  The oscillations observed in many time series, particularly in biomedicine,
exhibit morphological variations over time. These morphological variations are
caused by intrinsic or extrinsic changes to the state of the generating system,
henceforth referred to as dynamics. To model these time series (including and
specifically pathophysiological ones) and estimate the underlying dynamics, we
provide a novel wave-shape oscillatory model. In this model, time-dependent
variations in cycle shape occur along a manifold called the wave-shape
manifold. To estimate the wave-shape manifold associated with an oscillatory
time series, study the dynamics, and visualize the time-dependent changes along
the wave-shape manifold, we propose a novel algorithm coined Dynamic Diffusion
map (DDmap) by applying the well-established diffusion maps (DM) algorithm to
the set of all observed oscillations. We provide a theoretical guarantee on the
dynamical information recovered by the DDmap algorithm under the proposed
model. Applying the proposed model and algorithm to arterial blood pressure
(ABP) signals recorded during general anesthesia leads to the extraction of
nociception information. Applying the wave-shape oscillatory model and the
DDmap algorithm to cardiac cycles in the electrocardiogram (ECG) leads to
ectopy detection and a new ECG-derived respiratory signal, even when the
subject has atrial fibrillation.
