Novel measures based on the Kolmogorov complexity for use in complex system behavior studies and time series analysis

Abstract

Abstract:We propose novel metrics based on the Kolmogorov complexity for use in complex system behavior studies and time series analysis. We consider the origins of the Kolmogorov complexity and discuss its physical meaning. To get better insights into the nature of complex systems and time series analysis we introduce three novel measures based on the Kolmogorov complexity: (i) the Kolmogorov complexity spectrum, (ii) the Kolmogorov complexity spectrum highest value and (iii) the overall Kolmogorov complexity. The characteristics of these measures have been tested using a generalized logistic equation. Finally, the proposed measures have been applied to di erent time series originating from: a model output (the biochemical substance exchange in a multi-cell system), four di erent geophysical phenomena (dynamics of: river ow, long term precipitation, indoor Rn concentration and UV radiation dose) and the economy (stock price dynamics). The results obtained o er deeper insights into the complexity of system dynamics and time series analysis with the proposed complexity measures.

Randomness Representation of Turbulence in Canopy Flows Using Kolmogorov Complexity Measures

“…Thus, in the procedure of establishing a threshold for a criterion for coding, some information about the structure of the time series can be lost. In time series analysis of the canopy flow, we use two information measures based on the KC complexity: (i) the Kolmogorov complexity spectrum and (ii) the Kolmogorov complexity spectrum's highest value, which are introduced in [39] and briefly described. According to Definition 1 in [39], the time series {x i }, i = 1, 2, ..., N consists of either measured or calculated values.…”

“…In time series analysis of the canopy flow, we use two information measures based on the KC complexity: (i) the Kolmogorov complexity spectrum and (ii) the Kolmogorov complexity spectrum's highest value, which are introduced in [39] and briefly described. According to Definition 1 in [39], the time series {x i }, i = 1, 2, ..., N consists of either measured or calculated values. The Kolmogorov complexity spectrum of time series {x i }, i.e., the sequence {c i }, i = 1, 2, ..., N is obtained by the LZA algorithm applied N times on time series, where thresholds {x t,i } are all elements in {x i }.…”

“…Namely, the term complexity in physical systems has the connotation of an explicit measure of the probability of the state of the system. It is a mathematical measure that should not be equalized with entropy in statistical mechanics [39]. Thus, Shannon's entropy refers to dissimilarities between amplitudes in a time series, while the Kolmogorov complexity refers to the apparent sequence disorder of amplitudes in a time series.…”

Kolmogorov Complexity Based Information Measures Applied to the Analysis of Different River Flow Regimes

“…For example, in the procedure of establishing a threshold for a criterion for coding, some information about the structure of the time series can be lost. To get better insight into the nature of complex systems and time series analysis of the river flow we use three novel information measures based on the Kolmogorov complexity: (i) the Kolmogorov complexity spectrum, (ii) the Kolmogorov complexity spectrum highest value and (iii) the overall Kolmogorov complexity, which are introduced in [16]. All of them we will shortly describe in the rest of this subsection.…”

“…However, the KC complexity as an information measure cannot make distinction between time series with different amplitude variations and similar random components. Therefore, to get better insight into the nature of river flow time series analysis we introduce three novel information measures based on the Kolmogorov complexity: the Kolmogorov complexity spectrum, the KCM and KCO [16]. The purpose of this paper is to quantify the randomness of the river flow dynamics of seven rivers, in Bosnia and Herzegovina for the period 1965-1986, using the KCL, KCU, KCM and KCO information measures and the Kolmogorov complexity spectrum.…”