Extracting complexity waveforms from one-dimensional signals

BACKGROUND: Nonlinear methods provide a direct way of estimating complexity of one-dimensional sampled signals through calculation of Higuchi's fractal dimension (1<FD<2). In most cases the signal is treated as being characterized by one value of FD and consequently analyzed as one epoch or, if divided into more epochs, often only mean and standard deviation of epoch FD are calculated. If its complexity variation (or running fractal dimension), FD(t), is to be extracted, a moving window (epoch) approach is needed. However, due to low-pass filtering properties of moving windows, short epochs are preferred. Since Higuchi's method is based on consecutive reduction of signal sampling frequency, it is not suitable for estimating FD of very short epochs (N < 100 samples). RESULTS: In this work we propose a new and simple way to estimate FD for N < 100 by introducing 'normalized length density' of a signal epoch, [Image: see text] where y(n)(i) represents the ith signal sample after amplitude normalization. The actual calculation of signal FD is based on construction of a monotonic calibration curve, FD = f(NLD), on a set of Weierstrass functions, for which FD values are given theoretically. The two existing methods, Higuchi's and consecutive differences, applied simultaneously on signals with constant FD (white noise and Brownian motion), showed that standard deviation of calculated window FD (FD(w)) increased sharply as the epoch became shorter. However, in case of the new NLD method a considerably lower scattering was obtained, especially for N < 30, at the expense of some lower accuracy in calculating average FD(w). Consequently, more accurate reconstruction of FD waveforms was obtained when synthetic signals were analyzed, containig short alternating epochs of two or three different FD values. Additionally, scatter plots of FD(w )of an occipital human EEG signal for 10 sample epochs demontrated that Higuchi's estimations for some epochs exceeded the theoretical FD limits, while NLD-derived values did not. CONCLUSION: The presented approach was more accurate than the existing two methods in FD(t) extraction for very short epochs and could be used in physiological signals when FD is expected to change abruptly, such as short phasic phenomena or transient artefacts, as well as in other fields of science.