Ultrasonic chaotic excitations combined with sensor prediction algorithms have shown the ability to identify incipient damage (loss of preload) in a bolted joint. In this study we examine a physical experiment on a single-bolt aluminum lap joint as well as a three-dimensional physics-based simulation designed to model the behavior of guided ultrasonic waves through a similarly configured joint. A multiple bolt frame structure is also experimentally examined. In the physical experiment each signal is imparted to the structure through a macro-fiber composite (MFC) patch on one side of the lap joint and sensed using an equivalent MFC patch on the opposite side of the joint. The model applies the waveform via direct nodal displacement and 'senses' the resulting displacement using an average of the nodal strain over an area equivalent to the MFC patch. A novel statistical classification feature isdeveloped from information theory concepts of cross-prediction and interdependence. This damage detection algorithm is used to evaluate multiple damage levels and locations.

Orbits generated by discrete-time dynamical systems have some interesting combinatorial properties. In this paper we address the existence of forbidden order patterns when the dynamics is generated by piecewise monotone maps on one-dimensional closed intervals. This means that the points belonging to a sufficiently long orbit cannot appear in any arbitrary order. The admissible patterns are then (the inverses of) those permutations avoiding the so-called forbidden root patterns in consecutive positions. The last part of the paper studies and enumerates forbidden order patterns in shift systems, which are universal models in information theory, dynamical systems and stochastic processes. In spite of their simple structure, shift systems exhibit all important features of low-dimensional chaos, allowing to export the results to other dynamical systems via order-isomorphisms. This paper summarizes some results from [1] and [11]. Les orbites générées par des systèmes dynamiques à temps discret ont quelques propriétés combinatoires intéressantes. Dans cet article on adresse l’existence de motifs d’ordre exclus quand la dynamique est générée par des applications monotones à parts sur des intervalles fermés en une dimension. Ceci signifie que les points appartenant à une orbite suffisamment longue ne peuvent pas apparaître dans un ordre arbitraire. Les motifs admissibles sont alors (les inverses de) ces permutations qui évitent les motifs exclus fondamentaux en positions consécutives. La dernière partie de l’article étudie et énumère les motifs exclus dans les systèmes de déplacement, qui sont des modèles universels dans la théorie de l’information, les systèmes dynamiques et les processus stochastiques. Malgré leur structure simple, les systèmes de déplacement manifestent toutes les propriétés importantes du chaos en basse dimension, permettant exporter les résultats aux autres systèmes dynamiques via des isomorphismes d’ordre. Cet article résume quelques résultats de [1] et [11].

Forbidden ordinal patterns are ordinal patterns (or rank blocks) that cannot appear in the orbits generated by a map taking values on a linearly ordered space, in which case we say that the map has forbidden patterns. Once a map has a forbidden pattern of a given length L0, it has forbidden patterns of any length L≥L0 and their number grows superexponentially with L. Using recent results on topological permutation entropy, in this paper we study the existence and some basic properties of forbidden ordinal patterns for self-maps on n-dimensional intervals. Our most applicable conclusion is that expansive interval maps with finite topological entropy have necessarily forbidden patterns, although we conjecture that this is also the case under more general conditions. The theoretical results are nicely illustrated for n=2 both using the naive counting estimator for forbidden patterns and Chao's estimator for the number of classes in a population. The robustness of forbidden ordinal patterns against observational white noise is also illustrated.

The scope of this paper is two-fold. First, to present to the researchers in combinatorics an interesting implementation of permutations avoiding generalized patterns in the framework of discrete-time dynamical systems. Indeed, the orbits generated by piecewise monotone maps on one-dimensional intervals have forbidden order patterns, i.e., order patterns that do not occur in any orbit. The allowed patterns are then those patterns avoiding the so-called forbidden root patterns and their shifted patterns. The second scope is to study forbidden patterns in shift systems, which are universal models in information theory, dynamical systems and stochastic processes. Due to its simple structure, shift systems are accessible to a more detailed analysis and, at the same time, exhibit all important properties of low-dimensional chaotic dynamical systems (e.g., sensitivity to initial conditions, strong mixing and a dense set of periodic points), allowing to export the results to other dynamical systems via order-isomorphisms.

The Operationally Responsive Space (ORS) strategy hinges, in part, on realizing technologies which can facilitate the rapid deployment of satellites. Presently, preflight qualification testing and vehicle integration processes are time consumptive and pose as two significant hurdles which must be. overcome to effectively enhance US space asset deployment responsiveness. There is a growing demand for innovative embedded Structural Health Monitoring (SHM) technologies which can be seamlessly incorporated onto payload hardware and function in parallel with satellite construction to mitigate lengthy preflight checkout procedures. In this effort our work is focused on the development of a joint connectivity monitoring algorithm which can detect, locate, and assess preload in bolted joint assemblies. Our technology leverages inexpensive, lightweight, flexible thin-film macro-fiber composite (MFC) sensor/actuators with a novel online, data-driven signal processing algorithm. This algorithm inherently relies upon Chaotic Guided Ultrasonic Waves (CGUW) and a novel cross-prediction error classification technique. The efficacy of the monitoring algorithm is evaluated through a series of numerical simulations and experimentally in two test configurations. We conclude with a discussion surrounding further development of this approach into a commercial product as a real-time flight readiness indicator.

Permutation entropy quantifies the diversity of possible ordering of the successively observed values a random or deterministic system can take, just as Shannon entropy quantifies the diversity of the values themselves. When the observable or state variable has a natural order relation, making permutation entropy possible to compute, then the asymptotic rate of growth in permutation entropy with word length forms an alternative means of describing the intrinsic entropy rate of a source. Herein, extending a previous result on metric entropy rate, we show that the topological permutation entropy rate for expansive maps equals the conventional topological entropy rate familiar from symbolic dynamics. This result is not limited to one-dimensional maps.

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. Our estimator makes few assumptions about the underlying neural dynamics, shows excellent performance in experimentally relevant regimes, and uniquely provides confidence intervals bounding the range of information rates compatible with the observed spike train. We validate this estimator with simulations of spike trains and highlight how stimulus parameters affect its convergence in bias and variance. Finally, we apply these ideas to a recording from a guinea pig retinal ganglion cell and compare results to a simple linear decoder.

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. The chain has a probabilistic graph representation, and the cycles of the graph may be exhaustively enumerated with a classical deterministic algorithm, providing a global, comprehensive list of symbolic names for its periodic orbits. Once the symbolic codes of the periodic orbits are found, the partition is used to localize the orbits back in the original state space. Using the periodic orbits found, we can estimate several quantities of the attractor such as the Lyapunov exponent and topological entropy.

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.

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. We use a model weighting principle originally developed for lossless data compression, following the minimum description length principle. This weighting yields a direct estimator of the entropy rate, which, compared to existing methods, exhibits significantly less bias and converges faster in simulation. With Monte Carlo techinques, we estimate a Bayesian confidence interval for the entropy rate. In related work, we apply these ideas to estimate the information rates between sensory stimuli and neural responses in experimental data (Shlens, Kennel, Abarbanel, & Chichilnisky, in preparation).

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. Thus, chaotic communication by targeting need not consume any additional channel capacity beyond that required by the message itself and the message-bearing signal may appear identical to the uncontrolled oscillator. We also demonstrate control and data transmission at the channel capacity of the oscillator, the maximum possible data rate compatible with the grammar.

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. We construct an energy-like functional and use a nonequilibrium stochastic minimization algorithm to search through configuration space for the best assignment of symbols to observed data. As each observed point may be assigned a symbol, the partitions are not constrained to an arbitrary parametrization. We further show how to select particular generating partition solutions which also code low-order unstable periodic orbits in a given way, hence being able to enumerate through a number of potential generating partition solutions.

Permutation entropy quantifies the diversity of possible orderings of the values a random or deterministic system can take, as Shannon entropy quantifies the diversity of values. We show that the metric and permutation entropy rates—measures of new disorder per new observed value—are equal for ergodic finite-alphabet information sources (discrete-time stationary stochastic processes). With this result, we then prove that the same holds for deterministic dynamical systems defined by ergodic maps on n-dimensional intervals. This result generalizes a previous one for piecewise monotone interval maps on the real line [C. Bandt, G. Keller, B. Pompe, Entropy of interval maps via permutations, Nonlinearity 15 (2002) 1595–1602.] at the expense of requiring ergodicity and using a definition of permutation entropy rate differing modestly in the order of two limits. The case of non-ergodic finite-alphabet sources is also studied and an inequality developed. Finally, the equality of permutation and metric entropy rates is extended to ergodic non-discrete information sources when entropy is replaced by differential entropy in the usual way.

We demonstrate a technique for shaping current inputs for the direct modulation of a semiconductor laser for digital communication. The introduction of shaped current inputs allows for the suppression of relaxation oscillations and the avoidance of dynamical memory in the physical laser device, i.e., the output will not be influenced by previously communicated information. For the example of time-optimized bits, the possible performance enhancement for high data rate communications is shown numerically.

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. A general-purpose and robust statistical test procedure requires adapting to arbitrary dynamics which may have significant time correlation of undetermined form. Given a symbolization of the observed time series, the technology behind adaptive dictionary data compression algorithms offers a suitable estimate of reversibility, as well as a statistical likelihood test. The data compression methods create approximately independent segments permitting a simple and direct null test without resampling or surrogate data. We demonstrate the results on various time-series-reversible and irreversible systems.

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.SUBJECTS:Parameters for 3 males (ages 44, 45, 67) and 1 female (age 41) were recorded. One (1) subject (male age 45) had extensive training in this technique.INTERVENTIONS:This yogic technique is a 1 breath per minute (BPM) respiratory exercise with slow inspiration for 20 seconds, breath retention for 20 seconds, and slow expiration for 20 seconds, for 31 consecutive minutes.OUTCOME MEASURES:Fourteen beat-to-beat parameters were measured noninvasively and calculated for body surface area to yield: stroke index (SI), heart rate (HR), cardiac index, end diastolic index, peak flow, ejection fraction, thoracic fluid index, index of contractility, ejection ratio, systolic time ratio, acceleration index, and systolic, diastolic, and mean arterial pressures (MAPs). Left stroke work index (LSWI) and stroke systemic vascular resistance index (SSVRI) were calculated.RESULTS:We report on SI, HR, MAP, LSWI, and SSVRI and how they can help to describe hemodynamicstate changes. This technique induces dramatic shifts in all hemodynamic variables during the 1 BPM exercise and can produce unique changes in the postexercise resting period after long-term practice that appears to have a unique effect on the brain stem cardiorespiratory center regulating the Mayer wave (0.1-0.01 Hz) patterns of the cardiovascular system.CONCLUSIONS:Preclinical studies are warranted to examine the possible long-term effects of this technique that appear to reset a cardiorespiratory brain-stem pacemaker. We postulate that this effect may be the basis for the purported yogic health claim.

Many data-based statistical algorithms require that one find \textit{near or nearest neighbors} to a given vector among a set of points in that vector space, usually with Euclidean topology. The k-d data structure and search algorithms are the generalization of classical binary search trees to higher dimensional spaces, so that one may locate near neighbors to an example vector in $O(\log N)$ time instead of the brute-force O(N) time, with $N$ being the size of the data base. KDTREE2 is a Fortran 95 module, and a parallel set of C++ classes which implement tree construction and search routines to find either a set of $m$ nearest neighbors to an example, or all the neighbors within some Euclidean distance $r.$ The two versions are independent and function fully on their own. Considerable care has been taken in the implementation of the search methods, resulting in substantially higher computational efficiency (up to an order of magnitude faster) than the author's previous Internet-distributed version. Architectural improvements include rearrangement for memory cache-friendly performance, heap-based priority queues for large $m$searches, and more effective pruning of search paths by geometrical constraints to avoid wasted effort. The improvements are the most potent in the more difficult and slowest cases: larger data base sizes, higher dimensionality manifolds containing the data set, and larger numbers of neighbors to search for. The C++ implementation requires the Standard Template Library as well as the BOOST C++ library be installed.

Methods for estimating parameters of the Hindmarsh-Rose (HR) neuron model from a single time series are investigated. Two approaches, (1) synchronization based parameter estimation and (2) adaptive observer, are presented. Both methods are applied to membrane potential data recorded from a single lateral pyloric neuron synaptically isolated from other neurons.

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. We introduce a statistic and algorithm to refine empirical partitions for symbolic state reconstruction. This method optimizes an essential property of a generating partition, avoiding topological degeneracies, by minimizing the number of "symbolic false nearest neighbors." It requires only the observed time series and is sensible even in the presence of noise when no truly generating partition is possible.