Recurrence Period Density Entropy
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Recurrence period density entropy (RPDE) is a method, in the fields of
dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
,
stochastic processes In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appe ...
, and
time series analysis In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
, 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 Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
and time delayed
mutual information In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such ...
, except that it measures repetitiveness in the
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
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 Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
, Gaussianity or dynamical determinism. It has been successfully used to detect abnormalities in biomedical contexts such as
speech Speech is a human vocal communication using language. Each language uses Phonetics, phonetic combinations of vowel and consonant sounds that form the sound of its words (that is, all English words sound different from all French words, even if ...
signal.M. Little, P. McSharry, I. Moroz, S. Roberts (2006
Nonlinear, Biophysically-Informed Speech Pathology Detection
in 2006 IEEE International Conference on Acoustics, Speech and Signal Processing, 2006. ICASSP 2006 Proceedings.: Toulouse, France. pp. II-1080-II-1083.
M.A. Little, P.E. McSharry, S.J. Roberts, D.A.E. Costello, I.M. Moroz (2007
Exploiting Nonlinear Recurrence and Fractal Scaling Properties for Voice Disorder Detection
BioMedical Engineering OnLine, 6:23
The RPDE value \scriptstyle H_\mathrm is a scalar in the range zero to one. For purely periodic signals, \scriptstyle H_\mathrm=0, whereas for purely
i.i.d. In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is us ...
, uniform
white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
, \scriptstyle H_\mathrm \approx 1.


Method description

The RPDE method first requires the
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...
of a time series in
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
, which, according to stochastic extensions to Taken's embedding theorems, can be carried out by forming time-delayed vectors: :\mathbf_n= _n, x_, x_, \ldots, x_/math> for each value ''x''''n'' in the time series, where ''M'' is the
embedding dimension This is a glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary of ring theory and glossary of module theory. In this article, all rings are ...
, 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_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 A histogram is an approximate representation of the distribution of numerical data. The term was first introduced by Karl Pearson. To construct a histogram, the first step is to " bin" (or "bucket") the range of values—that is, divide the ent ...
. This histogram is normalised to sum to unity, to form an estimate of the recurrence period density function ''P''(''T''). The normalised
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
of this density: :H_\mathrm = -(\ln^ \sum_^ P(t) \ln is the RPDE value, where \scriptstyle T_\max is the largest recurrence value (typically on the order of 1000 samples). Note that RPDE is intended to be applied to both deterministic and stochastic signals, therefore, strictly speaking, Taken's original embedding theorem does not apply, and needs some modification.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


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 A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
or
linear prediction Linear prediction is a mathematical operation where future values of a discrete-time signal are estimated as a linear function of previous samples. In digital signal processing, linear prediction is often called linear predictive coding (LPC) and ...
. The recurrence period density is a sparse representation for nonlinear, non-Gaussian and nondeterministic signals, whereas the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
is only sparse for purely periodic signals.


See also

*
Recurrence plot In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for each moment i in time, the times at which the state of a dynamical system returns to the previous state at i, i.e., when the phase space trajectory visits rou ...
, a powerful visualisation tool of recurrences in dynamical (and other) systems. *
Recurrence quantification analysis Recurrence quantification analysis (RQA) is a method of nonlinear data analysis (cf. chaos theory) for the investigation of dynamical systems. It quantifies the number and duration of recurrences of a dynamical system presented by its phase space tr ...
, another approach to quantify recurrence properties.


References

{{Reflist


External links


Fast MATLAB code for calculating the RPDE value.
* http://www.recurrence-plot.tk/ Signal processing Entropy Stochastic processes Dynamical systems