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.

Highlights

  • Recent advances of technology in sensors and storage capacity mean that increasing volumes of high frequency data are being routinely captured for many applications

  • 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

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Summary

INTRODUCTION

Recent advances of technology in sensors and storage capacity mean that increasing volumes of high frequency data are being routinely captured for many applications. The original SPAR method[1] used three delay coordinates to construct a three-dimensional attractor, which, when projected onto a plane perpendicular to the [1, 1, 1] vector, gave an orbit with threefold rotational symmetry for periodic signals, where we use the term “orbit” to define the trajectory corresponding to one period of the signal We generalize this by using N ≥ 3 delay coordinates to give an embedding in an N-dimensional phase space.[7] for N > 3, any simple visualization of the phase space representation requires a somewhat arbitrary selection of two or three dimensions.

Delay coordinates and phase space reconstruction
Two-dimensional projections
The eigenvalues and eigenvectors of CN
The eigenvectors of RN
Real invariant subspaces
A general form for the two-dimensional projection
Alternative choices of the time delay
Relation to the discrete Fourier transform
Trigonometric polynomial interpolation
Circular attractors
Size of the attractor
Invariance of the attractor to a time rescaling of the signal
Fourier transform
Frequency response
APPLICATION TO REAL SIGNALS
Practicalities of attractor generation
Choice of the time delay
Filtering behavior of the attractor
Implementation as a digital filter
Clinical examples
CONCLUSIONS

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