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Series in
Signal and Information Processing, Vol. 23
edited by Hans-Andrea Loeliger
Christoph Reller
State-Space Methods in Statistical Signal Processing:
New Ideas and Applications.
1. Auflage/1st
edition 2013, XXIV, 264 Seiten/pages, € 64,00.
ISBN 3-86628-447-0,
978-3-86628-447-0
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 discrete-time state-space models. While state-space
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 message-passing 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 recursive-least-squares-type
algorithms that incorporate a forgetting factor. An application to outlier
detection in a noisy quasi-periodic signal with known period is shown.
Infinite impulse response systems are treated in depth with a focus on
the real-valued Jordan canonical form. For the autonomous second-order case,
analytic solutions for Gaussian messages across the whole state-space 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
state-transition matrix that is given in real-valued Jordan canonical form. The
same principles are used to estimate covariance matrices of a state-space
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 time-varying fundamental frequency of a quasi-periodic
signal.
The second part of this thesis starts with exposing connections between
model likelihood and scale factors of sum-product 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 sum-product 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 sum-product message passing is intimately connected with the
computation of likelihoods, likelihood functions, and log-likelihood 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 state-space
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 sum-product
message passing leads to the novel concept of likelihood filtering. In essence,
this is a message passing algorithm for computing efficiently
likelihood-related quantities for each member in the family under consideration.
This procedure can be considered as traditional sum-product 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
pulses-like events. Finally, we propose a hierarchical
likelihood filter architecture for general signal analysis.
Keywords: State-space model, factor graph, sum-product
message passing, parameter estimation, detection, recursive least squares, cyclo-stationary signal, quasi-periodic signal, frequency
estimation, expectation maximization, cyclic maximization, real Jordan
canonical form, variance estimation, parameter selection, hypothesis testing, glue
factor, likelihood filtering, change-point estimation, hierarchical likelihood
filtering.
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