longmemo (version 1.1-2)

specARIMA: 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)

Value

an object of class "spec" (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.

Arguments

eta

parameter vector eta = c(H, phi, psi).

p, q

integers giving AR and MA order respectively.

m

sample size determining Fourier frequencies.

Author

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

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

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)