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
is a scalar in the range zero to one. For purely periodic signals,
, 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, ...
,
.
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:
: