Compute Spectral Density of Functional Time Series

Description

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

Usage

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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

Value

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.

References

spec.pgram function of R.

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

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

Examples

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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)

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