Reply to Comments on “Simple measure for complexity”

Abstract

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 . . .

EEG analysis using wavelet-based information tools

“…As a consequence, it might not contain new information vis-a-vis the measure of order. Such an objection is discussed at length in Binder and Perry (2000), Crutchfield et al (2000), and Shiner et al (2000).…”

Complexity in the sunspot cycle

“…However, the nearlyflat background (∼ f −η with η ∼ 0.3) suggests that the stochastic fluctuations play a relevant role in the spatio-temporal complexity displayed by the solar activity cycle. Although the information entropy is a useful quantity to provide insights into the complex dynamics of a many degree-offreedom system, a more accurate characterization of the spatiotemporal complexity observed in a dynamical system could be given by the II order complexity measure Γ 11 introduced by Shiner et al (1999;see also Crutchfield et al 2000;Binder & Perry 2000;and Shiner et al 2000). From the definition of the information entropy S (t), it is possible to define a disorder measure Δ(t) (Landsberg 1984) by simply normalizing the information entropy to the maximum entropy S max , i.e.,…”

Extended Entropies and Disorder

“…a H max is the maximum possible entropy, which depends on the constraints imposed on the system and the question being addressed [50][51][52]. In the simplest case (no constraints other than normalization of the probabilities), H max = ln N , corresponding to the equiprobable distribution p i = 1/N, 1 ≤ i ≤ N .…”