longmemo (version 1.1-3)

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 $\lambda$.
$f^{\ast }(\lambda )=f(\lambda )/{\theta }_1$ adjusted such $\int \log (f^{\ast }(\lambda ))d\lambda =0$.

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 $2\ast \pi \ast j/n$, ($j=1,\dots ,(n-1)$), 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)