Chaos, Solitons and Fractals 130 (2020) 109394 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos Complementarity of information obtained by Kolmogorov and Aksentijevic–Gibson complexities in the analysis of binary time series A. Aksentijevic a,b , D.T. Mihailovi ´ c c,∗ , D. Kapor d , S. Crvenkovi ´ c d , E. Nikolic-Djori ´ c c , A. Mihailovi ´ c e a Department of Psychology, University of Roehampton, United Kingdom b Department of Psychological Sciences, Birkbeck, University of London, United Kingdom c Faculty of Agriculture, University of Novi Sad, Dositeja Obradovi´ ca Sq. 8, 21009 Novi Sad, Serbia d Faculty of Sciences and Mathematics, University of Novi Sad, Serbia e ACIMSI Centre for Meteorology and Environmental Modelling, University of Novi Sad, Serbia a r t i c l e i n f o Article history: Received 5 April 2019 Revised 29 June 2019 Accepted 16 August 2019 Keywords: Complexity Kolmogorov complexity Aksentijevic–Gibson complexity Lempel–Ziv algorithm Entropy Time series a b s t r a c t Understanding and measuring complexity is one of the emerging fields in physics and science more generally. The original impetus was given by Shannon’s information theory which quantifies disorder and uncertainty by means of relative probabilities of different outcomes and arrangements of symbols. Next came Kolmogorov complexity (KC) which defines complexity as the length of the shortest descrip- tion/algorithm needed to describe a string. Since this measure is non-computable it is calculated approx- imately by means of the Lempel–Ziv algorithm (LZA). KC has been used widely in different branches of physics and other sciences to provide overall estimates of the randomness of data structures, especially time series. Here, we consider the information measure Aksentijevic–Gibson complexity (AG), which de- fines complexity as amount of change at all levels of a pattern, and compare its performance with KC. We argue that KC and AG in their current implementations are complementary in that they focus on dif- ferent aspects of complexity—with the former providing efficient omnibus complexity estimates for long time series in different sciences and the latter precisely indexing data structure and locating regions of complexity change. The complementarity of these two measures was demonstrated on one deterministic (logistic equation), and four measured time series: physical (Rn-222 concentration), hydrological (stream- flow), meteorological (atmospheric noise) and economic (yield rate) time series, which in further text will be denoted as logistic, Radon, Rio Brazos, random and Imlek, respectively. In addition, we examine spatial transformations of a famous painting in order to demonstrate the sensitivity of AG complexity to spatial information. Finally, we discuss possible applications of the measure in different areas of science. © 2019 Elsevier Ltd. All rights reserved. 1. Introduction The computer has in turn changed the very nature of mathematical experience, suggesting for the first time that mathematics, like physics, may yet become an empirical discipline, a place where things are dis- covered because they are seen [1]. The spiritus movens of physics is curiosity about the universe underpinned by analytical power, which extends to problems tra- ditionally associated with other sciences and domains of enquiry. One such problem is the study of complex systems [2]. Accord- ing to Parisi [3], the study of complex systems represents the most recent revolution in physics. The key concept underpinning the study of complex systems is that of complexity. While complex- ∗ Corresponding author. E-mail address: guto@polj.uns.ac.rs (D.T. Mihailovi ´ c). ity by its very nature evades easy definition [4], one of the main problems is its presence in domains concerned with phenomena of vastly different properties and scales—from thermodynamics [5] to biology [6], economics [7] and psychology [8–10]. Another stum- bling block could be the reluctance to consider the possibility that complexity is ultimately subjective [11]. Its referent is the upper boundary of the knowable and this ultimately reflects the percep- tual and cognitive limitations of the observer. In addition, com- plexity implies structure and patterning—relational concepts that transcend/evade traditional quantitative approaches (e.g. informa- tion entropy [12]). Perhaps the main point of contention is the disagreement between those who equate complexity with disorder [13] and others who view complexity as a special domain lying between order and disorder, populated by structured yet difficult- to-analyse patterns [14]. Complexity measures in science can be divided into two broad categories—those that distinguish between complexity and disorder and others that do not [15]. https://doi.org/10.1016/j.chaos.2019.109394 0960-0779/© 2019 Elsevier Ltd. All rights reserved.