Fractal dimension of time series estimated by the surface division method

In: Wiadomości statystyczne / Glówny Urza̜d Statystyczny, Polskie Towarzystwo Statystyczne: czasopismo Głównego Urze̜du Statystycznego i Polskiego Towarzystwa = ˜Theœ Polish statistician, Band 64, Heft 9, S. 7-24

Abstract

One of the most important issues to be settled in the analysis of time series is determining their variability andidentifying the process of shaping their values. In the classical approach, volatility is most often identified with the variance of growth rates.However, risk can be characterisednot only by the variability, but also by the predictability of the changes which can be evaluatedusing thefractal dimension. The aim of this paper is to presentthe applicability of the fractal dimension estimated by the surface division method tothe assessment ofthe properties of time series. The paper presents a method for determining the fractal dimension, its interpretation, significance tables and an example of its application. Fractal dimension has been used here to describe the properties of the time series of the WIG stockexchange index in 2014–2018 and the time series of the growth rates of the largest listed Polish companiesin 2015–2018. The applied methodmakesit possible toclassify a time series into one of three classesof series: persistent, random or antipersistent. Specific cases showthe differences between the use of standard deviation and fractal dimension for riskassessment. Fractal dimension appears here to be a method for assessing the degree of stability of variations.