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S

Series in
Signal and Information Processing, Vol. 23
edited by HansAndrea Loeliger
Christoph Reller
StateSpace Methods in Statistical Signal Processing:
New Ideas and Applications.
1. Auflage/1st
edition 2013, XXIV, 264 Seiten/pages, € 64,00.
ISBN 3866284470,
9783866284470
This thesis is about several extensions of a general framework and about
the application of these extensions to various problems arising in signal
processing. The general framework is a graphical modeling technique, more
precisely factor graphs, which provides the basis for the development of
message passing algorithms. Such algorithms can be used to solve many
statistical inference problems, most notably, estimation and detection
problems.
Most of the problems addressed in this thesis are in some way linked to
one or several discretetime statespace models. While statespace
representations of systems are widely used in control theory and somewhat less
in statistics, a factor graph approach to such models seems to be neglected.
Indeed, this thesis shows how the interaction between these two topics leads to
a powerful framework for devising novel algorithms in a systematic yet
uncomplicated manner.
This thesis is partitioned into two parts. The first part focuses on Gaussian
message passing in linear models, and parameter estimation for such models. The
second part is concerned with messagepassing based computation of likelihoods
or related quantities. We start with the first part.
The factor graph representation of a linear model leads to Gaussian
message passing in the case of known model parameters. In a first extension we
consider several variants and enhancements of recursiveleastsquarestype
algorithms that incorporate a forgetting factor. An application to outlier
detection in a noisy quasiperiodic signal with known period is shown.
Infinite impulse response systems are treated in depth with a focus on
the realvalued Jordan canonical form. For the autonomous secondorder case,
analytic solutions for Gaussian messages across the whole statespace model are
derived and the relation between a forgetting factor and state noise is shown.
Continuous time signals and two interpolation models are touched upon. In a
further extension, the local factor graph view of three approximate inference
principles (cyclic maximization, expectation maximization, and local Taylor
approximation) is shown. These principles are applied to the estimation of a
statetransition matrix that is given in realvalued Jordan canonical form. The
same principles are used to estimate covariance matrices of a statespace
model. The resulting algorithms are iterative in nature and the resulting
messages are members of the exponential family. We show an application to the
estimation of the timevarying fundamental frequency of a quasiperiodic
signal.
The second part of this thesis starts with exposing connections between
model likelihood and scale factors of sumproduct messages in a factor graph. A
main result is the derivation of message passing update rules for two types of
such scale factors that arise in sumproduct message passing. First, different
types of general factors and general messages are considered. Then the setup is
narrowed down to linear factors and Gaussian messages.
Since sumproduct message passing is intimately connected with the
computation of likelihoods, likelihood functions, and loglikelihood ratios,
such quantities can be neatly expressed in terms of messages or message scale
factors. The latter need, however, not in all cases be computed, and this case
distinction is made precise.
Next, we consider a factor graph representation of linear statespace
models augmented with an additional factor  the “glue factor”  connecting
state variables of several models. This leads to the notion of a family of
factor graphs parametrized by the glue factor parameters and its position on
the time axis. A surprising variety of problems such as array processing and
pulse modeling can be treated in this framework.
The glue factor view of likelihood computation by means of sumproduct
message passing leads to the novel concept of likelihood filtering. In essence,
this is a message passing algorithm for computing efficiently
likelihoodrelated quantities for each member in the family under consideration.
This procedure can be considered as traditional sumproduct message passing on
several graphs, but without neglecting scale factors, followed by a likelihood
computation. Both offline (block based) and online algorithms are thus
formulated for estimation and detection of model changes and for locating
pulseslike events. Finally, we propose a hierarchical
likelihood filter architecture for general signal analysis.
Keywords: Statespace model, factor graph, sumproduct
message passing, parameter estimation, detection, recursive least squares, cyclostationary signal, quasiperiodic signal, frequency
estimation, expectation maximization, cyclic maximization, real Jordan
canonical form, variance estimation, parameter selection, hypothesis testing, glue
factor, likelihood filtering, changepoint estimation, hierarchical likelihood
filtering.
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