Spec function

Compute Spectral Density of Functional Time Series

This function estimates the spectral density operator of a Functional Time Series (FTS)

Spec(X, W = Epanechnikov_kernel, B.T = (dim(X)[1])^(-1/5),
  only.diag = FALSE, trace = FALSE, demean = TRUE, subgrid = FALSE,
  subgrid.density = 10, verbose = 0,
  subgrid.density.relative.to.bandwidth = TRUE)

Arguments

• X: A $T \times nbasis$ matrix of containing the coordinates of the FTS expressed in a basis. Each row corresponds to a time point, and each column corresponds to the coefficient of the corresponding basis function of the FTS.
• W: The weight function used to smooth the periodogram operator. Set by default to be the Epanechnikov kernel
• B.T: The bandwidth of frequencies over which the periodogram operator is smoothed. If B.T=0, the periodogram operator is returned.
• only.diag: A logical variable to choose if the function only computes the marginal spectral density of each basis coordinate (only.diag=TRUE). only.diag=FALSE by default, the full spectral density operator is computed .
• trace: A logical variable to choose if only the trace of the spectral density operator is computed. trace=FALSE by default.
• demean: A logical variable to choose if the FTS is centered before computing its spectral density operator.
• subgrid: A logical variable to choose if the spectral density operator is only returned for a subgrid of the Fourier frequencies, which can be useful in large datasets to reduce memory usage. subgrid=FALSE by default.
• subgrid.density: Only used if subgrid=TRUE. Specifies the approximate number of frequencies within the bandwidth over which the periodogram operator is smoothed.
• verbose: A variable to show the progress of the computations. By default, verbose=0.
• subgrid.density.relative.to.bandwidth: logical parameter to specify if subgrid.density is specified relative to the bandwidth parameter B.T

Returns

A list containing the following elements:

• spec: The estimated spectral density operator. The first dimension corresponds to the different frequencies over which the spectral density operators are estimated.
• omega: The frequencies over which the spectral density is estimated.
• m: The number of Fourier frequencies over which the periodogram operator was smoothed.
• bw: The equivalent Bandwidth used in the weight function W(), as defined in Bloomfield (1976, p.201).
• weight: The weight function used to smooth the periodogram operator.
• kappa.square: The L2 norm of the weight function W.

Examples

ma.scale1=c(-1.4,2.3,-2)
a1=Generate_filterMA(10, 10, MA.len=3, ma.scale=ma.scale1)
X=Simulate_new_MA(a1, T.len=512, noise.type='wiener')
ans=Spec(X, trace=FALSE, only.diag=FALSE)
plot(ans)
plot(Spec(X, trace=FALSE, only.diag=FALSE, subgrid=TRUE, subgrid.density=10,
subgrid.density.relative.to.bandwidth=FALSE))
rm(ans)

References

spec.pgram function of R.

Bloomfield, P. (1976) "Fourier Analysis of Time Series: AnIntroduction", Wiley.

Panaretos, V. M. and Tavakoli, S., "Fourier Analysis of Functional TimeSeries", Ann. Statist. Volume 41, Number 2 (2013), 568-603.