A Statistical Measure of Complexity

In this chapter, a statistical measure of complexity is introduced and some of its properties are discussed. Also, some straightforward applications are shown.

I present my viewpoint on complexity, stressing general arguments and using a rather simple language.

We discuss some aspects of the extension to continuous systems of a statistical measure of complexity introduced by Lopez-Ruiz, Mancini and Calbet (LMC) [Phys. Lett. A 209 (1995) 321]. In general, the extension of a magnitude from the discrete to the continuous case is not a trivial process and requires some choice. In the present study, several possibilities appear available. One of them is examined in detail. Some interesting properties desirable for any magnitude of comple...

Complexity is a multi-faceted phenomenon, involving a variety of features including disorder, nonlinearity, and self-organisation. We use a recently developed rigorous framework for complexity to understand measures of complexity. We illustrate, by example, how features of complexity can be quantified, and we analyse a selection of purported measures of complexity that have found wide application and explain whether and how they measure complexity. We also discuss some of the...

The concept of complexity appears in virtually all areas of knowledge. Its intuitive meaning shares similarities across fields, but disagreements between its details hinders a general definition, leading to a plethora of proposed measurements. While each might be appropriated to the problems it addresses, the lack of an underlying fundamental principle prevents the development of a unified theory. Here it is shown that the statistics of the amount of symmetry broken by system...

We review possible measures of complexity which might in particular be applicable to situations where the complexity seems to arise spontaneously. We point out that not all of them correspond to the intuitive (or "naive") notion, and that one should not expect a unique observable of complexity. One of the main problems is to distinguish complex from disordered systems. This and the fact that complexity is closely related to information requires that we also give a review of i...

Nearly 30 years ago, J.P. Crutchfield and K. Young proposed in Phys. Rev. Lett. {\bf 63}, 105 (1989) some supposedly novel measures of time series complexity, and their relations to existing concepts in nonlinear dynamical systems. At that time it seemed that the multiple faults of this paper would make it obsolete soon. Since this has not happened, and these faults still infest the literature on what is now called "computational mechanics", I want here to rectify the situati...

This article summarises a Web-book on "Complexity" that was developed to introduce undergraduate students to interesting complex systems in the biological, physical and social sciences, and the common tools, principles and concepts used for their study.

We extend previously proposed measures of complexity, emergence, and self-organization to continuous distributions using differential entropy. This allows us to calculate the complexity of phenomena for which distributions are known. We find that a broad range of common parameters found in Gaussian and scale-free distributions present high complexity values. We also explore the relationship between our measure of complexity and information adaptation.

At present, there is a great deal of confusion regarding complexity and its measures (reviews on complexity measures are found in, e.g. Lloyd, 2001 and Shalizi, 2006 and more references therein). Moreover, there is also confusion regarding the nature of life. In this situation, it seems the task of determining the fundamental complexity measures of life is especially difficult. Yet this task is just part of a greater task: obtaining substantial insights into the nature of bio...