Narrow Results

A novel method for the online detection of the modality of complex-valued nonlinear and nonstationary signals is introduced. This is achieved using a convex combination of complex nonlinear adaptive filters with different transient characteristics. To facilitate the online mode of operation, the convex mixing parameter λ within the proposed architecture is made gradient adaptive. Our focus is on the most important aspect of complex nonlinear modeling, that is, the identification of the split-complex and fully-complex nature of the signal in hand. The algorithms derived are robust and capable of tracking the changes in the modality of both benchmark and real world radar and wind complex vector fields.

Real systems often show complex behavior due to interaction among many elements composing a large-scale network. To model and predict these systems, it is desired to estimate network structures by using only time-series data observed as behavior of systems. Although several kinds of estimation techniques have been proposed, the optimum technique might be different according to properties of systems. To analyze the possibility, we estimate interactions of chaos coupled systems by four typical types of estimation techniques. For numerical simulations, we adopt the coupled map lattice, which is a model of large-scale complex systems, and we modify it so as to control the degree of synchronization and the instability of systems by changing the coupling strength and the topology of interaction among elements. As results, we can confirm that the optimum technique depends on properties of system, and then we clarify the reason from the viewpoint of synchronization and the Lyapunov exponents. Moreover, as an application, we predict future behavior of each element with new prediction model based on estimated interactions, and we demonstrate the efficiency of this prediction method.

We propose a new prediction method for nonlinear time series based on the paradigm of deterministic chaos. Introducing a stochastically equivalent dynamical system to an original map, a prediction method is derived by minimizing a random term that defines intervals in which a good prediction performance is obtained. The use of the present method is illustrated for some chaotic systems with particular emphasis on issues of choices of variable time steps that are necessary when discretizing the stochastic differential equation. Applying to some systems, it is found that the present method works better than traditional chaotic methods.

Some chaotic and a series of stochastic neural firings are multimodal. Stochastic multimodal firing patterns are of special importance because they indicate a possible utility of noise. A number of previous studies confused the dynamics of chaotic and stochastic multimodal firing patterns. The confusion resulted partly from inappropriate interpretations of estimations of nonlinear time series measures. With deliberately chosen examples the present paper introduces strategies and methods of identification of stochastic firing patterns from chaotic ones. Aided by theoretical simulation we show that the stochastic multimodal firing patterns result from the effects of noise on neuronal systems near to a bifurcation between two simpler attractors, such as a point attractor and a limit cycle attractor or two limit cycle attractors. In contrast, the multimodal chaotic firing trains are generated by the dynamics of a specific strange attractor. Three systems were carefully chosen to elucidate these two mechanisms. An experimental neural pacemaker model and the Chay mathematical model were used to show the stochastic dynamics, while the deterministic Wang model was used to show the deterministic dynamics. The usage and interpretation of nonlinear time series measures were systematically tested by applying them to firing trains generated by the three systems. We successfully identified the distinct differences between stochastic and chaotic multimodal firing patterns and showed the dynamics underlying two categories of stochastic firing patterns. The first category results from the effects of noise on the neuronal system near a Hopf bifurcation. The second category results from the effects of noise on the period-adding bifurcation between two limit cycles. Although direct application of nonlinear measures to interspike interval series of these firing trains misleadingly implies chaotic properties, definition of eigen events based on more appropriate judgments of the underlying dynamics leads to accurate identifications of the stochastic properties.

Extraction of the underlying dynamics of large biological systems is a crucial step to understanding their mechanisms and predicting their future behaviors. Some of the techniques developed in physics to analyze chaotical phenomena should be applied with caution when the data are short and noisy.

In this paper, we compared the Eckmann-Ruelle Linearization (ERL) local prediction method, and Neural Network (NN), fitting of a global nonlinear function, in the context of population dynamics. Data were generated by a discrete model and subjected to measurement noise. Prediction errors were analyzed according to noise level and length of the observations, using the above two methods

Where the noise level is high, ERL is more effective for prediction than NN. On the basis of these results, we proposed for short series of available observations, that additional data be generated by the Eckmann—Ruelle linearization method and these data be used as input for Neural Nets in order to obtain a global function for the observed dynamics. This gave more reliable results than when we apply the NN method directly .

Two examples are given, in ecology and epidemiology.

Predicting foreign exchange rates and stock market indices have been a well researched topic in the field of financial engineering. However, most methods suffer from serious drawback due to the inherent uncertainty in the data acquisition process. Here, we have analyzed the very nature of the time series data from a pure dynamic system point of view and explored the deterministic chaotic characteristic in it. In this research, the concept of chaos has been analyzed thoroughly and the relationships among chaos, stability and order have been explained with respect to the concept of time. A method of predicting time series data based on deterministic dynamically system has been presented in this monograph. The present research revolves around the concepts of embedding and fuzzy reconstruction. In this regard, the necessary and sufficient condition for this reconstruction of the state space of the dynamic system in a multi-dimensional Euclidean space has been substantiated in accordance to Theory of embedding. Finally, a fuzzy reconstruction method based on fuzzy multiple regression analysis method has been used to predict the foreign exchange rates with accuracy.

We perform out-of-sample prediction on both fixed and black market Chinese renminbi/US dollar, and black market rial/US dollar exchange rates by using the time-delay embedding technique and the local linear prediction method. We also predict an artificially generated chaotic time series with and without noise for the purpose of validation of the methods used in this study. In all examples tested, our prediction results significantly outperform those by the benchmark mean value predictor based on a statistic defined by Harvey et al. [11]. Another interesting result found in this paper is that one may use the embedding dimension as a measure of volatility of a financial asset.

An explicit formula is obtained for the nonlinear predictor of Y(t) = X(t)² − E(X(t)²), where X(t) is an N-ple Gaussian Markov process.