Hidden Markov Models  Traffic Modeling and Subspace Methods
Sofia Andersson
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology
2002
ISBN 9162852531
LUTFMS10182002

Abstract:

The main motivation for this thesis, however not the only one, is the search
for models for traffic in telecommunication networks. Traffic characterization
and modeling are of great importance in the analysis and dimensioning of
communication systems. During the last decades we have experienced an explosive
growth of our telecommunication networks. New important properties, both
on macroscopic and microscopic time scales, have been revealed. These new
features have a profound impact on network performance and thus needs to
be taken into consideration. In Paper A we study the microdynamics of some
data sets of traffic on the link level. Our approach is Poissonification,
a kind of random local smoothing of a point process. The idea is to use
Poissonification on empirical data materials in order to find the (small)
time scale where the process switch from a doubly stochastic to a homogeneous
behavior.

The models used and studied are hidden Markov models (HMMs). In Paper B we
study specifically the Markovmodulated Poisson process (MMPP). In fitting
an MMPP to some sets of observed traffic we concentrate on maximum likelihood
(ML) methods, but also use moment methods. The ML methods are computationally
more complex but concerning performance they prove to be superior to moment
methods.

As a possible candidate for a method to combine the simplicity of the moment
methods, and the accuracy of the likelihood methods, we have subspace methods.
To be able to use subspace methods for HMMs the process needs to be represented
in state space form. In Paper C of this thesis we show that it is possible
to represent an HMM as a state space system in innovation form. This
reformulation is complicated by the nonminimality within the state space
representation of the HMM. The reformulation involves deriving solutions
to algebraic Riccati equations which are usually treated under minimality
assumptions.
In Paper D we develop a subspace identification algorithm especially designed
for HMMs. Some of the crucial assumptions on the system, needed for the subspace
methods to work, fail to be met by HMMs and thus commonly used methods are
not directly applicable. Therefore we need to consider each of the steps
in existing algorithms with regard to the specific conditions of the HMM
and in this way develop a new algorithm. Consistency is proved of this algorithm.
However, we manage to estimate the system parameters only seen as a projection
on a certain subspace and only up to a similarity transformation.



Keywords:

hidden Markov model, Markovmodulated Poisson process, traffic analysis,
Poissonification, likelihood estimation, state space

representation, Riccati equation, subspace identification




