How to approach measuring of signal complexity on very short epochs? One of the ways could be to count the local extrema, where the signal complexity is expected to be directly proportional to their number. However, complexity is not equal if there are e.g. 3 local extrema on 5 or 50 samples. Therefore, if N_le denotes the number of local extrema and N number of samples, one may introduce 'local extrema density', L_ed, as the measure of complexity:

But this quantity has a substantial drawback: because both N_le and N are integers, L_ed tends to group such short signal epochs into 'quantized' classes. For example, if N = 4, the method is able to classify all epochs into only three classes of complexity, having L_ed = 0.00; 0.25; 0.50 (values of 0.75 and 1.00 should be excluded since we do not know whether edge samples are local extrema or not). Obviously, a continuous measure of signal complexity would be more suitable. We propose that such a quantity could be, 'normalized length density' (NLD). In fact, it is the signal length divided by the number of samples, and normalized for average signal amplitude and it is proportional (Fig. 2A) to the local extrema density:

where y_n(i) represents ith signal sample after amplitude normalization:

where

How to relate NLD with FD? For this purpose we used the same set of Weierstrass functions as in 8:

(γ > 1, 0<H<1), where FD = 2-H is given theoretically. A family of these functions was generated, with parameter values being γ = 1.1,1.2,...,5.0 (N_γ = 40), H = 0.99,0.98,...,0.01 (N_H = 99). The latter values resulted in an even distribution of FD values across their possible numerical range (1 <FD < 2). The whole set, therefore, numbered N_γN_H = 3960 functions. Sampling frequency and signal duration were chosen to resemble a biological signal: f_s = 256 samp/s, T = 30s, making the total number of samples within each Weierstrass function to be N = 7680. For each set of 99 Weierstrass functions, having a fixed value of γ, NLD_γ = φ_γ(FD) functions were calculated after amplitude normalization according to formulas (9) and (10) and presented as 40 thin black lines on Fig. 3. Since real biological signals are not characterized by any intrinsic parameter such as γ, by averaging all 40 NLD_γ values for a fixed FD, inverse of the final calibration curve

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.

For computational purposes, a mathematical model of this relation must be established. In this work we tested two models:

- logarithmic model: FD = a log (NLD - NLD_o) + C

- power model: FD = a (NLD - NLD_o)^k.

Nonlinear fitting was performed on the obtained points FD = f(NLD) (circles, Fig. 4) for both mathematical models. As can be seen, the power model showed better results (smaller square fitting error per point). A better matching of the power model to the experimental points could also be seen visually, especially for higher values of NLD and FD.

When extracting complexity waveforms from one-dimensional 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.

All natural signals are of variable complexity. Accuracy of any method for measuring FD could be estimated by scattering (standard deviation) of epoch (window) FD values, FD_w, if signals with constant FD are being analyzed (such as white noise, FD = 2, and fractal Brownian motion, FD = 1.5). Standard deviation of all measured FD_w increases with shortening of windows, causing numerical errors. We compared this scattering obtained by Higuchi's, consecutive differences and NLD method (power model version, WN) on these two types of noise (Fig. 5). Integral normalization was also performed, but since the results for mean(FD_w) obtained with IN version were less accurate than with WN, they are not presented. However, std(FD_w) obtained with IN version were also considerably smaller than those with Higuchi's or the CD method. As well, because more accurate results were obtained with power than with logarithmic model (lower fitting error on Fig. 4B), these and all following measurements were not performed with the logarithmic model. As expected, when the epoch shortened, std(FD_w) increased for all methods (Fig. 5B and 5D). The new NLD method showed a significantly lower FD_w scattering than Higuchi's or CD, especially for short epochs, reducing the numerical error. However, this improvement was obtained at the expense of an increased difference between mean FD_w and the theoretical FD value (Fig. 5, upper panels, A and C). Fortunately, when extracting complexity waveforms from one-dimensional signals, it is more important to have lower scattering of window FD values than the preciseness of their mean value. The latter contributes merely to the "DC" level of the extracted complexity waveform, while the waveform quality (signal-to-noise ratio i.e. window-to-window fluctuations) depends heavily on the standard deviation of the FD_w values. This statement will be illustrated on several examples.

In order to compare the new method with Higuchi's when dynamical change of signal FD occurs, two synthetic signals were constructed from different Weierstrass functions: one consisted of a series of 50 sample epochs, having two alternating FD values: 1.2 and 1.8 (Fig. 6A), while the other was formed by composing three FD values: 1.1, 1.5 and 1.9 (Fig. 6B). Both signals had γ = 3.4 and were 1000 samples long. This particular value for parameter γ was chosen because its corresponding calibration curve (one of the black lines on Fig. 3) was positioned closest to the average calibration curve (red line on Fig. 3), minimizing the induced systematic error (optimal "DC" level of FD(t) on Fig. 7A, C). Prior to further processing, each 50 sample epoch underwent amplitude normalization according to expressions (9) and (10), in order to eliminate influence of amplitude differences of componential Weierstrass functions on FD(t) extraction.

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).

The new method and the corresponding calibration curve were constructed by analyzing the formerly described set of Weierstrass functions. However, its applicability on real biomedical signals could only be tested if applied on such natural signals and the obtained results observed critically. We present on Fig. 8 how lower scattering of short window FD_wvalues, obtained with NLD method (compared to Higuchi's), looks on real biomedical signals. In this case a human occipital O1 derivation of the EEG activity was analyzed from a healthy adult in the relaxed awake state with closed eyes.

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 non-overlapping 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.

First, we explored how the initial calibration curve is modified when each of the power model parameters (a, NLD₀ and k) are being varied. In fact, according to the direction and location of the bias shown on Fig 8B, one needs to modify the calibration curve in such a way that FD values are decreased in the low FD region (left part of the plot), while they stay unchanged in the high FD region. Influence of parameters a and k on the calibration curve are presented on Fig. 9 (parameter NLD₀ was not varied, since it simply shifts the calibration curve horizontally). One can note that increase of k (dashed line) decreases FD in the low, while decrease of a (dotted line) decreases FD in the high FD region. Therefore, by increasing parameter k, the calibration curve is being modified as required – FD values decrease in the low, while they remain mostly unchanged in the high FD region.

Further, the optimal new value for k could be determined by analyzing a sufficiently large ensemble of biomedical, e.g. EEG signals (so that short epoch FD_w values populate more or less evenly all regions of the scatter plot on Fig. 8B) and by observing that all FD_w values fall between the required limits (1<FD<2). We analyzed 140 EEG signals, each 60s long, from ten adult wake healthy subjects. The electrodes were positioned at 14 locations (F7, F8, T3, T4, T5, T6, F3, F4, C3, C4, P3, P4, O1 and O2) according to the International 10–20 System with an average reference. Signals were sampled at a rate of 256 samples/s, band pass filtered between 0.5 and 70 Hz and artifacts were removed manually based on a visual inspection. Other details about data collection and preparation can be found in 11. For each signal, a series of 6 analyses (NLD method, epoch 10 samples) were performed, where parameter k was varied (0.2 – 0.45, step 0.05). For each of the 140 EEG signals, the minimal FD_w was plotted against the varied value of k and all 140 crossings with min(FD_w) = 1 were determined (Fig. 10). Optimal value for k was then calculated as the average of these crossing values: k_opt = 0.3523 (± 0.0122). Analogously, as parameter a influences the calibration curve in the high FD region, maximal FD_w values were also calculated and 140 crossings with max(FD_w) = 2 were determined. The obtained optimal value for a was a_opt = 1.8399 (± 0.0980). Finally, using these new parameters, as an example, we analyzed again the signal from Fig. 8. The corresponding new scatter plot, shown on Fig. 11, confirmed that these corrections (red points) eliminated the formerly detected bias in the low FD region.

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 (K-complexes, 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).