specFGN: Spectral Density of Fractional Gaussian Noise

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

Description

Calculation of the spectral density f of normalized fractional Gaussian noise with self-similarity parameter H at the Fourier frequencies 2*pi*j/m (j=1,...,(m-1)).

B.specFGN computes (approximations of) the B(λ, H) component of the spectrum f_H(λ).

Usage

  specFGN(eta, m, ...)
B.specFGN(lambd, H, k.approx=3, adjust = (k.approx == 3), nsum = 200)

Arguments

eta	parameter vector eta = c(H, *).
m	sample size determining Fourier frequencies.
...	optional arguments for B.specFGN(): k.approx, etc
lambd	numeric vector of frequencies in [0, pi]
H	Hurst parameter in (1/2, 1), (can be outside, here).
k.approx	either integer (the order of the Paxson approximation), or NULL, NA for choosing to use the slow direct sum (of nsum terms.)
adjust	logical indicating (only for k.approx == 3, the default) that Paxson's empirical adjustment should also be used.
nsum	if the slow sum is used (e.g. for k.approx = NA), the number of terms.

Details

Note that

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

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

Since longmemo version 1.1-0, a fast approximation is available (and default), using k.approx terms and an adjustment (adjust=TRUE in the default case of k.approx=3), which is due to the analysis and S code from Paxson (1997).

Value

specFGN() returns an object of class "spec" (see also spectrum) with components

freq	the Fourier frequencies om_j in (0,pi)) at which the spectrum is computed. Note that om_j = 2*pi*j/m for j=1,..., m-1, and m = floor((n-1)/2).
spec	the scaled values spectral density f(λ) values at the freq values of λ. f*(lambda) = f(lambda) / theta1 adjusted such \int \log(f^*(λ)) dλ = 0.
theta1	the scale factor θ_1.
H	the self-similarity parameter from input.
method	a character indicating the kind of model used.

B.specFGN() returns a vector of (approximate) values B(λ, H).

Author(s)

Jan Beran originally (using the slow sum); Martin Maechler, based on Vern Paxson (1997)'s code.

References

Jan Beran (1994). Statistics for Long-Memory Processes; Chapman & Hall, NY.

Vern Paxson (1997). Fast, Approximate Synthesis of Fractional Gaussian Noise for Generating Self-Similar Network Traffic; Computer Communications Review 27 5, 5–18.

See Also

The spectral estimate for fractional ARIMA, specARIMA; more generally, spectrum.

Examples

 str(rg.7  <- specFGN(0.7, m = 100))
 str(rg.5  <- specFGN(0.5, m = 100))# { H = 0.5 <--> white noise ! }

 plot(rg.7) ## work around plot.spec() `bug' in R < 1.6.0
 plot(rg.5, add = TRUE, col = "blue")
 text(2, mean(rg.5$spec), "H = 0.5 [white noise]", col = "blue", adj = c(0,-1/4))

## This was the original method in longmemo, upto version 1.0-0 (incl):
 rg.7.o <- specFGN(0.7, m = 100, k.approx=NA, nsum = 200)
 ## quite accurate (but slightly slower):
 rg.7f  <- specFGN(0.7, m = 100, k.approx=NA, nsum = 10000)
 ## comparing old and new default :
 all.equal(rg.7, rg.7.o)# different in about 5th digit
 all.equal(rg.7, rg.7f )# ==> new default is *more* accurate: 1.42 e-6
 ## takes about  7 sec {in 2011}:
 rg.7ff <- specFGN(0.7, m = 100, k.approx=NA, nsum = 500000)
 all.equal(rg.7f, rg.7ff)# ~ 10 ^ -7
 all.equal(rg.7  $spec, rg.7ff$spec)# ~ 1.33e-6 -- Paxson is accurate!
 all.equal(rg.7.o$spec, rg.7ff$spec)# ~ 2.40e-5 -- old default is less so

longmemo documentation built on March 26, 2020, 7:42 p.m.