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<p>Biomedical time series are non-invasive windows through which we may observe human
systems. Although a vast amount of information is hidden in the medical field's growing
collection of long-term, high-resolution, and multi-modal biomedical time series,
effective algorithms for extracting that information have not yet been developed.
We are particularly interested in the physiological dynamics of a human system, namely
the changes in state that the system experiences over time (which may be intrinsic
or extrinsic in origin). We introduce a mathematical model for a particular class
of biomedical time series, called the wave-shape oscillatory model, which quantifies
the sense in which dynamics are hidden in those time series. There are two key ideas
behind the new model. First, instead of viewing a biomedical time series as a sequence
of measurements made at the sampling rate of the signal, we can often view it as a
sequence of cycles occurring at irregularly-sampled time points. Second, the "shape"
of an individual cycle is assumed to have a one-to-one correspondence with the state
of the system being monitored; as such, changes in system state (dynamics) can be
inferred by tracking changes in cycle shape. Since physiological dynamics are not
random but are well-regulated (except in the most pathological of cases), we can assume
that all of the system's states lie on a low-dimensional, abstract Riemannian manifold
called the phase manifold. When we model the correspondence between the hidden system
states and the observed cycle shapes using a diffeomorphism, we allow the topology
of the phase manifold to be recovered by methods belonging to the field of unsupervised
manifold learning. In particular, we prove that the physiological dynamics hidden
in a time series adhering to the wave-shape oscillatory model can be well-recovered
by applying the diffusion maps algorithm to the time series' set of oscillatory cycles.
We provide several applications of the wave-shape oscillatory model and the associated
algorithm for dynamics recovery, including unsupervised and supervised heartbeat classification,
derived respiratory monitoring, intra-operative cardiovascular monitoring, supervised
and unsupervised sleep stage classification, and f-wave extraction (a single-channel
blind source separation problem).</p>
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