Adaptive, sine multitaper power spectral density estimation for R

by Andrew J Barbour, Jonathan Kennel, and Robert L Parker

*As of version 2.0, one can calculate the multivariate PSD (“cross spectrum”) between two signals.*

This is an R package for computing univariate power spectral density estimates with little or no tuning effort. We employ sine multitapers, allowing the number to vary with frequency in order to reduce mean square error, the sum of squared bias and variance, at each point. The approximate criterion of Riedel and Sidorenko (1995) is modified to prevent runaway averaging that otherwise occurs when the curvature of the spectrum goes to zero. An iterative procedure refines the number of tapers employed at each frequency. The resultant power spectra possess significantly lower variances than those of traditional, non-adaptive estimators. The sine tapers also provide useful spectral leakage suppression. Resolution and uncertainty can be estimated from the number of degrees of freedom (twice the number of tapers).

This technique is particularly suited to long time series, because it demands only one numerical Fourier transform, and requires no costly additional computation of taper functions, like the Slepian functions. It also avoids the degradation of the low-frequency performance associated with record segmentation in Welch’s method. Above all, the adaptive process relieves the user of the need to set a tuning parameter, such as time-bandwidth product or segment length, that fixes frequency resolution for the entire frequency interval; instead it provides frequency-dependent spectral resolution tailored to the shape of the spectrum itself.

`psd`

elegantly handles spectra with large dynamic range and mixed-bandwidth features|features typically found in geophysical datasets.

Bob and Andy have a paper in Computers & Geosciences to accompany this software (download a pdf, 1MB); it describes the theory behind the estimation process, and how we apply it in practice. If you find `psd`

useful in your research, we kindly request you cite our paper. See also:

`citation("psd")`

You can to install the package and it’s dependencies with CRAN (from within the `R`

environment):

`install.packages("psd")`

then load the package library

`library(psd)`

We have included a dataset to play with, namely `Tohoku`

, which represents recordings of high-frequency borehole strainmeter data during teleseismic waves from the 2011 Mw 9.0 Tohoku earthquake (original data source). Access and inspect these data with:

```
data(Tohoku)
print(str(Tohoku))
```

The ‘preseismic’ data has interesting spectral features, so we subset it, and analyze the areal strain (the change in borehole diameter):

```
Dat <- subset(Tohoku, epoch=="preseismic")
Areal <- ts(Dat$areal)
```

For the purposes of improving the accuracy of the spectrum, we remove a linear trend:

`Dat <- prewhiten(Areal, plot=FALSE)`

Now we can calculate the adaptive PSD:

```
mtpsd <- pspectrum(Dat[['prew_lm']], plot=TRUE)
print(class(mtpsd))
```

In the previous example the `plot=TRUE`

flag produces a comparison with a basic periodogram, but we can also visualize the spectrum with builtin plotting methods:

`plot(mtpsd, log="dB")`

The spectral uncertainty can be easily calculated:

```
sprop <- spectral_properties(mtpsd)
with(sprop, {
plot(taper/max(taper), type="h", ylim=c(0,2), col="dark grey")
lines(stderr.chi.lower)
lines(stderr.chi.upper)
})
```

Should you wish to install the development version of this software, the remotes library will be useful:

```
library(remotes)
install_github("abarbour/psd")
```