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):
AR.max == 0Remove (optionally) a mean and/or linear trend.
AR.max > 0Remove an autoregressive model
In the second case,
the time series is
filtered in the time domain with a finite-impulse-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.