DEVELOPMENT OF STATISTICAL ENTROPIC AND INFORMATIONAL METHODS FOR THE INVESTIGATION OF COMPLEX DYNAMICAL SYSTEMS

Project leaders

Luciano Telesca, Alejandro Ramirez-rojas

Agreement

MESSICO - CONACYT-expired - Consejo Nacional de Ciencia y Tecnologia

Call

CNR/CONACYT 2012-2014

Department

ICT

Thematic area

Engineering, ICT and technologies for energy and transportation

Status of the project

New

Research proposal

Complex dynamical systems are a topic of great contemporary research interest. Complexity measures are considered as general indicators of pattern, structure, and correlation in systems or processes. Several mathematical notions have been proposed for quantifying the concepts of complexity, including the algorithmic information theory of Kolmogorov and Chaitin (Kolmogorov, 1965; Chaitin, 1966), the classical information theory of Shannon and Weaver (1949),  Fisher information (Frieden, 2004), the logical (Bennet, 1988) and the thermodynamical (Lloyd and Pagels, 1988) depths, the statistical measure of complexity defined by López-Ruiz-Mancini-Calbet (LMC) (López-Ruiz et al., 1995; Catalán et al., 2002), the simple two-parameter disorder-order derived measure of complexity according to Shiner-Davison-Landsberg (SDL) (Shiner et al., 1999), among others. Some of them share rigorous connections with others as well as with Bayes and information theory. Although there is no general agreement about the definition of complexity, its quantitative characterization has been an important subject of research and it has received considerable attention, becoming a challenging task, being used in several fields, from dynamical systems to disordered systems, from spatial patterns to language, from multielectronic systems to cellular automata, neuronal networks, self-organization, DNA analyses, social sciences. Therefore, the characterization of complexity is not unique and the utility of each definition depends on the type of system or process, the level of the description, and the scale of the interactions among elementary particles.
Fundamental concepts such as uncertainty or randomness are frequently employed in the definitions of complexity, although some other concepts such as clustering, order, localization or organization might be also important for characterizing the complexity of systems or processes. Besides the well-known Fisher Information Measure (Frieden, 2004) and the Shannon Entropy (de Araujo, 2003) for describing the complex interactions within dynamical systems, it has been recently becoming a challenging task the development of statistical methods based on the Tsallis' Entropy (Tsallis et al., 1998)  and the Natural Time Domain (NTD) Entropy (Varotsos et al., 2007). The Tsallis' Entropy generalizes the Boltzmann-Gibbs entropy and describes the so-called non-extensive dynamical systems, in which all-length scale interactions are allowed. For instance, the shock fragmentation, in which high energies are involved, is well described by concept that leads to the emergence of long-range interactions between all parts of the object being fragmented. Among the several applications, the leader of the Italian team has applied Tsallis' statistics to investigate the complex mechanism of relative displacement of fault plates, which is the main cause of earthquakes (Telesca, 2010; Telesca, 2011). The NTD method allows following the evolution of a complex dynamical system and identifying when it enters into a critical stage (Abe et al., 2005). For instance, the effectiveness of the NTD method in distinguishing seismo-electric signals from artificial noises (dichotomous electrical disturbances recorded at a  measuring site due to the nearby man-made electric sources)  was shown by Varotsos et al. (2009).
In this context,  the present project intends to produce an advancement in the development and application of statistical methods based on the notion of entropy and information theory in order to better explore and characterize the dynamics of complex systems and processes and to obtain a well-assessed notion of complexity. The potential of Fisher Information Measure, the Shannon entropy, the Tsallis' Entropy and the NTD Entropy method will be explored and, in particular, applied to geophysical systems. Both the teams, in fact, manage a large amount of geophysical data that will be used to develop and test the methods.  It will be also explored the theoretical link between the such entropic and informational  formalism and the fractal/multifractal formalism, the last representing a well-know methodological approach in investigating the complexity of dynamical systems. The use of such methods will lead to the definition and estimation of parameters, which are important for characterizing the dynamics of complex real systems.
The cooperation between the involved Italian and Mexican institutions offers a real synergistic advantage of exchange of statistical/mathematical know-how. Furthermore, the project will constitute the base of a solid scientific relationship that may be maintained with the joint participation in future projects. Each involved institution has a valuable level of expertise in all the aspects of the  project, which will constitute a platform to share and combine their knowledge. Both the Italian and the Mexican teams have an on-going investigation in the field of Tsallis' Entropy and NTD method, along with an advanced speciality in statistical analysis and methodological  approaches, also focused on the Fisher Information Measure, Shannon Entropy and fractal/multifractal formalism.

Research goals

The specific objectives of the project are: 1) to develop and apply entropy-based and information-based methods to time series of geophysical processes to describe and characterize their complex dynamics;  2) to organize testing procedures and define confidence bands for the entropy-based and information-based estimated parameters by using simulated series generated by means of different surrogate methods (random shuffling, Adjusted Amplitude Fourier Transform (AAFT) method); 3) to understand the link between the entropic formalism and the fractal/multifractal formalism in order to deepen the theoretical framework for the further development of entropy-based statistical methods; 4) to disseminate the results of the projects in international peer-reviewed journals, in international conference and in a project-oriented website.

Last update: 07/06/2025