Blind Construction of Optimal Nonlinear Recursive Predictors for Discrete Sequences

We present a new method for nonlinear prediction of discrete random sequences under minimal structural assumptions. We give a mathematical construction for optimal predictors of such processes, in the form of hidden Markov models. We then describe an algorithm, CSSR (Causal-State Splitting Reconstruction), which approximates the ideal predictor from data. We discuss the reliability of CSSR, its data requirements, and its performance in simulations. Finally, we compare our approach to existing methods using variable-length Markov models and cross-validated hidden Markov models, and show theoretically and experimentally that our method delivers results superior to the former and at least comparable to the latter.Comment: 8 pages, 4 figure

An Algorithm for Pattern Discovery in Time Series

We present a new algorithm for discovering patterns in time series and other sequential data. We exhibit a reliable procedure for building the minimal set of hidden, Markovian states that is statistically capable of producing the behavior exhibited in the data -- the underlying process's causal states. Unlike conventional methods for fitting hidden Markov models (HMMs) to data, our algorithm makes no assumptions about the process's causal architecture (the number of hidden states and their transition structure), but rather infers it from the data. It starts with assumptions of minimal structure and introduces complexity only when the data demand it. Moreover, the causal states it infers have important predictive optimality properties that conventional HMM states lack. We introduce the algorithm, review the theory behind it, prove its asymptotic reliability, use large deviation theory to estimate its rate of convergence, and compare it to other algorithms which also construct HMMs from data. We also illustrate its behavior on an example process, and report selected numerical results from an implementation.Comment: 26 pages, 5 figures; 5 tables; http://www.santafe.edu/projects/CompMech Added discussion of algorithm parameters; improved treatment of convergence and time complexity; added comparison to older method

The Computational Structure of Spike Trains

Neurons perform computations, and convey the results of those computations through the statistical structure of their output spike trains. Here we present a practical method, grounded in the information-theoretic analysis of prediction, for inferring a minimal representation of that structure and for characterizing its complexity. Starting from spike trains, our approach finds their causal state models (CSMs), the minimal hidden Markov models or stochastic automata capable of generating statistically identical time series. We then use these CSMs to objectively quantify both the generalizable structure and the idiosyncratic randomness of the spike train. Specifically, we show that the expected algorithmic information content (the information needed to describe the spike train exactly) can be split into three parts describing (1) the time-invariant structure (complexity) of the minimal spike-generating process, which describes the spike train statistically; (2) the randomness (internal entropy rate) of the minimal spike-generating process; and (3) a residual pure noise term not described by the minimal spike-generating process. We use CSMs to approximate each of these quantities. The CSMs are inferred nonparametrically from the data, making only mild regularity assumptions, via the causal state splitting reconstruction algorithm. The methods presented here complement more traditional spike train analyses by describing not only spiking probability and spike train entropy, but also the complexity of a spike train's structure. We demonstrate our approach using both simulated spike trains and experimental data recorded in rat barrel cortex during vibrissa stimulation.Comment: Somewhat different format from journal version but same conten

Discovering Functional Communities in Dynamical Networks

Many networks are important because they are substrates for dynamical systems, and their pattern of functional connectivity can itself be dynamic -- they can functionally reorganize, even if their underlying anatomical structure remains fixed. However, the recent rapid progress in discovering the community structure of networks has overwhelmingly focused on that constant anatomical connectivity. In this paper, we lay out the problem of discovering_functional communities_, and describe an approach to doing so. This method combines recent work on measuring information sharing across stochastic networks with an existing and successful community-discovery algorithm for weighted networks. We illustrate it with an application to a large biophysical model of the transition from beta to gamma rhythms in the hippocampus.Comment: 18 pages, 4 figures, Springer "Lecture Notes in Computer Science" style. Forthcoming in the proceedings of the workshop "Statistical Network Analysis: Models, Issues and New Directions", at ICML 2006. Version 2: small clarifications, typo corrections, added referenc

Measuring Shared Information and Coordinated Activity in Neuronal Networks

Most nervous systems encode information about stimuli in the responding activity of large neuronal networks. This activity often manifests itself as dynamically coordinated sequences of action potentials. Since multiple electrode recordings are now a standard tool in neuroscience research, it is important to have a measure of such network-wide behavioral coordination and information sharing, applicable to multiple neural spike train data. We propose a new statistic, informational coherence, which measures how much better one unit can be predicted by knowing the dynamical state of another. We argue informational coherence is a measure of association and shared information which is superior to traditional pairwise measures of synchronization and correlation. To find the dynamical states, we use a recently-introduced algorithm which reconstructs effective state spaces from stochastic time series. We then extend the pairwise measure to a multivariate analysis of the network by estimating the network multi-information. We illustrate our method by testing it on a detailed model of the transition from gamma to beta rhythms.Comment: 8 pages, 6 figure

Automatic Filters for the Detection of Coherent Structure in Spatiotemporal Systems

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 version