Globally enumerating unstable periodic orbits for observed data using symbolic dynamics.

The unstable periodic orbits of a chaotic system provide an important skeleton of the dynamics in a chaotic system, but they can be difficult to find from an observed time series. We present a global method for finding periodic orbits based on their symbolic dynamics, which is made possible by several recent methods to find good partitions for symbolic dynamics from observed time series. The symbolic dynamics are approximated by a Markov chain estimated from the sequence using information-theoretical concepts.

Estimating information rates with confidence intervals in neural spike trains.

Information theory provides a natural set of statistics to quantify the amount of knowledge a neuron conveys about a stimulus. A related work (Kennel, Shlens, Abarbanel, & Chichilnisky, 2005) demonstrated how to reliably estimate, with a Bayesian confidence interval, the entropy rate from a discrete, observed time series. We extend this method to measure the rate of novel information that a neural spike train encodes about a stimulus--the average and specific mutual information rates.

Variance estimators for the Lempel-Ziv entropy rate estimator.

Lempel and Ziv's 1976 algorithm provides an easy-to-compute way to automatically estimate the entropy rate for symbolic time series, requiring no free parameters. Here we derive an analytical variance estimate for the Lempel-Ziv entropy rate estimator that is easily computable from observations with negligible extra effort beyond the entropy rate itself, and compare to another procedure, a time-series-based bootstrap method. These provide a justified "error bar" quantifying the size of expected fluctuations on the estimate itself, given by the single time series of symbols.

Complete and flexible replacement of chaotic uncertainty with transmitted information.

Natural chaos can be described as an information source emitting symbolic sequences with positive entropy. We use two algorithmic techniques from data compression in a nonstandard way along with a control scheme to replace the natural uncertainty in chaotic systems with an arbitrary digital message. Unlike previous targeting-based control, the controlled, deterministic, transmission appears statistically identical to natural chaos, with a message modulated on it at the intrinsic Kolmogorov-Sinai information generation rate of the chaotic oscillator.

Statistically relaxing to generating partitions for observed time-series data.

We introduce a relaxation algorithm to estimate approximations to generating partitions for observed dynamical time series. Generating partitions preserve dynamical information of a deterministic map in the symbolic representation. Our method optimizes an essential property of a generating partition: avoiding topological degeneracies.

Estimating entropy rates with Bayesian confidence intervals.

The entropy rate quantifies the amount of uncertainty or disorder produced by any dynamical system. In a spiking neuron, this uncertainty translates into the amount of information potentially encoded and thus the subject of intense theoretical and experimental investigation. Estimating this quantity in observed, experimental data is difficult and requires a judicious selection of probabilistic models, balancing between two opposing biases.

Hemodynamic observations on a yogic breathing technique claimed to help eliminate and prevent heart attacks: a pilot study.

Objective: This pilot study investigated the hemodynamics of a yogic breathing technique claimed "to help eliminate and prevent heart attacks due to abnormal electrical events to the heart," and to generally "enhance performance of the central nervous system (CNS) and to help eliminate the effects of traumatic shock and stress to the CNS."

Design: Parameters for (4) subjects were recorded during a preexercise resting period, a 31-minute exercise period, and a postexercise resting period.

Settings/location: Parameters for subjects were recorded in a laboratory at the University of California, San Diego.

Testing time symmetry in time series using data compression dictionaries.

Time symmetry, often called statistical time reversibility, in a dynamical process means that any segment of time-series output has the same probability of occurrence in the process as its time reversal. A technique, based on symbolic dynamics, is proposed to distinguish such symmetrical processes from asymmetrical ones, given a time-series observation of the otherwise unknown process. Because linear stochastic Gaussian processes, and static nonlinear transformations of them, are statistically reversible, but nonlinear dynamics such as dissipative chaos are usually statistically irreversible, a test will separate large classes of hypotheses for the data.

Estimating good discrete partitions from observed data: symbolic false nearest neighbors.

A symbolic analysis of observed time series requires a discrete partition of a continuous state space containing the dynamics. A particular kind of partition, called "generating," preserves all deterministic dynamical information in the symbolic representation, but such partitions are not obvious beyond one dimension. Existing methods to find them require significant knowledge of the dynamical evolution operator.

Context-tree modeling of observed symbolic dynamics.

Modern techniques invented for data compression provide efficient automated algorithms for the modeling of the observed symbolic dynamics. We demonstrate the relationship between coding and modeling, motivating the well-known minimum description length (MDL) principle, and give concrete demonstrations of the "context-tree weighting" and "context-tree maximizing" algorithms. The predictive modeling technique obviates many of the technical difficulties traditionally associated with the correct MDL analyses.

False neighbors and false strands: a reliable minimum embedding dimension algorithm.

The time-delay reconstruction of the state space of a system from observed scalar data requires a time lag and an integer embedding dimension. We demonstrate a reliable method to estimate the minimum necessary embedding dimension that improves upon previous methods by correcting for systematic effects due to temporal oversampling, autocorrelation, and changing time lag. The method gives a sharp and reliable indication of the proper dimension.