Michael Kremliovsky

Estimating statistics for detecting determinism using global dynamical models

Abstract Classification of time series using a dynamical system ansatz is potentially powerful, however assessing performance for noisy experimental data is problematic. Here, we develop a rigorous statistical framework for calculating classification probabilities using global dynamical models, and analytically derive some asymptotic properties. We illustrate the method numerically by attempting to detect “determinism” in a noisy data set.

Estimating dynamical models using generalized correlation functions

Abstract We develop a method for estimating closed-form nonlinear dynamical models from observed time series, which expresses the unknown coefficients as functions of generalized higher-order data correlations. Besides robust numerical properties, this method often yields analytic coefficient representations which provide theoretical insight into general model properties.

CHARACTERIZATION OF DOLPHIN ACOUSTIC ECHO-LOCATION DATA USING A DYNAMICAL CLASSIFICATION METHOD

Many methods for time series analysis derived from nonlinear dynamical systems theory have been developed in the last decade, and have demonstrated remarkable results in a variety of simulated, experimental, and real applications. Classification of time series based on the underlying dynamical generator is also potentially powerful, and we have previously presented a method for dynamical classification based on empirically estimated sets of nonlinear ordinary differential equations, i.e. global dynamical models. A particularly useful area of application for such methods may be biologic and medical data analysis, where few quantitative methods exist for the highly complex time evolutions. Here, as an example of the application of such classification methods, we present an analysis of data of the acoustic pulse trains produced by dolphins as they attempt to echo-locate objects in an ocean environment, which is derived from a controlled experimental framework.

Using delay differential equations as dynamical classifiers

We show how direct estimation of delay differential equations can be used for time series analysis of observed signals corrupted by noise. The specific application we discuss is to detection (identifying the presence of a deterministic signal) and classification (assignment of a particular signal to a known class). These ideas have potentially widespread application to areas such as remote sensing, voice recognition and image processing.

Classifying Complex, Deterministic Signals

When attempting to analyze, model, or predict time series, one often finds that the data is so complex or noisy that the presence or location of a desired signal or signals is unknown. Therefore, the first step in such an analysis is often to implement some scheme for detecting or classifying signals in otherwise long stretches of noisy background. Detection/classification of signals is thus one of the principal areas of signal processing, and the utilization of nonlinear information has long been considered as a way of improving performance beyond standard linear (e.g., spectral) techniques. Here, we develop a method for using global models of chaotic dynamical systems theory to define a signal classification processing chain which is sensitive to nonlinear correlations in the data. We use it to demonstrate classification in high noise regimes, using short data segments which mimic real-world processing restrictions, and also show that classification probabilities can be directly computed from ensemble statistics in the model coefficient space. We also develop a modification for non-stationary signals (i.e. pulses) using non-autonomous ODEs. Finally, we demonstrate the technique by analyzing actual open ocean acoustic data from marine biological sources such as whales and dolphins.

Signal classification using global dynamical models, Part I: Theory

Detection and classification of signals is one of the principal areas of signal processing, and the utilization of nonlinear information has long been considered as a way of improving performance beyond standard linear (e.g. spectral) techniques. Here, we develop a method for using global models of chaotic dynamical systems theory to define a signal classification processing chain, which is sensitive to nonlinear correlations in the data. We use it to demonstrate classification in high noise regimes (negative SNR), and argue that classification probabilities can be directly computed from ensemble statistics in the model coefficient space. We also develop a modification for non‐stationary signals (i.e. transients) using non‐autonomous ODEs. In Part II of this paper, we demonstrate the analysis on actual open ocean acoustic data from marine biologics.

Signal classification using global dynamical models, Part II: SONAR data analysis

In Part I of this paper, we described a numerical method for nonlinear signal detection and classification which made use of techniques borrowed from dynamical systems theory. Here in Part II of the paper, we will describe an example of data analysis using this method, for data consisting of open ocean acoustic (SONAR) recordings of marine mammal transients, supplied from NUWC sources. The purpose here is two‐fold: first to give a more operational description of the technique and provide rules‐of‐thumb for parameter choices; and second to discuss some new issues raised by the analysis of non‐ideal (real‐world) data sets. The particular data set considered here is quite non‐stationary, relatively noisy, is not clearly localized in the background, and as such provides a difficult challenge for most detection/classification schemes.

Classification of EEG Signals From a Music Perception Experiment Using Empirical Dynamical Models

In this paper we describe a data classification method as applied to a set of EEG signals provided to the international workshop on Nonlinear Techniques in Physiological Time Series Analysis (Dresden, 1995). This method is derived from the theory of nonlinear dynamics, and consists of a classification processing chain which utilizes estimated sets of nonlinear ODEs as a data model. The investigation consisted of a blind analysis of the EEG recordings from a single human subject, taken while the subject was exposed to music segments of varying and controlled dynamical complexity. From this analysis, we find firstly that there appears to be dynamical similarity between many of the sensors, but that most sensors do not provide significant classification capability. However, at least one sensor is seen to demonstrate a strong classification performance, and this was used to generate a relative classification scheme for the data. This scheme shows significant distinction between certain musical samples, and from this we infer several general conclusions about the sensitivity of the particular subject to the musical data classes.