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Measure of regularity of nonlinear time series

`approxEnt = approximateEntropy(X)`

`approxEnt = approximateEntropy(X,lag)`

`approxEnt = approximateEntropy(X,[],dim)`

`approxEnt = approximateEntropy(X,lag,dim)`

`approxEnt = approximateEntropy(___,Name,Value)`

estimates the approximate entropy with additional options specified by one or more
`approxEnt`

= approximateEntropy(___,`Name,Value`

)`Name,Value`

pair arguments.

Approximate entropy is computed in the following way,

The

`approximateEntropy`

function first generates a delayed reconstruction*Y*for N data points with embedding dimension_{1:N}*m*, and lag*τ*.The software then calculates the number of within range points, at point

*i*, given by,$${N}_{i}={\displaystyle \sum _{i=1,i\ne k}^{N}1\left({\Vert {Y}_{i}-{Y}_{k}\Vert}_{\infty}<R\right)}$$

where

**1**is the indicator function, and*R*is the radius of similarity.The approximate entropy is then calculated as $$approxEnt={\Phi}_{m}-{\Phi}_{m+1}$$ where,

$${\Phi}_{m}={\left(N-m+1\right)}^{-1}{\displaystyle \sum _{i=1}^{N-m+1}\mathrm{log}\left({N}_{i}\right)}$$

[1] Pincus, Steven M. "Approximate
entropy as a measure of system complexity." *Proceedings of the National
Academy of Sciences*. 1991 88 (6) 2297-2301;
doi:10.1073/pnas.88.6.2297.

[2] U. Rajendra Acharya, Filippo
Molinari, S. Vinitha Sree, Subhagata Chattopadhyay, Kwan-Hoong Ng, Jasjit S. Suri.
"Automated diagnosis of epileptic EEG using entropies." *Biomedical Signal
Processing and Control* Volume 7, Issue 4, 2012, Pages 401-408, ISSN
1746-8094.

[3] Caesarendra, Wahyu &
Kosasih, P & Tieu, Kiet & Moodie, Craig. "An application of nonlinear feature
extraction-A case study for low speed slewing bearing condition monitoring and
prognosis." *IEEE/ASME International Conference on Advanced Intelligent
Mechatronics: Mechatronics for Human Wellbeing, AIM 2013*.1713-1718.
10.1109/AIM.2013.6584344.

[4] Kantz, H., and Schreiber, T.
*Nonlinear Time Series Analysis*. Cambridge:
Cambridge University Press, 2003.