Produces power spectral density estimates through iterative refinement of the optimal number of sine-tapers at each frequency. This optimization procedure is based on the method of Riedel and Sidorenko (1995), which minimizes the Mean Square Error (sum of variance and bias) at each frequency, but modified for computational stability.

Adaptive, sine multitaper power spectral density estimation for R

by Andrew J Barbour and Robert L Parker

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 I have published 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.

```
citation("psd")
```

Firstly you'll need to install the package and it's dependencies
from CRAN
(from within the `R`

environment):

```
install.packages("psd", dependencies=TRUE)
```

then load the package library

```
library(psd)
```

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

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

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

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

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

For the purposes of spectral estimation, 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))
```

We can visualize the spectrum with builtin methods:

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

and the spectral uncertainty:

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

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

```
install.packages("devtools", dependencies=TRUE)
library(devtools)
install_github("abarbour/psd")
```