specARIMA {longmemo}

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)

Arguments

eta

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

p, q

integers giving AR and MA order respectively.

m

sample size determining Fourier frequencies.

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" (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)