For a description of Fractional Gaussian Noise, click here.
GRAPHICAL OUTPUT
These are time series plots of FGN corresponding to different values of H.
IMPLEMENTATION
Fractional Gaussian Noise series have been simulated using a version of the Durbin-Levinson Algorithm, implemented in S-Plus using C routines. This algorithm is described for example in Chapter 8.2 of Time Series: Theory and Methods, by P.J. Brockwell and R.A. Davis, Springer-Verlag, New York, 2nd edition, 1991. The source code for this algorithm is available here.
FARIMA
A more detailed description of FARIMA series.
SAMPLE RUN of
arima.fracdiff.sim.
OTHER
GRAPHICAL
OUTPUTS
IMPLEMENTATION
Following is a brief description of some of the variables used in the calls
to the functions.
ETHERNET DATA
They were first analyzed in W. E. Leland, M. S. Taqqu, W. Willinger
and D. V. Wilson, "On the self-similar nature of Ethernet traffic (Extended
version)", IEEE/ACM Transactions on Networking, 1994, 2, pp. 1-15.
Ethernet traffic sets and information can be obtained at the
Internet traffic archive,
here
.
IMPLEMENTATION
The byte and packet data sets are
available.
FARIMA series are fractionally differenced auto-regressive moving average
series. They are used at great length in time series analysis. For
reference see Brockwell and Davis, Time Series: Theory and Methods,
Springer-Verlag, 1991. . S-Plus has a function for simulating
FARIMA series, arima.fracdiff.sim. This function uses a version of the
Durbin-Levinson algorithm to produce the series. Since the algorithm
involves variances and covariances, it is inappropriate when stable series
are used.
# Denotes comments added after the session.
# In S-Plus:
> X11() # Enable graphics window.
> source("farima.generate") # Read in the program.
> temp_arima.fracdiff.sim(model=list(d=0.3),n=10000)
# Generates a Gaussian FARIMA(0,d,0) series
# with d=0.3, length=10000.
> tsplot(temp) # Time series plot of the simulated series.
> q() # Quit S-Plus.
Graphical output.
A different way of simulating the FARIMA series can be used. It is
especially appropriate if stable series are involved. The algorithm (
farima.generate )
generates whatever innovations are needed, and then passes them through a
differencing filter. Ideally this filter should have a summation up to
infinity. In practice, it is truncated to a value n. The
function available here currently can only generate FARIMA(1,d,1) series
with stable innovations,
but it should be easy to modify for whatever innovations and parameters are
necessary. To get a FARIMA(0,d,0), set theta =0, phi =0 below.
One should choose larger n's for larger d's. For example, we
have used n = 2000 for d=0.4, n=500 for d=0.3. The
farima.generate.pareto function generates FARIMA(0,d,0) with Pareto
innovations. It uses the parameters n, N, d, alpha, sigma , and is
part of the source code below.
The Ethernet series used here are part of a data set collected at Bellcore in
August of 1989. They correspond to one "normal" hour's worth of traffic,
collected every 10 milliseconds, thus resulting in a length of 360,000.
One data set measures the number of bytes per unit time, and one measures
the number of packets per unit time. These data sets have been widely
used.
