crossspectra.md

crossspectra.jl

Cross-spectra objects created by FourierAnalysis are incapsulated in the following structure:

CrossSpectra

Categories: data objects, FDobjects.

struct CrossSpectra
    y           :: Union{Vector{LowerTriangular}, Vector{Hermitian}}
    sr          :: Int
    wl          :: Int
    DC          :: Bool
    taper       :: String
    flabels     :: Vector{T} where T<:Union{Real, Int}
    nonlinear   :: Bool
    smoothing   :: Smoother
    tril        :: Bool
end

Fields:

y: a real vector of LowerTriangular or Hermitian matrices (see Linear Algebra Julia standard package), depending on the tril field, as explained below, holding the cross-spectral matrices.

sr: the sampling rate of the data on which the cross-spectra have been estimated.

wl: the FFT window length used for estimating the cross-spectra.

DC: if true, the first matrix holds the cross-spectra of the DC level, otherwise it holds the cross-spectra of the first positive frequency. Thus, if DC is false, the number of matrices in y is equal to wl÷2 (integer division), otherwise it is equal to (wl÷2)+1 (see Overview). In all constructors it is false by default.

taper: the time-domain tapering window used for FFT computation, as a string, with parameters in parentheses for Slepian's dpss. See tapers.jl.

flabels: a vector holding all Fourier discrete frequencies in Hz. Those are the frequency labels for the matrices in y. If DC is true, the first label is 0, otherwise it is the first positive frequency, which is equal to the frequency resolution sr/wl.

nonlinear: if true, the amplitude information has been eliminated from the DFT coefficients, that is, the coefficients have been normalized by their modulus before being averaged to obtain the cross-spectra. This leads to non-linear estimates (Congedo, 2018; Pascual-Marqui 2007) where the diagonal elements of the cross-spectral matrices (the spectra) are 1.0 for all frequencies. In all constructors it is false by default.

smoothing: a Smoother flag indicating whether the cross-spectral matrices have been smoothed across adjacent frequencies. If no smoothing has been applied, it is equal to noSmoother, which is the default in all constructors.

tril: if false, the cross-spectral matrices in y are full Hermitian matrices, otherwise they are LowerTriangular matrices holding only the lower triangles of the cross-spectra. In all constructors it is false by default.

Note: In Julia the fields are accessed by the usual dot notation, e.g., you may verify that for CrossSpectra object S, length(S.flabels) == length(S.y)== (S.wl/2)+S.DC.

A vector of CrossSpectra objects is of type CrossSpectraVector.

Methods for CrossSpectra and CrossSpectraVector objects

method	CrossSpectra	CrossSpectraVector
bands	✔	✔
extract	✔	✔
mean	✔	✔
smooth	✔	✔
sameParams		✔
isLinear		✔
isNonLinear		✔

Generic Constructors:

In order to construct a CrossSpectra object from multivariate data using the Welch method, FourierAnalysis provides two crossSpectra constuctors, which is what you will use in practice most of the time.

Manual constructors are also possible, for which you have to provide appropriate arguments. The default manual constructor of CrossSpectra objects is

CrossSpectra(y, sr, wl, DC, taper, flabels, nonlinear, smoothing, tril)

Other constructors are also provided:

CrossSpectra(y, sr, wl, DC, taper, nonlinear)

enables construction giving only y, sr, wl, DC, taper and nonlinear argument. flabels is generated automatically, smoothing is set to noSmoother and tril is set to false;

CrossSpectra(y, sr, wl, taper)

as above, but setting by default also DC and nonlinear to false.

Constructors from data:

crossSpectra

References:

M. Congedo (2018), Non-Parametric Synchronization Measures used in EEG and MEG, Technical Report. GIPSA-lab, CNRS, University Grenoble Alpes, Grenoble INP.

R. Pascual-Marqui (2007), Instantaneous and lagged measurements of linear and nonlinear dependence between groups of multivariate time series: frequency decomposition, arXiv:0711.1455.