Characterizing attractors using local intrinsic dimensions calculated by singular-value decomposition and information-theoretic criteria.

Abstract

An algorithm to estimate the average local intrinsic dimension (〈${\mathit{d}}_{\mathrm{LID}}$〉) of an attractor using signal versus noise separation methods based on information-theoretic criteria is explored in this work. Using noisy sample data the 〈${\mathit{d}}_{\mathrm{LID}}$〉 is computed from an eigenanalysis of local attractor regions, indicating the local orthogonal directions along which the data are clustered. The 〈${\mathit{d}}_{\mathrm{LID}}$〉 algorithm requires the separation of signal eigenvalues, i.e., the dominant eigenvalues, from the noise eigenvalues for which thresholding mechanisms based on two information-theoretic criteria are used. Singular-value decomposition is used to calculate the eigenvalues as well as to determine the rank of the local-phase-space data matrix. The two specific information-theoretic criteria which we consider to separate signal and noise are the Akaike information criterion (AIC) [H. Akaike, IEEE Trans. Auto. Control 19, 716 (1974)] and the minimum description length (MDL) of Rissanen [Automatica 14, 465 (1978)] and Schwarz [Ann. Stat. 6, 461 (1978)]. Several test cases are presented demonstrating the use of the AIC and MDL to calculate 〈${\mathit{d}}_{\mathrm{LID}}$〉. Results are then compared to the correlation dimension as computed by the Grassberger-Procaccia method [Physica D9, 189 (1983)]. The AIC and MDL separation techniques are found to be inappropriate at higher signal-to-noise ratios (SNR), above \ensuremath{\sim}12 dB, and the MDL produces results in good overall agreement with the accepted correlation dimension in the range of SNR between 5 and 12 dB. Two of the primary benefits of the 〈${\mathit{d}}_{\mathrm{LID}}$〉 technique are that it will yield reasonable and fairly accurate estimates of the dimension of attractors and that it can probe small length scales on the attractor even for moderate SNR. Other thresholding schemes are being pursued for SNR's above 12 dB.