WelchFactory

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../../_images/WelchFactory.png
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 Hanning 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.Hanning(), 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)

Initialize self. See help(type(self)) for accurate signature.

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 Hanning 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.