NONLINEAR TIME SERIES/SYSTEM ANALYSIS (STATE-SPACE, DYNAMICS, INTERVENTION, RESILIENCE, KALMAN)

A discussion of nonlinear dynamics, demonstrated by the familiar automobile, is followed by the development of a systematic method of analysis of a possibly nonlinear time series using difference equations in the general state-space format. This format allows recursive state-dependent parameter estimation after each observation thereby revealing the dynamics inherent in the system in combination with random external perturbations.^ The one-step ahead prediction errors at each time period, transformed to have constant variance, and the estimated parametric sequences provide the information to (1) formally test whether time series observations y(,t) are some linear function of random errors (ELEM)(,s), for some t and s, or whether the series would more appropriately be described by a nonlinear model such as bilinear, exponential, threshold, etc., (2) formally test whether a statistically significant change has occurred in structure/level either historically or as it occurs, (3) forecast nonlinear system with a new and innovative (but very old numerical) technique utilizing rational functions to extrapolate individual parameters as smooth functions of time which are then combined to obtain the forecast of y and (4) suggest a measure of resilience, i.e. how much perturbation a structure/level can tolerate, whether internal or external to the system, and remain statistically unchanged. Although similar to one-step control, this provides a less rigid way to think about changes affecting social systems.^ Applications consisting of the analysis of some familiar and some simulated series demonstrate the procedure. Empirical results suggest that this state-space or modified augmented Kalman filter may provide interesting ways to identify particular kinds of nonlinearities as they occur in structural change via the state trajectory.^ A computational flow-chart detailing computations and software input and output is provided in the body of the text. IBM Advanced BASIC program listings to accomplish most of the analysis are provided in the appendix. ^

Application of Signal Advance Technology to Electrophysiology

Medical instrumentation used in diagnosis and treatment relies on the accurate detection and processing of various physiological events and signals. While signal detection technology has improved greatly in recent years, there remain inherent delays in signal detection/ processing. These delays may have significant negative clinical consequences during various pathophysiological events. Reducing or eliminating such delays would increase the ability to provide successful early intervention in certain disorders thereby increasing the efficacy of treatment. In recent years, a physical phenomenon referred to as Negative Group Delay (NGD), demonstrated in simple electronic circuits, has been shown to temporally advance the detection of analog waveforms. Specifically, the output is temporally advanced relative to the input, as the time delay through the circuit is negative. The circuit output precedes the complete detection of the input signal. This process is referred to as signal advance (SA) detection. An SA circuit model incorporating NGD was designed, developed and tested. It imparts a constant temporal signal advance over a pre-specified spectral range in which the output is almost identical to the input signal (i.e., it has minimal distortion). Certain human patho-electrophysiological events are good candidates for the application of temporally-advanced waveform detection. SA technology has potential in early arrhythmia and epileptic seizure detection and intervention. Demonstrating reliable and consistent temporally advanced detection of electrophysiological waveforms may enable intervention with a pathological event (much) earlier than previously possible. SA detection could also be used to improve the performance of neural computer interfaces, neurotherapy applications, radiation therapy and imaging. In this study, the performance of a single-stage SA circuit model on a variety of constructed input signals, and human ECGs is investigated. The data obtained is used to quantify and characterize the temporal advances and circuit gain, as well as distortions in the output waveforms relative to their inputs. This project combines elements of physics, engineering, signal processing, statistics and electrophysiology. Its success has important consequences for the development of novel interventional methodologies in cardiology and neurophysiology as well as significant potential in a broader range of both biomedical and non-biomedical areas of application.