Recurrence period density entropy

Recurrence period density entropy (RPDE) is a method, in the fields of dynamical systems, stochastic processes, and time series analysis, for determining the periodicity, or repetitiveness of a signal.

Overview

Recurrence period density entropy is useful for characterising the extent to which a time series repeats the same sequence, and is therefore similar to linear autocorrelation and time delayed mutual information, except that it measures repetitiveness in the phase space of the system, and is thus a more reliable measure based upon the dynamics of the underlying system that generated the signal. It has the advantage that it does not require the assumptions of linearity, Gaussianity or dynamical determinism. It has been successfully used to detect abnormalities in biomedical contexts such as speech and ECG signals (Little et al. 2006, 2007).

The RPDE value $scriptstyle\; H\_mathrm\{norm\}$ is a scalar in the range zero to one. For purely periodic signals, $scriptstyle\; H\_mathrm\{norm\}=0$, whereas for purely i.i.d., uniform white noise, $scriptstyle\; H\_mathrm\{norm\}\; approx\; 1$ (Little et al. 2007).

Method description

The RPDE method first requires the embedding of a time series in phase space, which, according to stochastic extensions to Taken's embedding theorems, can be carried out by forming time-delayed vectors:

:$mathbf\{X\}\_n=\; [x\_n,\; x\_\{n+\; au\},\; x\_\{n+2\; au\},\; ldots,\; x\_\{n+(m-1)\; au\}]$

for each value "x"_"n" in the time series, where "m" is the embedding dimension, and τ is the embedding delay. These parameters are obtained by systematic search for the optimal set (due to lack of practical embedding parameter techniques for stochastic systems) (Stark et al. 2003). Next, around each point $scriptstyle\; mathbf\{X\}\_n$ in the phase space, an $varepsilon$-neighbourhood (an "m"-dimensional ball with this radius) is formed, and every time the time series returns to this ball, after having left it, the time difference "T" between successive returns is recorded in a histogram. This histogram is normalised to sum to unity, to form an estimate of the recurrence period density function "P"("T"). The normalised entropy of this density:

:$H\_mathrm\{norm\}\; =\; -(ln\{T\_max)\}^\{-1\}\; sum\_\{t=1\}^\{T\_max\}\; P(t)\; ln\{P(t)\}$

is the RPDE value, where $scriptstyle\; T\_max$ is the largest recurrence value (typically on the order of 1000 samples) (Little et al. 2007).

[
$scriptstylevarepsilon$ is created. All recurrences into this neighbourhood are tracked, and the time interval "T" between recurrences is recorded in a histogram. This histogram is normalised to create an estimate of the recurrence period density function "P"("T"). The normalised entropy of this density is the RPDE value $scriptstyle\; H\_mathrm\{norm\}$.]

RPDE in practice

RPDE has the ability to detect subtle changes in natural biological time series such as the breakdown of regular periodic oscillation in abnormal cardiac function which are hard to detect using classical signal processing tools such as the Fourier transform or linear prediction. The recurrence period density is a sparse representation for nonlinear, non-Gaussian and nondeterministic signals, whereas the Fourier transform is only sparse for purely periodic signals.

References

* M. Little, P. McSharry, S. Roberts, D. Costello, I. Moroz (2007), [http://www.biomedical-engineering-online.com/content/6/1/23 Exploiting Nonlinear Recurrence and Fractal Scaling Properties for Voice Disorder Detection] , Biomed Eng Online, 6(1):23
* M. Little, P. McSharry, S. Roberts, I. Moroz (2006), Nonlinear, Biophysically-Informed Speech Pathology Detection. 2006 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2006. ICASSP-2006., Toulouse, France, IEEE Press
* J. Stark, D. S. Broomhead, M. E. Davies and J. Huke (2003) Delay Embeddings for Forced Systems. II. Stochastic Forcing. Journal of Nonlinear Science, 13(6):519-577
* cite journal
author=N. Marwan, M. C. Romano, M. Thiel, J. Kurths
title=Recurrence Plots for the Analysis of Complex Systems
journal=Physics Reports
volume=438
issue=5-6
year=2007
url=http://dx.doi.org/10.1016/j.physrep.2006.11.001
doi=10.1016/j.physrep.2006.11.001

See also

* Recurrence plot, a powerful visualisation tool of recurrences in dynamical (and other) systems.
* Recurrence quantification analysis, another approach to quantify recurrence properties.

External links

* [http://www.eng.ox.ac.uk/samp Systems Analysis, Modelling and Prediction (SAMP), University of Oxford] [http://www.eng.ox.ac.uk/samp/rpde_soft.html Fast MATLAB code] for calculating the RPDE value.
* http://www.recurrence-plot.tk/