artfima (version 1.5)

artfimaSDF: Computation of theoretical spectral density function (SDF)

Description

Computes the theoretical SDF at the Fourier frequencies for a time series of length n. Used for Whittle MLE. Assumes model parameters are valid for a stationary process.

Usage

artfimaSDF(n = 100, d = 0, lambda = 0, phi = numeric(0), theta = numeric(0),  obj = NULL, plot=c("loglog", "log", "none"))

Arguments

n

length of time series

d

ARTFIMA difference parameter, any real value. When d=numeric(0), reduces to ARMA and lambda is ignored.

lambda

ARTFIMA tempered decay parameter. When lambda=numeric(0), reduces to ARFIMA

phi

AR coefficients

theta

MA coefficients, Box-Jenkins definition

obj

object of class artfima

plot

type of plot, "log-log", "log" or "none"

Value

vector of length floor(n/2) containing the values of the SDF at the Fourier frequencies, 2*pi*c(1/n, floor(n/2)/n, 1/n).

Warning

This function serves as a utility function for Whittle estimation so, for speed, we skip the checking if the parameters d, phi, or lambda are valid parameters for a stationary process.

Details

The Fourier frequencies, 2*pi*c(1/n, floor(n/2)/n, 1/n), are used in the definition of the SDF. The SDF is normalized so that the area over (0, 0.5) equals the variance of the time series assuming unit innovation variance. The periodogram is normalized in the same way, so the mean of the periodogram is an estimate of the variance of the time series. See example below.

References

TBA

See Also

artfimaTACVF, Periodogram

Examples

phi <- 0.8
n <- 256
set.seed(4337751)
z <- artsim(n, phi=phi)
VarZ <- mean((z-mean(z))^2)
Ip <- Periodogram(z)
length(Ip)
x <- (1/n)*(1:length(Ip))
plot(x, Ip, xlab="frequency", ylab="Spectral density & Periodogram", 
     main=paste("AR(1), phi =", phi), type="l", col=rgb(0,0,1,0.5))
n <- 5000
y <- artfimaSDF(n, phi=phi)
x <- (1/n)*(1:length(y))
lines(x, y, type="l", lwd=1.25)
h <- x[2]-x[1] #step length
SimpsonsRule <- function(h, y) {
  n <- length(y)
  h/3*sum(y * c(1, rep(c(4,2), n-1), 1))
}
AreaApprox <- SimpsonsRule(h, y)
text(0.2, 50, labels=paste("Area under SDF using Simpson's Rule =", 
                           round(AreaApprox,4)))
TVarZ <- 1/(1-phi^2)
text(0.2, 40, labels=paste("Theoretical AR Variance =", round(TVarZ,4)))
text(0.2, 30, labels=paste("mean(Ip) =", round(mean(Ip),4)))
text(0.2, 20, labels=paste("sample variance =", round(VarZ,4)))