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We present a new, non-parametric forecasting method for data where continuous values are observed discretely in space and time. Our method, "light-cone reconstruction of states" (LICORS), uses physical principles to identify predictive states which are local properties of the system, both in space and time. LICORS discovers the number of predictive states and their predictive distributions automatically, and consistently, under mild assumptions on the data source. We provide an algorithm to implement our method, along with a cross-validation scheme to pick control settings. Simulations show that CV-tuned LICORS outperforms standard methods in forecasting challenging spatio-temporal dynamics. Our work provides applied researchers with a new, highly automatic method to analyze and forecast spatio-temporal data., Comment: Main text: 30 pages; supplementary material: 12 pages; 5+2 figures

Cyclic cellular automata (CCA) are models of excitable media. Started from random initial conditions, they produce several different kinds of spatial structure, depending on their control parameters. We introduce new tools from information theory that let us calculate the dynamical information content of spatial random processes. This complexity measure allows us to quantitatively determine the rate of self-organization of these cellular automata, and establish the relationship between parameter values and self-organization in CCA. The method is very general and can easily be applied to other cellular automata or even digitized experimental data., Comment: 10 pages, 6 figures. This was a preliminary report on the research whose final results appeared in nlin.AO/0409024. However, this report includes certain algorithmic details and discussion of related literature omitted from the paper for reasons of space

Most current methods for identifying coherent structures in spatially-extended systems rely on prior information about the form which those structures take. Here we present two new approaches to automatically filter the changing configurations of spatial dynamical systems and extract coherent structures. One, local sensitivity filtering, is a modification of the local Lyapunov exponent approach suitable to cellular automata and other discrete spatial systems. The other, local statistical complexity filtering, calculates the amount of information needed for optimal prediction of the system's behavior in the vicinity of a given point. By examining the changing spatiotemporal distributions of these quantities, we can find the coherent structures in a variety of pattern-forming cellular automata, without needing to guess or postulate the form of that structure. We apply both filters to elementary and cyclical cellular automata (ECA and CCA) and find that they readily identify particles, domains and other more complicated structures. We compare the results from ECA with earlier ones based upon the theory of formal languages, and the results from CCA with a more traditional approach based on an order parameter and free energy. While sensitivity and statistical complexity are equally adept at uncovering structure, they are based on different system properties (dynamical and probabilistic, respectively), and provide complementary information., Comment: 16 pages, 21 figures. Figures considerably compressed to fit arxiv requirements; write first author for higher-resolution versions

Despite broad interest in self-organizing systems, there are few quantitative, experimentally-applicable criteria for self-organization. The existing criteria all give counter-intuitive results for important cases. In this Letter, we propose a new criterion, namely an internally-generated increase in the statistical complexity, the amount of information required for optimal prediction of the system's dynamics. We precisely define this complexity for spatially-extended dynamical systems, using the probabilistic ideas of mutual information and minimal sufficient statistics. This leads to a general method for predicting such systems, and a simple algorithm for estimating statistical complexity. The results of applying this algorithm to a class of models of excitable media (cyclic cellular automata) strongly support our proposal., Comment: Four pages, two color figures

In this chapter, I review the main methods and techniques of complex systems science. As a first step, I distinguish among the broad patterns which recur across complex systems, the topics complex systems science commonly studies, the tools employed, and the foundational science of complex systems. The focus of this chapter is overwhelmingly on the third heading, that of tools. These in turn divide, roughly, into tools for analyzing data, tools for constructing and evaluating models, and tools for measuring complexity. I discuss the principles of statistical learning and model selection; time series analysis; cellular automata; agent-based models; the evaluation of complex-systems models; information theory; and ways of measuring complexity. Throughout, I give only rough outlines of techniques, so that readers, confronted with new problems, will have a sense of which ones might be suitable, and which ones definitely are not., Comment: 96 pages, 8 figures. Versions 2 and 3: corrects minor typographical errors. Version 4: Expanded examples, updated references (through late 2004), matches published version up to changes in formatting

It is increasingly common to encounter time-varying random fields on networks (metabolic networks, sensor arrays, distributed computing, etc.). This paper considers the problem of optimal, nonlinear prediction of these fields, showing from an information-theoretic perspective that it is formally identical to the problem of finding minimal local sufficient statistics. I derive general properties of these statistics, show that they can be composed into global predictors, and explore their recursive estimation properties. For the special case of discrete-valued fields, I describe a convergent algorithm to identify the local predictors from empirical data, with minimal prior information about the field, and no distributional assumptions., Comment: 20 pages, 5 figures. For the conference "Discrete Models of Complex Systems" (Lyon, June, 2003). v2: Typos fixed, regenerated figures should now produce readable PDF output

Particle-like objects are observed to propagate and interact in many spatially extended dynamical systems. For one of the simplest classes of such systems, one-dimensional cellular automata, we establish a rigorous upper bound on the number of distinct products that these interactions can generate. The upper bound is controlled by the structural complexity of the interacting particles---a quantity which is defined here and which measures the amount of spatio-temporal information that a particle stores. Along the way we establish a number of properties of domains and particles that follow from the computational mechanics analysis of cellular automata; thereby elucidating why that approach is of general utility. The upper bound is tested against several relatively complex domain-particle cellular automata and found to be tight., Comment: 17 pages, 12 figures, 3 tables, http://www.santafe.edu/projects/CompMech/papers/ub.html V2: References and accompanying text modified, to comply with legal demands arising from on-going intellectual property litigation among third parties. V3: Accepted for publication in Physica D. References added and other small changes made per referee suggestions