Consecutive Differences as a Method of Signal Fractal Analysis

Abstract

We propose a new method for calculating fractal dimension (DF) of a signal y(t), based on coefficients m (n) y , mean absolute values of its nth order derivatives (consecutive finite differences for sampled signals). We found that logarithms of m (n) y , n = 2, 3, . . . , n max , exhibited linear dependence on n:with stable slopes and Y -intercepts proportional to signal DF values. Using a family of Weierstrass functions, we established a link between Y -intercepts and signal fractal dimension:and calculated parameters A(n max ) and B(n max ) for n max = 3, . . . , 7. Compared to Higuchi's algorithm, advantages of this method include greater speed and eliminating the need to choose value for k max , since the smallest error was obtained with n max = 3. 283 Fractals 2005.13:283-292. Downloaded from www.worldscientific.com by EMORY UNIVERSITY on 07/30/15. For personal use only.

Spectral and Fractal Analysis of Cerebellar Activity After Single and Repeated Brain Injury

“…One of the applications of the chaos theory in brain activity analysis is fractal dimension estimation in the time domain as a measure of signal complexity. Several methods have been developed for measuring fractal dimension of EEG time series (Accardo et al, 1997;Esteller et al, 1999;Klonowski et al, 2003Klonowski et al, , 2006Kalauzi et al, 2005) and in heart rate variability (Hernandez Caceres et al, 2004).…”

“…Fractal analysis was performed also by estimating fractal dimension values (CD) of cerebellar cortical signals before and after brain injury employing our recently published method of consecutive differences (Kalauzi et al, 2005). It is a method of calculating fractal dimension of a sampled signal y(t) based on coefficients m (n) y -the mean absolute values of consecutive finite differences nth order for sampled signals.…”

Multifractal analysis of meteorological time series to assess climate impacts

“…Therefore, more subtle methods are being developed and applied in order to project changes in meteorological para meters in climatic zones with less extreme climate change dynamics. These include fractal analysis and chaotic evolution analysis of the atmospheric system (Kantelhardt et al 2002, Higuchi 1988, Kalauzi et al 2005.…”

“…Recently, the combined fractal and chaos methods have been incorporated for processing of geophysical, climatic and meteorological time series (Sivakumar 2000, Kalauzi et al 2005, Gutiérrez et al 2006, Chaudhuri 2006). This new approach delivered information about the impact of climate variability on ecological systems.…”

“…on a daily basis). Kalauzi et al (2005) elaborated the consecutive differences method, which was used to analyse monthly meteo ro logical records of rainfall, evaporation, relative humid ity, minimum temperature, sunshine duration and evaporation/precipitation ratio from the Amazonian area in Ecuador and Verano, Italy (Kalauzi et al 2009). Applying fast Fourier transform (FFT) to fractal dimension courses showed that fractal rhythms followed almost the same oscillation patterns as those corresponding to meteorological dynamics.…”