A comparative classification of complexity measures

Wackerbauer, Renate; Witt, Annette; Atmanspacher, Harald; Kurths, Jürgen; Scheingraber, H.

doi:10.1016/0960-0779(94)90023-x

Cited by 235 publications

(132 citation statements)

References 28 publications

Supporting

Mentioning

129

Contrasting

Unclassified

Order By: Relevance

“…Our statement was based on the original exposition by Lloyd and Pagels [8] and other discussions (e.g. [17]). The results of Crutchfield and Shalizi [9] would indeed seem to indicate that thermodynamic depth is a monotonically increasing function of "disorder", given their insistence that the choice of states to be made should be the "causal states" of "ǫ -machines".…”

mentioning

confidence: 99%

Reply to Comments on “Simple measure for complexity”

Shiner

Davison²,

Landsberg³

2000

Phys. Rev. E

View full text Add to dashboard Cite

We respond to the comment by Crutchfield, Feldman and Shalizi and that by Binder and Perry, pointing out that there may be many maximum entropies, and therefore "disorders" and "simple complexities". Which ones are appropriate depend on the questions being addressed. "Disorder" is not restricted to be the ratio of a nonequilibrium entropy to the corresponding equilibrium entropy; therefore, "simple complexity" need not vanish for all equiibrium systems, nor must it be nonvanishing for a nonequilibrium system. 05.20.-y, 05.90.+m We are pleased that our contribution on a "simple measure for complexity" [1] (hereafter referred to as SDL) is of sufficient interest to have generated two comments, one by Crutchfield, Feldman and Shalizi [2] (CFS) and another by Binder and Perry [3] (BP). In SDL we proposed∆ ≡ S/S max , Ω ≡ 1 − ∆.(2) as a "simple measure for complexity". α and β are (constant) parameters, S is the Boltzmann-Gibbs-Shannon entropy [4], and S max , the maximum entropy. ∆ was introduced earlier by one of us as a measure for disorder, and Ω is referred to as "order" [5,6]. CFS raise several points: I. Since S max is the equilibrium entropy, ∆ and Γ αβ vanish for all equilibrium systems, and neither can "distinguish between two-dimensional Ising systems at low temperature, high temperature, or the critical temperature . . .

show abstract

mentioning

confidence: 99%

Reply to Comments on “Simple measure for complexity”

Shiner

Davison²,

Landsberg³

2000

Phys. Rev. E

View full text Add to dashboard Cite

show abstract

“…These include Renyi entropies, effective complexity, ε complexity etc. (Rapp and Jimenez-Montano, 2001;Wackerbauer and Scheingraber, 1994). Here we will not discuss all these methods but focus on the approach based on the study of attractors organization (or testing of topology of phase space images of unknown dynamics).…”

Section: Phase Space Structures As Image Of Dynamicsmentioning

confidence: 99%

Identification of Complex Processes Based on Analysis of Phase Space Structures

Matcharashvili

Chélidzé

Janiashvili³

Imaging for Detection and Identification

View full text Add to dashboard Cite

Abstract. The problem of investigation of temporal and/or spatial behavior of highly nonlinear or complex natural systems has long been of fundamental scientific interest. At the same time it is presently well understood that identification of dynamics of processes in complex natural systems, through their qualitative description and quantitative evaluation, is far from a purely academic question and has an essential practical importance. This is quite understandable as systems with complex dynamics abound in nature and examples can be found in very different areas such as medicine and biology (rhythms, physiological cycles, epidemics), atmosphere (climate and weather change), geophysics (tides, earthquakes, volcanoes, magnetic field variations), economy (financial markets behavior, exchange rates), engineering (friction, fracturing), communication (electronic networks, internet packet dynamics) etc. The past two decades of research on qualitative and especially quantitative investigations of dynamics of real processes of different origin brought significant progress in the understanding of behavior of natural processes. At the same time serious drawbacks have also been revealed. This is why exhaustive investigation of dynamical properties of complex processes for scientific, engineering or practical purposes is now recognized as one of the main scientific challenges. Much attention is paid to elaboration of appropriate methods aiming to measuring the complexity of both global and local dynamical behaviors from the observed data sets -time series. This chapter presents a short overview of modern methods of qualitative and quantitative evaluation of dynamics of complex natural processes such as calculation of Lyapunov exponents and fractal dimensions, recurrence plots and recurrence quantification analysis. Other related methods are also described. The traditional approach to studying dynamical behavior of complex nonlinear systems is to reconstruct from observation scalar time series state or phase space plot. This graph indicates how the systems behavior changes over the time. We focus on the methods of identification and quantitative evaluation of complex dynamics that are based on the testing of evolutionary and geometric properties of phase space graphs as images of investigated complex dynamics. For practical examples of the application c 2006 Springer. Printed in the Netherlands.of nonlinear methods for identification of complex natural processes, our results on medical, geophysical, hydrological, and stick-slip time series analysis are presented.

show abstract

“…3 The most significant and easy case is a = 2/3 as the graph of the tent map intersects the diagonal there, see Figure 1; so, we consider…”

Section: Figurementioning

confidence: 99%

Randomness, chaos, and structure

2009

View full text Add to dashboard Cite

We show how a simple scheme of symbolic dynamics distinguishes a chaotic from a random time series and how it can be used to detect structural relationships in coupled dynamics. This is relevant for the question at which scale in complex dynamics regularities and patterns emerge. © 2009 Wiley Periodicals, Inc. Complexity 15: 29-35, 2009 1. SYMBOLIC DYNAMICS DISTINGUISHING CHAOTIC FROM RANDOM DYNAMICS Random Sequences and Dynamical IteratesA ccording to many popular accounts, chaotic dynamics seem to blur the distinction between determinism and randomness. Although following a fixed rule, it is characteristic of chaotic dynamics that in the longer term no prediction of the iterates of given initial values is possible, and it therefore seems that sequences of points generated by chaotic dynamics are difficult, if not impossible, to distinguish from random sequences. Of course, this is not so, and one may exploit regularities in the relationships between subsequent points in the sequence to extract useful information about the underlying dynamics. By now, very sophisticated methods have been successfully developed, and we refer to [1] for a good account of the state of the art, describing both the older linear and the more recent nonlinear tools, in particular phase space and other embedding methods, together with a rich spectrum of applications.It is the purpose of the present article to analyze the relationship between randomness and chaos in an elementary manner using simple symbolic dynamics, and to utilize this to elucidate the formation of higher level structures through the coordination of lower level nonlinear dynamics, as initiated in our earlier contribution [2].The baseline situation is a sequence x n , with n ∈ N as usual, of points randomly drawn from the unit interval [0, 1], independently of each other and all distributed according to the uniform density. The latter means that for each subinterval of [0, 1], the probability of finding x n , for given n, in that interval is equal to its length.We then consider the tent mapf (x) = 2x for 0 ≤ x ≤ 1/2 2 − 2x for 1/2 ≤ x ≤ 1 (1)

show abstract

A comparative classification of complexity measures

Cited by 235 publications

References 28 publications

Reply to Comments on “Simple measure for complexity”

Reply to Comments on “Simple measure for complexity”

Identification of Complex Processes Based on Analysis of Phase Space Structures

Randomness, chaos, and structure

Contact Info

Product

Resources

About