Complementarity of information obtained by Kolmogorov and Aksentijevic–Gibson complexities in the analysis of binary time series

Abstract 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 description/algorithm needed to describe a string. Since this measure is non-computable it is calculated approximately 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 defines 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 different 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 (streamflow), 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.