Abstract
Symmetric Projection Attractor Reconstruction (SPAR) provides an intuitive visualization and simple quantification of the morphology and variability of approximately periodic signals. The original method takes a three-dimensional delay coordinate embedding of a signal and subsequently projects this phase space reconstruction to a two-dimensional image with threefold symmetry, providing a bounded visualization of the waveform. We present an extension of the original work to apply delay coordinate embedding in any dimension 𝑁≥3 while still deriving a two-dimensional output with some rotational symmetry property that provides a meaningful visualization of the higher dimensional attractor. A generalized result is developed for taking 𝑁≥3 delay coordinates from a continuous periodic signal, where we determine invariant subspaces of the phase space that provide a two-dimensional projection with the required rotational symmetry. The result in each subspace is shown to be equivalent to following each pair of coefficients of the trigonometric interpolating polynomial of 𝑁 evenly spaced points as the signal is translated horizontally. Bounds on the mean and the frequency response of our new coordinates are derived. We demonstrate how this aids our understanding of the attractor properties and its relationship to the underlying waveform. Our generalized result is then extended to real, approximately periodic signals, where we demonstrate that the higher dimensional SPAR method provides information on subtle changes in different parts of the waveform morphology.
Data science is challenged to derive meaningful information from large volumes of data. The Symmetric Projection Attractor Reconstruction (SPAR) method1 provides an innovative approach to the analysis of approximately periodic time-series data by generating a unique, bounded, visualization of the signal that encapsulates waveform shape and variability from a three-dimensional time delay embedding of a signal. This paper extends this work to present a general form giving a similar visualization derived from an embedding of the signal in any dimension 𝑁≥3, and the result is shown to be closely related to trigonometric polynomial interpolation. While the theoretical results are stated in the context of periodic signals, significantly, we show that the properties of this theoretical generalization can be translated to the analysis of real, approximately periodic signals where we demonstrate the potential clinical utility of the SPAR approach in the novel visualization and quantification of physiological signals.