Symmetric projection attractor reconstruction: Embedding in higher dimensions

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.

Detecting beats in the photoplethysmogram: benchmarking open-source algorithms

“…A c c e p t e d M a n u s c r i p t The PPG is segmented into 20 s windows and time delay coordinates are used to represent it in 7-dimensional phase space with the time delay set to one seventh of the average inter-beat interval. The Symmetric Projection Attractor Reconstruction method is then used to construct an appropriate 2-dimensional projection of the phase space [35], [36]. Beats are identified as times at which the orbit crosses the x-axis.…”

Photoplethysmogram beat detection using Symmetric Projection Attractor Reconstruction

“…Using Takens’ method for reconstructing attractors using time delay coordinates ( Takens, 1981 ), an attractor can be reconstructed in an N -dimensional phase space from a single signal x ( t ) by using a vector of delay coordinates given by where τ > 0 is a fixed delay and N ≥ 2 is the embedding dimension. The embedding dimension was initially chosen as N = 3 for ease of visualisation in Aston et al (2018) although this was later generalised to any embedding dimension N ≥ 3 in Lyle and Aston (2021) .…”

“…More generally, the embedding dimension can be set to any integer N ≥ 3. Lyle and Aston (2021) showed how to generate similar two-dimensional attractors ( v N , k , w N , k ), but with multiple attractors for each dimension N > 4 which we call projections, indexed by k = 1, … , ⌊( N − 1)/2⌋. The ( N , k ) attractor has an approximate m -fold rotational symmetry where m = N /gcd ( N , k ).…”

“…For higher dimensions, the attractors become more circular in nature ( Lyle and Aston, 2021 ) which is an advantage. However, if N is too large, there may be very few data points between each of the delay coordinates if the sampling frequency of the data is relatively low and so there may be a mismatch between the expected positions of the delay coordinates and data points available.…”

“…In this study we propose a novel beat detection algorithm for PPG signals based on the generation of Symmetric Projection Attractor Reconstruction (SPAR) attractors ( Aston et al, 2018 ; Lyle and Aston, 2021 ). The SPAR method derives attractors from signals using a number of equally spaced points running through the signal to generate a bounded attractor in a two-dimensional phase space.…”