Autoregressive prewhitening of a series for aMTM analysis

The amtm_prewhiten function prewhitens an evenly spaced series with an AR(p) filter whose coefficients are obtained from the autocovariance sequence of a low-variance aMTM spectrum (NW = 4, K = 7, zero padding to 2^(ceiling(log2(N)) + nfft_add)) through the Yule-Walker equations. The one-step predictions minus the observations form the prewhitened series; the first 2*order values are dropped and the mean is removed. The spectrum of the prewhitened series can be "recoloured" by dividing it by \(|1 - \sum_j \phi_j e^{-i 2\pi f j \Delta t}|^2\). Translated from the "Python PreWhiten Filter Example" (prewhiten(), correct_spec_for_pw()) distributed with amtm_signaldetect (MIT licence), itself based on an R prewhiten() routine (Wesley Burr / David Riegert).

Usage

amtm_prewhiten(data = NULL, order = 2, nw = 4, k = 7, nfft_add = 2)

Arguments

data: Series to prewhiten: a matrix or data frame with depth/time in column 1 and the proxy value in column 2, evenly spaced (extra columns are ignored, rows with non-finite values are dropped and the rows are sorted by depth/time). Required (Default=NULL gives an error).
order: Order p of the autoregressive (AR) prewhitening filter (integer, >= 1; the series must have more than 2*order + 2 values). Higher orders remove more of the red-noise background but also more of the signal; the first 2*order samples are lost. Default=2.
nw: Time-bandwidth product of the multitaper spectrum from which the autocovariance sequence (ACVS) is computed (numeric, > 0, typically 2 to 4). Default=4.
k: Number of DPSS tapers of that spectrum (integer, >= 1, at most about 2*nw - 1). Default=7.
nfft_add: Zero-padding exponent increment (integer, >= 0): the spectrum is computed on 2^(ceiling(log2(N)) + nfft_add) frequencies, N being the number of samples. Larger values give a finer frequency grid and a longer, smoother ACVS. Default=2.

Value

A list with data (2-column matrix of the prewhitened series, with the depth/time values of the retained samples), ar_coef (AR coefficients phi_1..phi_p) and acvs (autocovariance sequence used).

References

Percival, D. B., & Walden, A. T. (2020). Spectral Analysis for Univariate Time Series. Cambridge University Press. doi:10.1017/9781139235723

Author

Michiel Arts (R translation); original Python code: Hector A. Salinas et al.

Examples

# \donttest{
set.seed(3)
x <- as.numeric(stats::filter(rnorm(400), 0.9, method = "recursive"))
pw <- amtm_prewhiten(cbind(seq_along(x), x), order = 1)
pw$ar_coef
#> [1] 0.8846462
# }