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S

Series in Signal and Information Processing, Vol. 31
edited by HansAndrea Loeliger
Federico
Wadehn
State Space Methods with Applications
in Biomedical Signal Processing
1^{st }Edition
2019. XVIII, 196 pages. € 64,00.
ISBN 9783866286405
Abstract
For more than a decade, the
modelbased approach to signal processing based on state space models (SSMs)
and factor graphs is being pursued at the Signal and Information Processing
Laboratory of ETH Zurich and other groups around the world. This thesis
contributes to the theory and methods of this approach, and showcases some
biomedical applications; thereby, further highlighting its usefulness. The
proposed models and estimation algorithms, however, are applicable beyond the ﬁeld
of biomedical signal processing.
In Part 1 of this thesis we
cover modeling, inference, and learning with SSMs and factor graphs. First, we
introduce the topic of probabilistic modeling of signals and systems with SSMs
and factor graphs. Then, we discuss how to perform inference in linear Gaussian
models by Gaussian message passing. When applied to linear SSMs, Gaussian
message passing generalizes Kalman ﬁltering and can be used for diverse
tasks such as input and state estimation as well as output smoothing. Using
this message passing perspective, two matrixinversionfree Kalman smoothers
are derived (the MBF and BIFM smoothers) and subsequently extended to their
respective square root versions. Square root Gaussian message passing is
particularly suitable for applications where numerical stability issues arise.
After having described inference algorithms, we cover learning algorithms for
linear models and SSMs, with a particular focus on learning with sparsity. For
this, we rely on variational representations of sparsitypromoting priors
called normals with unknown variances (NUV). The NUV representation suggests
estimation algorithms based on alternating maximization and expectation
maximization (EM). The latter approach commonly requires a round of Gaussian
message passing at each EM iteration. Combining SSMs with sparsity opens up
many applications, from outlierrobust estimation, to event detection, to signal separation. To handle
signals whose variances or sparsity patterns have temporal correlations, a
model denoted by state space models with dynamical and sparse variances is
proposed. For this model we present approximate inference algorithms based on
EM and variational message passing. The theoretical part of this thesis
concludes with a duality perspective on inference and learning in factor graphs.
Dual factor graphs represent a dual optimization problem or equivalently a dual
estimation problem and are based on the Legendre transformation on factor
graphs and the Fenchel duality theorem. Dualizing factor graphs of SSMs results
in dual SSMs. These can be used to show that the MBF and BIFM smoothers are
dual algorithms. We then proceed to deriving the so called Hamiltonian system
using dual SSMs. The Hamiltonian system can be seen as a transformed form of
the Karush–Kuhn–Tucker (KKT) conditions of optimality for state estimation in
SSMs. The Hamiltonian equations can be solved by iterative algorithms.
Algorithms that iteratively solve the Hamiltonian system are alternatives to
Kalman smoothing and are a promising approach for estimation in largescale SSMs,
because these primaldual algorithms do not require the storage of covariance
matrices. Finally, the variational NUV representation of sparsitypromoting
priors is derived via the Legendre transform on factor graphs. Subsequently, we
show how to use Hamiltonian iterations and NUVs for outlierrobust estimation
in SSMs.
In Part 2, various state
space methods are used for analyzing physiological signals and systems. In
particular, two cardiovascular signal processing applications are presented.
One is an approach for robustly detecting heart beats in photoplethysmogram
recordings using autonomous state space models and localized model ﬁtting.
In the other, we use Gaussian message passing for input estimation in SSMs to
detect heart beats in ballistocardiogram recordings. Finally, we use sparse
input estimation in SSMs for estimating neural controller signals and for
performing a modelbased separation of the diﬀerent types of eye
movements (saccades, smooth pursuit, and ﬁxations) present in
freeviewing eye movement recordings.
Keywords: Biomedical signal processing: cardiovascular
signals; eye movements; physiological models; Statistical signal processing:
duality factor graphs; Gaussian message passing; robust estimation; sparse
estimation; state space models; system identiﬁcation.
Stichworte: Statistische Signalverarbeitung: Dualität;
Faktorgraphen;GaussianMessagePassing;RobusteSchätzung;SparseSchätzung;
Zustandsraummodelle; Systemidentiﬁkation; Biomedizinische
Signalverarbeitung: Kardiovaskuläre Signale; Augenbewegungen.
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