pdPgram {pdSpecEst}

Multitaper HPD periodogram matrix

Description

Given a multivariate time series, pdPgram computes a multitapered HPD periodogram matrix based on averaging raw Hermitian PSD periodogram matrices of tapered multivariate time series segments.

Usage

pdPgram(X, B, method = c("multitaper", "bartlett"), bias.corr = F,
  nw = 3)

Arguments

X

an (n,d)-dimensional matrix corresponding to a multivariate time series, with the d columns corresponding to the components of the time series.

B

depending on the argument method, either the number of orthogonal DPSS tapers, or the number of non-overlapping segments to compute Bartlett's averaged periodogram. By default, B = d, such that the averaged periodogram is guaranteed to be positive definite.

method

the tapering method, either "multitaper" or "bartlett" explained in the Details section below. Defaults to "multitaper".

bias.corr

should an asymptotic bias-correction under the affine-invariant Riemannian metric be applied to the HPD periodogram matrix? Defaults to FALSE.

nw

a positive numeric value corresponding to the time-bandwidth parameter of the DPSS tapering functions, see also dpss, defaults to nw = 3.

Details

If method = "multitaper", pdPgram calculates a (d,d)-dimensional multitaper periodogram matrix based on B DPSS (Discrete Prolate Spheroidal Sequence or Slepian) orthogonal tapering functions as in dpss applied to the d-dimensional time series X. If method = "bartlett", pdPgram computes a Bartlett spectral estimator by averaging the periodogram matrices of B non-overlapping segments of the d-dimensional time series X. Note that Bartlett's spectral estimator is a specific (trivial) case of a multitaper spectral estimator with uniform orthogonal tapering windows.
In the case of subsequent periodogram matrix denoising in the space of HPD matrices equipped with the affine-invariant Riemannian metric, one should set bias.corr = T, thereby correcting for the asymptotic bias of the periodogram matrix in the manifold of HPD matrices equipped with the affine-invariant metric as explained in (Chau and von Sachs 2019) and Chapter 3 of (Chau 2018). The pre-smoothed HPD periodogram matrix (i.e., an initial noisy HPD spectral estimator) can be given as input to the function pdSpecEst1D to perform intrinsic wavelet-based spectral matrix estimation. In this case, set bias.corr = F (the default) as the appropriate bias-corrections are applied internally by the function pdSpecEst1D.

Value

A list containing two components:

freq

vector of n/2 frequencies in the range [0,0.5) at which the periodogram is evaluated.

P

a (d, d, n/2)-dimensional array containing the (d,d)-dimensional multitaper periodogram matrices at frequencies corresponding to freq.

References

Chau J (2018). Advances in Spectral Analysis for Multivariate, Nonstationary and Replicated Time Series. phdthesis, Universite catholique de Louvain.

Chau J, von Sachs R (2019). “Intrinsic wavelet regression for curves of Hermitian positive definite matrices.” Journal of the American Statistical Association. doi: 10.1080/01621459.2019.1700129.

See Also

pdPgram2D, dpss

Examples

## ARMA(1,1) process: Example 11.4.1 in (Brockwell and Davis, 1991)
Phi <- array(c(0.7, 0, 0, 0.6, rep(0, 4)), dim = c(2, 2, 2))
Theta <- array(c(0.5, -0.7, 0.6, 0.8, rep(0, 4)), dim = c(2, 2, 2))
Sigma <- matrix(c(1, 0.71, 0.71, 2), nrow = 2)
ts.sim <- rARMA(200, 2, Phi, Theta, Sigma)
ts.plot(ts.sim$X) # plot generated time series traces
pgram <- pdPgram(ts.sim$X)