A Robust Algorithm for Optimisation and Customisation of Fractal Dimensions of Time Series Modified by Nonlinearly Scaling Their Time Derivatives: Mathematical Theory and Practical Applications

Abstract

Standard methods for computing the fractal dimensions of time series are usually tested with continuous nowhere differentiable functions, but not benchmarked with actual signals. Therefore they can produce opposite results in extreme signals. These methods also use different scaling methods, that is, different amplitude multipliers, which makes it difficult to compare fractal dimensions obtained from different methods. The purpose of this research was to develop an optimisation method that computes the fractal dimension of a normalised (dimensionless) and modified time series signal with a robust algorithm and a running average method, and that maximises the difference between two fractal dimensions, for example, a minimum and a maximum one. The signal is modified by transforming its amplitude by a multiplier, which has a non-linear effect on the signal's time derivative. The optimisation method identifies the optimal multiplier of the normalised amplitude for targeted decision making based on fractal dimensions. The optimisation method provides an additional filter effect and makes the fractal dimensions less noisy. The method is exemplified by, and explained with, different signals, such as human movement, EEG, and acoustic signals.

A Method for Quantifying the Emotional Intensity and Duration of a Startle Reaction with Customized Fractal Dimensions of EEG Signals

“…that the fractal dimension derived from the actual raw EEG signal with Fuss' method [9] was identical to the one obtained with Higuchi's method [3]. As there were two options for the multiplier m, i.e.…”

“…Fuss' method or Modified Amplitude Fractal Dimension Method (MAFDM [9]) was able to detect the emotional reaction after being startled by a scary image (Figure 9). The image was shown at 20 s into the video for 0.5 s, and less than 0.5 s later, the fractal dimensions started to increase.…”

“…The raw EEG signal was processed with Fuss' method [9] by maximizing the difference between regions of high and low fractal dimensions. This was achieved by identifying potential regions with high fractal dimensions from the running average filter, and expected regions with low fractal dimensions from 0 to 20 s (before the scary image), and after the high dimension regions.…”

“…The entire 60 s signal was processed with a window width of 1 s (255 data), and the average fractal dimension was calculated for high and low dimension regions as well as of the entire signal. For this purpose, and according to Fuss' method [9], the amplitude of the signal was modified with a multiplier m and maximum, average and minimum fractal dimensions, FD max , FD avg , FD min , were plotted against the decadic logarithm of the multiplier m. In order to find the optimal multiplier m, the following differentials and ratio were calculated:…”

“…The aim of this research is to trigger a sudden emotion with a startle reaction by using a video method, and by applying a new fractal dimension processing method [9] for assessment of EEG signals.…”