Autoregressive (parametric) spectral density

Computes the parametric AR spectral density of a one-dimensional signal from an autoregressive model of order order fitted by Yule-Walker (via arYuleWalker): \(S(f) = \sigma^2 / |1 - \sum_k \phi_k e^{-i 2\pi f k}|^2\), evaluated on n_freq normalized frequencies in \([0, 0.5]\). Unlike the periodogram / Welch estimate, the AR spectrum is smooth and low-variance, with sharp peaks at resonant frequencies, using only order parameters. It matches stats::spec.ar (including its small-sample innovation-variance convention \(N/(N-p-1)\)).

Usage

arSpectrum(x, order = 8L, n_freq = 500L)

Arguments

x

A numeric vector (the time series).

order

Autoregressive order \(p\) (default 8).

n_freq

Number of frequencies in \([0, 0.5]\) (default 500).

Value

A list with freq (normalized frequencies, cycles/sample), spec (the AR spectral density), and order.

References

Percival, D.B., Walden, A.T. (1993). Spectral Analysis for Physical Applications. Cambridge University Press.

See also

arYuleWalker, spec.ar

Examples

set.seed(1)
s <- arSpectrum(as.numeric(arima.sim(list(ar = c(0.6, -0.3)), 500)), order = 2)
s$freq[which.max(s$spec)]   # peak (normalized) frequency
#> [1] 0.1452906