Abstract
"Symmetric Projection Attractor Reconstruction (SPAR) is a recently developed innovative technique that generates a bounded visualisation of an approximately periodic signal, encapsulating its morphology and variability, and providing a simple way to quantify the waveform features.
The original SPAR method employed time delay embedding to generate a three-dimensional phase space reconstruction of the signal, which was then projected to a plane where it exhibited three-fold rotational symmetry. Our work extends this to a delay coordinate embedding in any dimension N ≥ 3 for continuous, periodic signals, whilst still determining a two-dimensional invariant subspace with some rotational symmetry property, and we then explore the properties of our resulting `attractor' image. 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, and we demonstrate how this aids our understanding of the attractor behaviour, and its relationship to the underlying waveform.
Our generalised result is then applied to approximately periodic signals, where we show that the higher dimensional SPAR method provides information on subtle morphology changes in the waveform. We discuss the implementation of SPAR for real signals, and demonstrate how appropriate metrics can be extracted to quantify the attractor shape and variability.
The electrocardiogram (ECG) is widely used to capture the electrical activity of the heart, supporting diagnosis and patient monitoring. As a multi-component signal, it is a good candidate for SPAR analysis. We present five preliminary clinical studies where we demonstrate that the SPAR approach allows us to discriminate between distinct biological groups, supplementing traditional techniques of analysis. Our work therefore supports the further mathematical exploration of the SPAR method in parallel with extending its implementation for the ECG, particularly in the areas of patient stratification and risk management."