Description Usage Arguments Details Value References See Also Examples
Given a multivariate time series, pdPgram2D
computes a multitapered HPD timevarying periodogram matrix based on
averaging raw Hermitian PSD timevarying periodogram matrices of tapered multivariate time series segments.
1 2 
X 
an (n,d)dimensional matrix corresponding to a multivariate time series,
with the 
B 
depending on the argument 
tf.grid 
a list with two components 
method 
the tapering method, either 
nw 
a positive numeric value corresponding to the timebandwidth parameter of the tapering functions,
see also 
bias.corr 
should an asymptotic biascorrection under the affineinvariant Riemannian metric be applied to
the HPD periodogram matrix? Defaults to 
If method = "dpss"
, pdPgram2D
calculates a (d,d)dimensional multitaper timevarying
periodogram matrix based on sliding B DPSS (Discrete Prolate Spheroidal Sequence or Slepian) orthogonal tapering functions
as in dpss
applied to the ddimensional time series X
. If B ≥ d, the
multitaper timevarying periodogram matrix is guaranteed to be positive definite at each timefrequency 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 nonoverlapping 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 timevarying periodogram matrix replacing the DPSS
tapers by orthogonal Hermite tapering functions as in e.g., \insertCiteBB96pdSpecEst.
In the case of subsequent periodogram matrix denoising in the space of HPD matrices equipped with the
affineinvariant 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 affineinvariant metric as explained in
\insertCiteCvS17pdSpecEst and Chapter 3 and 5 of \insertCiteC18pdSpecEst. The presmoothed HPD periodogram matrix
(i.e., an initial noisy HPD spectral estimator) can be given as input to the function pdSpecEst2D
to perform
intrinsic waveletbased timevarying spectral matrix estimation. In this case, set bias.corr = F
(the default) as the
appropriate biascorrections are applied internally by the function pdSpecEst2D
.
A list containing two components:

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

a (d,d,m_1,m_2)dimensional array with 
1 2 3 4 5 6 7 8 9 10 11  ## 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)

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