specARIMA: Spectral Density of Fractional ARMA Process

In longmemo: Statistics for Long-Memory Processes (Book Jan Beran), and Related Functionality

Spectral Density of Fractional ARMA Process

Description

Calculate the spectral density of a fractional ARMA process with standard normal innovations and self-similarity parameter H.

Usage

specARIMA(eta, p, q, m, spec.only = FALSE)

Arguments

eta	parameter vector eta = c(H, phi, psi).
p, q	integers giving AR and MA order respectively.
m	sample size determining Fourier frequencies.
spec.only	logical indicating that only the numeric vector of spectral density estimates (the spec component of the full list) is returned.

Details

at the Fourier frequencies 2*\pi*j/n, (j=1,\dots,(n-1)), cov(X(t),X(t+k)) = (sigma/(2*pi))*integral(exp(iuk)g(u)du).

— or rather – FIXME –

1. cov(X(t),X(t+k)) = integral[ exp(iuk)f(u)du ]

2. f() = theta1 * f*() ; spec = f*(), and integral[log(f*())] = 0

Value

an object of class "spec" (by default, when spec.only is false) see also spectrum) with components

freq	the Fourier frequencies (in (0,\pi)) at which the spectrum is computed, see freq in specFGN.
spec	the scaled values spectral density f(\lambda) values at the freq values of \lambda. f^*(\lambda) = f(\lambda) / \theta_1 adjusted such \int \log(f^*(\lambda)) d\lambda = 0.
theta1	the scale factor \theta_1.
pq	a vector of length two, = c(p,q).
eta	a named vector c(H=H, phi=phi, psi=psi) from input.
method	a character indicating the kind of model used.

Author(s)

Jan Beran (principal) and Martin Maechler (fine tuning)

References

Beran (1994) and more, see ....

See Also

The spectral estimate for fractional Gaussian noise, specFGN. In general, spectrum and spec.ar.

Examples

 str(r.7  <- specARIMA(0.7, m = 256, p = 0, q = 0))
 str(r.5  <- specARIMA(eta = c(H = 0.5, phi=c(-.06, 0.42, -0.36), psi=0.776),
                       m = 256, p = 3, q = 1))
 plot(r.7)
 plot(r.5)

longmemo documentation built on Aug. 8, 2025, 6:15 p.m.