The Matlab code for MLE of the parameters of fBt Homepage

Matlab code for the ML estimation of the parameters of fractional Brownian traffic

Description of the Package

The Matlab routines available here perform the maximum likelihood estimation (MLE) of the parameters of fractional Brownian traffic. The MLE is performed in the time domain, using either geometrical or linear sampling, with different approximations for the inverse and determinant of the covariance matrix in the Gausian likelihood function.

Copyright Information

The program library has resulted from the work conducted within the framework of the COM² project funded by the Academy of Finland. The programs can be used and modified freely. No claims are made about the correctness of the programs and no liability is taken for any damage caused by the use of the programs in the library.

The following individuals have contributed to the library: Attila Vidács and Jorma Virtamo.

The Code

For DOS users: mle.zip

For UNIX users: mle.tar.gz

(Or the ASCII source files can be accessed one-by-one: README.TXT, mle.m, alph.m, lev.m and l_deriv.m.)

Reports of bugs in the program will be gratefully received by vidacs@ttt-atm.ttt.bme.hu

Notes for use

Whithin Matlab run the M-file mle.m (e.g., by starting Matlab in the actual working directory, and typing 'mle').

Inputs: No command line arguments are passed to the routine. Instead, switches must be set in the mle.m file header.

Outputs: The MLE estimates (using the actual pre-selected methods) for m, a and H are printed, together with their standard deviations (and confidence intervals).

Documentation

This page is only intended to make the routines available and to give some basic instructions in their use, not to provide detailed documentation!

A more detailed description is given within the source codes of the routines. Furthermore, the following papers deal with the subject:

A. Vidács and J.T. Virtamo, "ML Estimation of the Parameters of FBM Traffic with Geometrical Sampling", submitted for publication, March, 1999.

A. Vidács and J.T. Virtamo, "Time Domain MLE of the Parameters of FBM Traffic", internal document, Lab. of Telecommunications Technology, Helsinki University of Technology, Espoo, Finland, May, 1999.

Theoretical notes

Fractional Brownian motion is a popular model for long-range dependent traffic. Norros suggested the following model:

X(t) = m·t + \sqrt(a)·Z(t)

where X(t) represents the amount of traffic arrived in (0,t). The model has three parameters, m is the mean input rate, a is a variance parameter and 0.5 < H < 1 is the self-similarity parameter of Z(t), which is a normalized fractional Brownian motion.

Assume the traffic has been observed at n time instants forming the vector t = (t₁, t₂, ..., t_n)^t. Since X(t) is Gaussian, the joint distribution of the observed traffic values (X₁, X₂, ..., X_n) is n dimensional Gaussian with mean m·t and covariance matrix G(n x n). Thus, a Gaussian Maximum Likelihood Estimation (MLE) can be applied (in the time domain) to estimate the model parameters m, a and H. Explicit expressions for the ML estimates of m and a in terms of H can be given, as well as the expression for the log-likelihood function from which the estimate of H is obtained as the minimizing argument.

The Hurst parameter H describes the scaling behavior of the traffic. Therefore, in order to determine its value from measured traffic, the sample points have to cover several time scales. With the ordinary linear sampling this leads to the requirement of very large number of sample points. In order to use the measurements more efficiently we introduce a geometric sequence of sampling points:

t_i = qⁱ, with some 0 < q < 1.

In addition to distributing the sample points in a better way on different time scales, geometric sampling fits neatly with the self-similar behavior of the fBm traffic. By a proper descaling the traffic process is stationary on this grid leading to a Toeplitz-type covariance matrix, and allows us to develop approximations to the inverse and determinant of the covariance matrix G.

In practice, exact MLE poses computational problems. To avoid these problems, one can use approximate methods to calculate the inverse and the determinant of the covariance matrix G. For example, a simple linear approximation can be used to compute an approximate inverse and its determinant for G. Nevertheless, the use of this approximation in the likelihood function yields a good estimate for H, while the accuracy of the estimate of a sufers more. (An improved approximation can be achieved if we approximate the inverse of G by a band-matrix.)

However, the geometrical sampling does indeed give a better estimate for H than the traditional linear sampling. For the estimation of a the geometrical sampling does not give any direct advantage, but as its estimator actually depends on the estimator of H, the overall accuracy obtained with geometrical sampling is better. (As for m, in fact, the ML estimate gives almost negligible reduction in its variance when compared to the sample mean as an estimate for m.)