Non-empty box counting algorithm for calculating fractal dimensions and its applications in EEG analysis

NON-EMPTY BOX COUNTING ALGORITHM FOR CALCULATING FRACTAL DIMENSIONS AND ITS APPLICATIONS IN EEG ANALYSIS Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai, PR China N. Xu, Shanghai Institute of Physiology, Academia Sinica, Shanghai, PR China |. H. Xu, Shanghai Institute of Biochemistry, Academia Sinica, Shanghai, PR China The non-empty box counting algorithm for calculating fractal (generalized) dimensions is suggested. The algorithm is based on the following assumption: although we cannot possess a data set large enough, so that all non-empty boxes will be visited, but the number of non-empty boxes N(8) for a limited data set will increase exponentially, with capacity D O as the exponent, i.e., Do N(E) ~ 8 and similarly for other Dq's. This assumption is justified with many numerical tests. For example, for the logistic map and Henon map, the error is about 5% for 1000 to 2000 points, and 1% for 10000 to 20000 points. With this algorithm we need only very limited storase places (no more than total number of the known data), and the computing time is also greatly reduced (one to two orders lower than that for calculating a correlation integral). Moreover, it is possible to get all required generalized dimensions of any order. This algorithm is therefore especially convenient for an experimental data set. The analysis of EEG data sets obtained in our experiments shows some regularities of the brain dynamics.

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