Extracting complexity waveforms from onedimensional signals
Abstract
Background
Nonlinear methods provide a direct way of estimating complexity of onedimensional 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 lowpass 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,
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 NLDderived 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.
Keywords
Fractal Dimension Amplitude Normalization Synthetic Signal Weierstrass Function Complexity WaveformBackground
where f_{ s }represents the sampling frequency, N_{ w }number of samples in the window.
According to equation (2), in order to extract accurately the waveform Q(t), i.e. avoid the attenuation of as many Fourier components as possible, it is desirable to have large values of f_{ c }, according to (2) small values of N_{ w }(short windows) for a given sampling frequency. Importance and limitations of estimating nonlinear properties of biological (particularly EEG) signals for short epochs has already been recognized (cf. [1, 2]).
Although these filtering properties were derived for oscillations (waveforms) of signal amplitudes, we showed in our previous work dealing with meteorological data (cf. [3, 4]) that they are also valid for oscillations of signal complexity, expressed quantitatively as "running fractal dimension" Q(t) = FD(t) (cf. [5]).
we were able to calculate parameters A(n_{ max }) and B(n_{ max }) for n_{ max }= 3,...,7. In this method, the need to choose a value for k_{ max }is eliminated. More, introducing n_{ max }instead of k_{ max }did not mean substituting one indeterminacy with another, since the smallest numerical error, on the used set of Weierstrass function, was obtained with n_{ max }= 3.
Results
Normalized length density
was calculated and presented as a thick red line on Fig. 3. As this and its inverse function FD = φ^{1}(NLD) = f(NLD) turned to be monotonous, the latter fulfilled the necessary condition to be used as a calibration curve in further calculations.

logarithmic model: FD = a log (NLD  NLD_{ o }) + C

power model: FD = a (NLD  NLD_{ o })^{ k }.
When extracting complexity waveforms from onedimensional signals, a problem arises which is not present when conventional FD measurements are applied (averaging of window FD values for the whole signal). Namely, since in every natural signal both amplitude and complexity simultaneously vary, it is essential to eliminate, as much as possible, influence of amplitude variations on complexity measurements. Fortunately, Higuchi's method is invariant to amplitude variations. However, this is not the case when NLD analysis is performed. Two procedures are possible for signal amplitude normalization: a) to normalize signal amplitudes for the whole signal, by applying formulas (9) and (10) before the NLD procedure. Such a procedure is in accordance with the way the calibration curves, presented on Fig. 3 and Fig. 4B were obtained after amplitude normalization of the Weierstrass functions. However, this version (integral normalization, IN) is not entirely immune on signal amplitude variations that necessarily occur while the window of analysis moves along the signal. b) According to the other version of the method, amplitude normalization is to be performed on every part of the analyzed signal selected by the moving window (window normalization, WN). Each of these two versions of the NLD method showed more or less accurate results in extracting complexity waveforms for very short signal epochs, depending on the measurement conditions. Therefore, the problem of current signal amplitude variations and elimination of their influence on FD(t) is still to be elucidated in future studies.
Dependence of accuracy of FDcalculation on epoch length
Extraction of complexity waveforms from synthetic signals
The complexity waveforms of these synthetic signals were analyzed with both power model (WN, IN) versions of the NLD method (Fig. 7A, C and Higuchi's method (Fig. 7B, D), all methods using 5 sample moving epochs, step 2 samples. When compared with the ideal output (series of rectangular impulses/stairs shown in red on all four panels), greater accuracy of the IN version of the new method (WN not shown), presented on panels A and C of Fig. 7, is obvious. Expressed quantitatively, square error per sample for the first waveform (Fig. 7A, B) was 0.0466 in case of NLD, while 0.5998 for the Higuchi's method (≈12.9 times higher). For the second waveform (Fig. 7C, D), the corresponding figures were 0.0463 and 0.3902 (≈8.4 times higher).
Final adjustment of the calibration curve based on the analysis of natural (EEG) signals
Fig. 8A presents part of this signal in time domain (4s), while total signal duration was 60s. It was analyzed simultaneously by Higuchi's and the NLD method (power model, WN) with moving nonoverlapping short epochs of 10 samples. Resulting values of FD_{ w }are given in Fig. 8B in form of a scatter plot. One can observe that points obtained with Higuchi's method are "spilling over" the allowed limits (1<FD<2), while in case of the NLD analysis they remain within the marked boundaries. Values calculated with NLD are, however, overestimated in the low FD region. We tried to correct this systematic error by constructing a more accurate calibration curve than the one presented on Fig. 4, which was derived only from the Weierstrass functions.
The method described in this work could find its applicability in those situations where signal complexity changes occur and are of special interest for the researchers. There are two possible cases of such changes:
a) short or intermittent disturbances of the existing signal complexity: external artefacts or internal phasic phenomena, e.g. those occurring in NREM (Kcomplexes, delta bursts) and in REM sleep (muscle twitches, central and peripheral phasic events such as PGO waves, bursts of autonomic nerves, surges of blood pressure, etc.), as well as microarousal (cf. [12]);
b) slowly evolving FD variations, caused by physiological processes themselves.
In the first case, it is to be expected from the present approach to detect such occurrences, while slower FD variations should be extracted as a function of time (complexity waveforms).
In both cases, it is to be expected that extracted signal complexity changes should be more accurately measured (i.e. less "noisy") if performed by the NLD method than by the existing ones. As well, the new method could be tested in other areas where changes of signal complexity occur, such as engineering, geology, meteorology, astronomy (cf. [13]).
Conclusion
The new NLD method is more accurate than Higuchi's or consecutive differences for extracting signal complexity waveforms (FD(t), current changes of signal FD), when using short signal epochs (<30 samples). This result follows the fact that standard deviation of its moving window FD values (FD_{ w }) is considerably smaller than those obtained by the two other methods. However, this improvement is achieved at the expense of lower accuracy in measuring mean FD_{ w }. Initially detected overestimation of the FD_{ w }values, calculated by NLD in the low FD region, was due to an imperfect calibration curve FD = f(NLD), derived from a set of Weierstrasss functions. The bias was corrected by a procedure of modification of power model parameters, based on the analysis of natural (EEG) signals. The present approach might be used whenever "running" signal FD changes are of interest, e.g. in physiological signal analysis for automatic detection of short phasic phenomena or transient artefacts, as well as in other areas where changes of signal complexity occurs, such as engineering, geology, meteorology, astronomy (cf. [12]).
Notes
Acknowledgements
This work was supported by projects OI 145062 and OI 143027 of Ministry of Science and Technological Development of the Republic of Serbia and the project "Homeostatic mechanisms and the regulation system of behavior" of the Serbian Academy of Sciences and Arts. We thank Dr Aleksandra Vuckovic for providing us with the EEG signals analyzed in this work.
Supplementary material
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