Markov chains are a natural and well understood tool for describing one-dimensional patterns in time or space. We show how to infer k-th order Markov chains, for arbitrary k, from finite data by applying Bayesian methods to both parameter estimation and model-order selection. Extending existing results for multinomial models of discrete data, we connect inference to statistical mechanics through information-theoretic (type theory) techniques. We establish a direct relationship between Bayesian evidence and the partition function which allows for straightforward calculation of the expectation and variance of the conditional relative entropy and the source entropy rate. Finally, we introduce a novel method that uses finite data-size scaling with model-order comparison to infer the structure of out-of-class processes.
We have developed a method for reconstructing equations of motion for systems where all the necessary variables have not been observed. This technique can be applied to systems with one or several such hidden variables, and can be used to reconstruct maps or differential equations. The effects of experimental noise are discussed through specific examples. The control of nonlinear systems containing hidden variables is also discussed.
We present findings in an experiment where we obtain stationary ramified transportation networks in a macroscopic nonbiological system. Our purpose here is to introduce the phenomenology of the experiment. We describe the dynamical formation of the network which consists of three growth stages: (I) strand formation, (II) boundary formation, and (III) geometric expansion. We find that the system forms statistically robust network features, like the number of termini and the number of branch points. We also find that the networks are usually trees, meaning that they lack closed loops; indeed, we find that loops are unstable in the network. Finally, we find that the final topology of the network is sensitive to the initial conditions of the particles, in particular to its geometry.pattern formation ͉ self-organization P attern formation, loosely speaking, is the study of order in open dissipative systems (1); this includes dynamic selforganization, characteristic of fluid and chemical systems (2), and inhomogeneous growth, characteristic in some physical (3-6) and biological (7-9) systems. Of more recent interest, not generally categorized under pattern formation, is the evolution of complex networks (10). Although this latter study has so far focused on abstract topological issues, it may soon bear important connections with the former studies, especially as it pertains to branched (what we shall refer to as ramified) patterns used for transportation throughout nature. Indeed, several researchers have attempted to include either spatial (11, §) or flow (12) constraints to the study of complex networks. Meanwhile, efficient transportation of resources through real fractal networks was already an important insight into understanding the allometric scaling of all organisms (13,14).Another example of a transportation network, this time nonbiological, was studied in experiments on an electromechanical system (15-17), where conducting particles selforganize into dendritic patterns under the inf luence of an electric field for the purpose of collecting and transporting charge. The authors were concerned with formulating a variational principle concerning the stability of patterns in open dissipative systems. In those studies, the authors concluded that in order for the patterns to be stable, they must be (locally) minimal in dissipation. The experiment has also been studied in simulation with the idea that fractals are generated by a dynamical rule: Particles always move to regions of higher gradient in potential until they stick next to a boundary point (18). All studies simplified the system by dealing only with the two-dimensional Poisson equation, and by assuming the source of charge was independent of both space and time: ٌ 2 ϭ S(r ជ, t)͞ oil Ϸ S 0 ͞ oil , where is the electric potential, S is the source term, and oil is the conductivity of the oil medium. The limitations of these approximations are unknown. Moreover, the proof showing that the dissipation is minimal relies on showing that the potential energy is al...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.