL^1(\Rset^n)$ (ie $\int_{\Rset^n} |C^{stat}_{i,j}(\vect{\tau})|\di{\vect{\tau}}\, < +\infty$ ), with $C^{stat}(\vect{\tau}) = C(\vect{s}, \vect{s}+\vect{\tau})$ as this quantity does not depend on $\vect{s}$ .

The bilateral spectral density function $S : \Rset^n \rightarrow \mathcal{H}^+(d)$ exists and is defined as the Fourier transform of the covariance function $C^{stat}$ :

$\forall \vect{f} \in \Rset^n, \,S(\vect{f}) = \int_{\Rset^n}\exp\left\{ -2i\pi <\vect{f},\vect{\tau}> \right\} C^{stat}(\vect{\tau})\di{\vect{\tau}}$

where $\mathcal{H}^+(d) \in \mathcal{M}_d(\Cset)$ is the set of d-dimensional positive definite hermitian matrices.

The Welch estimator is a non parametric estimator based on the segmentation of the time series into blockNumber segments possibly overlapping (size of overlap overlap). The length of each segment is deduced.

Examples

Create a time series from a stationary second order process:

>>> import openturns as ot
>>> myTimeGrid = ot.RegularGrid(0.0, 0.1, 2**8)
>>> model = ot.CauchyModel([5.0], [3.0])
>>> gp = ot.SpectralGaussianProcess(model, myTimeGrid)
>>> myTimeSeries = gp.getRealization()

Estimate the spectral model with WelchFactory:

>>> mySegmentNumber = 10
>>> myOverlapSize = 0.3
>>> myFactory = ot.WelchFactory(ot.Hann(), mySegmentNumber, myOverlapSize)
>>> myEstimatedModel_TS = myFactory.build(myTimeSeries)

Change the filtering window:

>>> myFactory.setFilteringWindows(ot.Hamming())

Methods

`build`(*args)	Estimate the spetral model.
`getBlockNumber`()	Accessor to the block number.
`getClassName`()	Accessor to the object's name.
`getFFTAlgorithm`()	Accessor to the FFT algorithm used for the Fourier transform.
`getFilteringWindows`()	Accessor to the filtering window.
`getId`()	Accessor to the object's id.
`getName`()	Accessor to the object's name.
`getOverlap`()	Accessor to the overlap rate.
`getShadowedId`()	Accessor to the object's shadowed id.
`getVisibility`()	Accessor to the object's visibility state.
`hasName`()	Test if the object is named.
`hasVisibleName`()	Test if the object has a distinguishable name.
`setBlockNumber`(blockNumber)	Accessor to the block number.
`setFFTAlgorithm`(fft)	Accessor to the FFT algorithm used for the Fourier transform.
`setFilteringWindows`(window)	Accessor to the filtering window.
`setName`(name)	Accessor to the object's name.
`setOverlap`(overlap)	Accessor to the block number.
`setShadowedId`(id)	Accessor to the object's shadowed id.
`setVisibility`(visible)	Accessor to the object's visibility state.

buildAsUserDefinedSpectralModel

__init__(*args)¶

build(*args)¶

Estimate the spetral model.

Available usages:

build(myTimeSeries)

build(myProcessSample)

Parameters:

myTimeSeriesTimeSeries: One realization of the process.
myProcessSampleProcessSample: Several realizations of the process.

Returns:

mySpectralModelUserDefinedSpectralModel: The spectral model estimated with the Welch estimator.

getBlockNumber()¶

Accessor to the block number.

Returns:

blockNumberint

The number of blocks used in the Welch estimator.

By default, blockNumber = 1.

getClassName()¶

Accessor to the object’s name.

Returns:

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

getFFTAlgorithm()¶

Accessor to the FFT algorithm used for the Fourier transform.

Returns:

fftAlgoFFT: The FFT algorithm used for the Fourier transform.

getFilteringWindows()¶

Accessor to the filtering window.

Returns:

filteringWindowFilteringWindows

The filtering window used.

By default, the Hann one.

getId()¶

Accessor to the object’s id.

Returns:

idint: Internal unique identifier.

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getOverlap()¶

Accessor to the overlap rate.

Returns:

overlapfloat, $0 \leq overlap \leq 0.5$ .

The overlap rate of the time series.

By default, overlap = 0.5.

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns:

idint: Internal unique identifier.

getVisibility()¶

Accessor to the object’s visibility state.

Returns:

visiblebool: Visibility flag.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns:

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

setBlockNumber(blockNumber)¶

Accessor to the block number.

Parameters:

blockNumberpositive int: The number of blocks used in the Welch estimator.

setFFTAlgorithm(fft)¶

Accessor to the FFT algorithm used for the Fourier transform.

Parameters:

fftAlgoFFT: The FFT algorithm used for the Fourier transform.

setFilteringWindows(window)¶

Accessor to the filtering window.

Parameters:

filteringWindowFilteringWindows: The filtering window used.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setOverlap(overlap)¶

Accessor to the block number.

Parameters:

blockNumberint, $0 \leq overlap \leq 0.5$ .: The overlap rate of the times series.

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters:

idint: Internal unique identifier.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters:

visiblebool: Visibility flag.

Examples using the class¶

Estimate a scalar ARMA process

Estimate a spectral density function

Estimate a stationary covariance function

OpenTURNS

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

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WelchFactory¶

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