OFFPRINT Permutation complexity of spatiotemporal dynamics

We call permutation complexity the kind of dynamical complexity captured by any quantity or functional based on order relations, like ordinal patterns and permutation entropies. These mathematical tools have found interesting applications in time series analysis and abstract dynamical systems. In this letter we propose to extend the study of permutation complexity to spatiotemporal systems, by applying some of its tools to a time series obtained by coarsegraining the dynamics and to state vectors at fixed times, considering the latter as sequences. We show that this approach allows to quantify the complexity and to classify different types of dynamics in cellular automata and in coupled map lattices. Furthermore, we show that our analysis can be used to discriminate between different types of spatiotemporal dynamics registered in magnetoencephalograms. Copyright c © EPLA, 2010 Introduction. – Permutation entropy was introduced in [1] as a complexity measure for time series. Roughly speaking, permutation entropy replaces the probabilities of length-L symbol blocks in the definition of Shannon’s entropy by the probabilities of length-L ordinal patterns —a digest of the ups and downs of L consecutive elements of a time series. Permutation entropy was later extended, both in metric and topological versions, to one-dimensional dynamical systems in ref. [2], and to higher-dimensional systems in refs. [3,4]. Since then, different techniques based on the analysis of ordinal patterns, that we refer to as permutation complexity analysis, have found a number of interesting applications: Estimation of metric and topological entropy [4,5], complexity analysis of time series [6], detection of determinism in noisy time series [7,8], recovery of control parameters in symbolic sequences of unimodal maps [9] and characterization of synchronization [10]. In all these applications, computational simplicity and robustness against observational noise have been a crucial advantage. In this letter we propose to extend the study of permutation complexity to spatiotemporal systems. We apply our ideas to two well-known models of spatiotemporal dynamics, cellular automata (CA) and coupled map lattices (CML), as well as to the experimental spatiotemporal data provided by magnetoencephalograms (MEGs). Our analysis is divided into two (complementary) parts. The first part is an analysis of a time series obtained by coarse-graining the dynamics considered. The second part is an analysis of the state vector at fixed times. We show that the first part provides an estimation of the dynamical complexity, and its combination with the second one provides further insights, especially when it comes to the classification of different types of dynamical behavior. Considering this, and similarly to recent works where tools of network analysis were used for time series analysis [11,12], our work reveals a link between the analysis of spatially extended systems and novel ideas from time series analysis. Models of spatiotemporal systems. – The analysis that we propose can be applied to spatiotemporal data of the form {xt} T t=1 = {x1,x2, . . . ,xT }, where xt = (xt(1), xt(2), . . . , xt(N)) is the state vector at time t of a system of N sites, each component xt(i) being the state of the i-th site at time t. The numerical data that we consider in this letter are generated with one-dimensional CA and CMLs featuring a next-neighbor local rule of evolution, xt+1(i) = f(xt(i− 1), xt(i), xt(i+1)), (1) and the periodic boundary condition xt(1) = xt(N +1). Some tools of permutation complexity. – The basic tools that we use in our analysis are the following. Consider for simplicity that {zn} M n=1 = {z1, z2, . . . , zM} is a sequence of real numbers. We say that a length-L block