In this paper, we propose algorithm of deterministic nonlinear prediction, or a modified version of the method of analogues which was originally proposed by E.N. Lorenz (J. Atom. Sci., 26, 636-646, 1969), and apply it to the artificial time series data produced from nonlinear dynamical systems and further corrupted by superimposed observational noise. The prediction performance of the present method are investigated by calculating correlation coefficients, root mean square errors and signature errors and compared with the prediction algorithm of local linear approximation method. As a result, it is shown that the prediction performance of the proposed method are better than those of the local linear approximation especially in case that the amount of noise is large.

In this paper, we discuss the principle of the clumsy painter method proposed for extracting interested regions from image signals automatically. We theoretically clarify the reason why the clumsy painter method is effective so well. We compare its algorithm with the opening operation in mathematical morphology, and prove that the clumsy painter method has the advantage over the opening operation in mathematical morphology on removing uninterested regions from image signals. Simulating these two methods on two simple geometrical models, we show that the extracted redults by the opening operation are included in those by the clumsy painter method.

Electroencephalographic (EEG) potentials are analysed by the Lyapunov spectrum in order to evaluate the orbital instability peculiar to deterministic chaos quantitatively. First, the Lyapunov spectra are estimated to confirm the existence of chaotic behavior in EEG data by the optimal approximation of Jacobian matrix in the reconstructed statespace. Second, the same method is applied to a neural network model with chaotic dynamics, the macroscopic average activity of which is analysed as a simple model of EEG data. The first analysis shows that the largest Lyapunov exponent is actually positive in the EEG data. On the other hand, the second analysis on the chaotic neural network shows that the positive largest Lyapunov exponent can be obtained by observing only the macroscopic average activity. Thus, these results indicate the possibility that one can know the existence of chaotic dynamics in the brain by analysing the Lyapunov spectrum of the macroscopic EEG data.

In this paper, performance of chaos and burst noises injected to the Hopfield Neural Network for quadratic assignment problems is investigated. For the evaluation of the noises, two methods to appreciate finding a lot of nearly optimal solutions are proposed. By computer simulations, it is confirmed that the burst noise generated by the Gilbert model with a laminar part and a burst part achieved the good performance as the intermittency chaos noise near the three-periodic window.

Deterministic nonlinear prediction is applied to both artificial and real time series data in order to investigate orbital-instabilities, short-term predictabilities and long-term unpredictabilities, which are important characteristics of deterministic chaos. As an example of artificial data, bimodal maps of chaotic neuron models are approximated by radial basis function networks, and the approximation abilities are evaluated by applying deterministic nonlinear prediction, estimating Lyapunov exponents and reconstructing bifurcation diagrams of chaotic neuron models. The functional approximation is also applied to squid giant axon response as an example of real data. Two metnods, the standard and smoothing interpolation, are adopted to construct radial basis function networks; while the former is the conventional method that reproduces data points strictly, the latter considers both faithfulness and smoothness of interpolation which is suitable under existence of noise. In order to take a balance between faithfulness and smoothness of interpolation, cross validation is applied to obtain an optimal one. As a result, it is confirmed that by the smoothing interpolation prediction performances are very high and estimated Lyapunov exponents are very similar to actual ones, even though in the case of periodic responses. Moreover, it is confirmed that reconstructed bifurcation diagrams are very similar to the original ones.

We propose a novel segmentation algorithm which combines an image segmentation method into small regions with chaotic neurodynamics that has already been clarified to be effective for solving some combinatorial optimization problems. The basic algorithm of an image segmentation is the variable-shape-bloch-segmentation (VB) which searches an opti-mal state of the segmentation by moving the vertices of quadran-gular regions. However, since the algorithm for moving vertices is based upon steepest descent dynamics, this segmentation method has a local minimum problem that the algorithm gets stuck at undesirable local minima. In order to treat such a problem of the VB and improve its performance, we introduce chaotic neurodynamics for optimization. The results of our novel method are compared with those of conventional stochastic dynamics for escaping from undesirable local minima. As a result, the better results are obtained with the chaotic neurodynamical image segmentation.

We analyse a mathematical neuron model with chaotic dynamics, or a chaotic neuron model by the generalized dimensions and the f(α) spectrum. The results show that the multi-fractal structure of a chaotic neuron model can be quantified by the f(α) spectrum.

Some plants have air purification ability. This purification ability of plants is considered a promising method for indoor air purification because of the low cost and high purification performance. Therefore, several studies have been carried out to investigate the relationship between the air purification ability of plants and environmental conditions. Nevertheless, the purification mechanism and process have not been clarified yet. In this paper, we investigated the air purification process in plants by bioelectrical potential analysis using linear and nonlinear analysis methods. First, we showed that two types of plants have a high air purification ability; Schefflera and Boston fern. Next, we measured AC bioelectrical potential during the purifying process of plants for pollutant gas. Then, we evaluated the power spectra of time series data of the bioelectrical potential. We found that the power spectra shifted to a lower level after gas injection over all frequencies. Thus, the higher power spectrum came from possible higher physiological activities of the plant. Finally, we introduced a nonlinear analysis method from the dynamical system theory. We transformed the time series data of the potential to a higher dimensional state space using a delay coordinate, which is often used in the field of nonlinear time series analysis. The results show that the orbits in the reconstructed state space have a large variation in gas injection. These experimental results suggest that the measurement of bioelectrical potential could become a useful method for evaluating the air purification ability of plants.

We analyze additive effects of nonlinear dynamics for conbinatorial optimization. We apply chaotic time series as noise sequence to neural networks for 10-city and 20-city traveling salesman problems and compare the performance with stochastic processes, such as Gaussian random numbers, uniform random numbers, 1/fα noise and surrogate data sets which preserve several statistics of the original chaotic data. In result, it is shown that not only chaotic noise but also surrogates with similar autocorrelation as chaotic noise exhibit high solving abilities. It is also suggested that since temporal structure of chaotic noise characterized by autocorrelation affects abilities for combinatorial optimization problems, effects of chaotic sequence as additive noise for escaping from undesirable local minima in case of solving combinatorial optimization problems can be replaced by stochastic noise with similar autocorrelation.

In this paper, we propose a new strategy of estimating correlation dimensions in combination with the method of surrogate data, which is a kind of statistical control usually introduced to avoid spurious estimates of nonlinear statistics, such as fractal dimensions, Lyapunov exponents and so on. In the case of analyzing time series with the method of surrogate data, it is desirable to decide values of estimated nonlinear statistics of the original data and surrogate data sets as exactly as possible. However, when dimensional analysis is applied to possible attractors reconstructed from real time series, it is very dangerous to decide a single value as the estimated dimensions and desirable to analyze its scaling property for avoiding spurious estimates. In order to solve this defficulty, a dimension estimator algorithm and the method of surrogate data are combined by introducing Monte Carlo hypothesis testing. In order to show effectiveness of the new strategy, firstly artificial time series are analyzed, such as the Henon map with additive noise, filtered random numbers and filtered random numbers transformed by a static monotonic nonlinearity, and then experimental time series are also examined, such as wolfer's sunspot numbers and the fluctuations in a farinfrared laser data.