pdPgram2D {pdSpecEst}

Multitaper HPD time-varying periodogram matrix

Description

Given a multivariate time series, pdPgram2D computes a multitapered HPD time-varying periodogram matrix based on averaging raw Hermitian PSD time-varying periodogram matrices of tapered multivariate time series segments.

Usage

pdPgram2D(X, B, tf.grid, method = c("dpss", "hermite"), nw = 3,
  bias.corr = F)

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 or Hermite tapering functions. By default, B = d, such that the multitaper periodogram is guaranteed to be positive definite.

tf.grid

a list with two components tf.grid$time and tf.grid$frequency specifying the rectangular grid of time-frequency points at which the multitaper periodogram is evaluated. tf.grid$time should be a numeric vector of rescaled time points in the range (0,1). tf.grid$frequency should be a numeric vector of frequency points in the range (0,0.5), with 0.5 corresponding to the Nyquist frequency.

method

the tapering method, either "dpss" or "hermite" explained in the Details section below. Defaults to method = "dpss".

nw

a positive numeric value corresponding to the time-bandwidth parameter of the tapering functions, see also dpss, defaults to nw = 3. Both the DPSS and Hermite tapers are rescaled with the same time-bandwidth parameter.

bias.corr

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

Details

If method = "dpss", pdPgram2D calculates a (d,d)-dimensional multitaper time-varying periodogram matrix based on sliding B DPSS (Discrete Prolate Spheroidal Sequence or Slepian) orthogonal tapering functions as in dpss applied to the d-dimensional time series X. If B \ge d, the multitaper time-varying periodogram matrix is guaranteed to be positive definite at each time-frequency point in the grid expand.grid(tf.grid$time, tf.grid$frequency). In short, the function pdPgram2D computes a multitaper periodogram matrix (as in pdPgram) in each of a number of non-overlapping time series segments of X, with the time series segments centered around the (rescaled) time points in tf.grid$time. If method = "hermite", the function calculates a multitaper time-varying periodogram matrix replacing the DPSS tapers by orthogonal Hermite tapering functions as in e.g., (Bayram and Baraniuk 1996).
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 and 5 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 pdSpecEst2D to perform intrinsic wavelet-based time-varying spectral matrix estimation. In this case, set bias.corr = F (the default) as the appropriate bias-corrections are applied internally by the function pdSpecEst2D.

Value

A list containing two components:

tf.grid

a list with two components corresponding to the rectangular grid of time-frequency points at which the multitaper periodogram is evaluated.

P

a (d,d,m_1,m_2)-dimensional array with m_1 = length(tf.grid$time) and m_2 = length(tf.grid$frequency) corresponding to the (d,d)-dimensional tapered periodogram matrices evaluated at the time-frequency points in tf.grid.

References

Bayram M, Baraniuk R (1996). “Multiple window time-frequency analysis.” In Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis, 173–176.

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

pdPgram, dpss

Examples

## Coefficient matrices
Phi1 <- array(c(0.4, 0, 0, 0.8, rep(0, 4)), dim = c(2, 2, 2))
Phi2 <- array(c(0.8, 0, 0, 0.4, 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)

## Generate piecewise stationary time series
ts.Phi <- function(Phi) rARMA(2^9, 2, Phi, Theta, Sigma)$X
ts <- rbind(ts.Phi(Phi1), ts.Phi(Phi2))

pgram <- pdPgram2D(ts)