The R-S multitapers do not exhibit the remarkable
spectral-leakage suppression properties of the Thomson
prolate tapers, so that in spectra with large dynamic
range, power bleeds from the strong peaks into
neighboring frequency bands of low amplitude -- spectral
leakage. Prewhitening can ameliorate the problem, at
least for red spectra [see Chapter 9, Percival and Walden
(1993)]. The value of the AR.max argument is made absolute,
after which this function has essentially two modes of
operation (detailed below): [object Object],[object Object] In the second case, the time
series is filtered in the time domain with a
finite-impluse-response filter of AR.max terms.
The filter is found by solving the Yule-Walker equations
for which it is assumed the series was generated by an
autoregressive process, up to order AR.max.
Mean and trend (AR.max == 0){
Power spectral density estimates can become badly biased
(especially at lower frequencies) if a signal of the form
$f(x) = A x + B$ is not removed from the series. If
detrend=TRUE a model of this form is removed over
the entire series using a linear least-squares estimator;
in this case a mean value is removed regardless of the
logical state of demean. To remove only a
mean value, set detrend=FALSE and (obviously)
demean=TRUE.
}
Auto Regressive (AR) innovations
(AR.max > 0){
When an autoregressive model is removed from a
non-stationary series, the residuals are known as
'innovations', and may be stationary (or very-nearly
stationary). This function fits an AR model [order at
least 1, but up to and including AR(AR.max)] to
the series by solving the Yule-Walker equations; however,
AIC is used to estimate the highest significant order,
which means that higher-order components may not
necessarily be fit. The resulting innovations can be used
to better estimate the stationary component of the
original signal, and possibly in an interactive editing
method.
Note that the method used here--solving the Yule-Walker
equations--is not a true maximum likelihood estimator;
hence the AIC is calculated based on the variance
estimate (no determinant). From ?ar: In
ar.yw the variance matrix of the innovations is
computed from the fitted coefficients and the
autocovariance of x.
A quick way to determine whether this may be needed for
the series is to run acf on the series, and see if
significant non-zero lag correlations are found. A
warning is produced if the fit returns an AR(0) fit,
indicating that AR prewhitening most likely inappropriate
for the series, which is apparently stationary (or very
nearly so). (The innovations could end up having
higher variance than the input series in such a
case.)
Note that AR.max is restricted to the range
$[1,N-1]$ where $N$ is the series length.
}