A Statistical Measure of Complexity

The definition of complexity through Statistical Complexity Measures (SCM) has recently seen major improvements. Mostly, effort is concentrated in measures on time series. We propose a SCM definition for spatial dynamical systems. Our definition is in line with the trend to combine entropy with measures of structure (such as disequilibrium). We study the behaviour of our definition against the vectorial noise model of Collective Motion. From a global perspective, we show how ...

A general notion of information-related complexity applicable to both natural and man-made systems is proposed. The overall approach is to explicitly consider a rational agent performing a certain task with a quantifiable degree of success. The complexity is defined as the minimum (quasi-)quantity of information that's necessary to complete the task to the given extent -- measured by the corresponding loss. The complexity so defined is shown to generalize the existing notion ...

Information Theory provides a fundamental basis for analysis, and for a variety of subsequent methodological approaches, in relation to uncertainty quantification. The transversal character of concepts and derived results justifies its omnipresence in scientific research, in almost every area of knowledge, particularly in Physics, Communications, Geosciences, Life Sciences, etc. Information-theoretic aspects underlie modern developments on complexity and risk. A proper use an...

In this paper, we present some results on information, complexity and entropy as defined below and we discuss their relations with the Kolmogorov-Sinai entropy which is the most important invariant of a dynamical system. These results have the following features and motivations: -we give a new computable definition of information and complexity which allows to give a computable characterization of the K-S entropy; -these definitions make sense even for a single orbit and ...

Concepts used in the scientific study of complex systems have become so widespread that their use and abuse has led to ambiguity and confusion in their meaning. In this paper we use information theory to provide abstract and concise measures of complexity, emergence, self-organization, and homeostasis. The purpose is to clarify the meaning of these concepts with the aid of the proposed formal measures. In a simplified version of the measures (focusing on the information produ...

In the 21st century, many of the crucial scientific and technical issues facing humanity can be understood as problems associated with understanding, modelling, and ultimately controlling complex systems: systems comprised of a large number of non-trivially interacting components whose collective behaviour can be difficult to predict. Information theory, a branch of mathematics historically associated with questions about encoding and decoding messages, has emerged as somethi...

The science of complexity is far from being fully understood and even its foundations are not well established. On the other hand, during the last decade, the random motion of particles or waves - the so-called diffusion - has been known better. In this paper, we discuss how simple ideas of diffusion can be used to deal with the description of most complex structure.

The LMC complexity, an indicator of complexity based on a probabilistic description, is revisited. A straightforward approach allows us to establish the time evolution of this indicator in a near-equilibrium situation and gives us a new insight for interpreting the LMC complexity for a general non equilibrium system. Its relationship with the Renyi entropies is also explained. One of the advantages of this indicator is that its calculation does not require a considerable comp...

In this chapter, a statistical measure of complexity and the Fisher-Shannon information product are introduced and their properties are discussed. These measures are based on the interplay between the Shannon information, or a function of it, and the separation of the set of accessible states to a system from the equiprobability distribution, i.e. the disequilibrium or the Fisher information, respectively. Different applications in discrete and continuous systems are shown. S...

Disorder in point patterns can be quantified by means of the complexity, rather than in terms of geometric attributes of pattern structure. A complexity-based disorder-quantifying statistic indicates the practical difficulties associated with modeling processes that produce jammed patterns, particularly with the assessment of model fit and with the simulation of high-intensity hard-core patterns.