title: Modifications of the Takens-Ellner algorithm for medium- and high-dimensional signals.
abstract: The paper presents the modifications of the classic Takens-Ellner (TE) algorithm that are necessary to estimate the dimensional complexity (d) of the medium- and high-dimensional signals. The main idea is the fitting of the fourth-degree polynomial to the relation d=fn(W) (where W is window width) in order to find the point of minimum slope in this relation. This point corresponds to the plateau area being the feature of the low-dimensional signals and representing the range of the optimal W used in the calculations. The point of minimum slope is represented by the local minimum in the first derivative function (third-degree polynomial). The exclusion of the attractor pairs of points lying approximately closer than the autocorrelation time removes the tendency to down-estimate d observed in the classic TE algorithm. The procedure of the choice of the embedding parameters was modified: The lag (L) was calculated for the given embedding dimension (m) and W was calculated by using the formula L=W/(m-1) in order to obtain the desired values of W for generating the precise relation d=fn(W) being the basis for the polynomial fitting. The cubic interpolation of the signal is proposed for the noninteger L's. Three signals possessing the dimensional complexities d(A)≈4, d(B)≈6, and d(C)≈8 and being the sums of two, three, and four Lorenz signals, respectively, were analyzed. Their lengths were 65,536 points. The applied algorithm gives a precise estimation of d for every signal A, B, and C. Also presented are the results of (a) the method for the estimation of the cubic interpolation error, (b) the analysis for signals contaminated with the white noise, and (c) the comparison between the original signals and the surrogates.
authors.affiliation: Department of Biophysics, Poznań University of Medical Sciences, ul. Fredry 10, Poznań, PL-61-701, Poland. kmichalak@ump.edu.pl
authors.forename: Krzysztof Piotr
authors.initials: KP
authors.lastname: Michalak
mesh_terms: D000465:Algorithms; D017711:Nonlinear Dynamics