Multiscale correlation dimension method

We propose a new method as an extension of the correlation dimension analysis by combining it with multiscale analysis taking into consideration the features in multiple time scales. We introduce and demonstrate multiscale correlation dimension analysis (MSCD) on several chaotic and stochastic time series in detail. We also study the choice of effective scaling filter as an alternative to the overlapping coarse-graining procedure we used for MSCD analysis and suggest the Gaussian filter according to its favorable performance and experiment it by assigning it for the second part of the study. Based on MSCD analysis, we further investigate CD and Hurst exponent relationship in multiscale on the same set of time series. We unveil a remarkable consistent patterns for the stochastic time series and describe it in a functional form. Consequently, the observed distinguishing patterns imply to opening up a new way of characterizing chaotic and stochastic time series.

Modelling and forecasting the hydroecological systems pollution dynamics by using a chaos theory methods: II. advanced chaos data on pollution of the Small Carpathians river’s watersheds

This paper goes on our advanced quantitative studying results of a pollution dynamics for variations hydroecological systems, namely, the nitrates etc concentrations dynamics for a number of the Small Carpathians river’s watersheds in the Eastern Slovakia. The different methods and algorithms of the chaos theory (chaos-geometric approach) and dynamical systems theory have been used in the advanced versions. New more exact data on chaotic behaviour of the nitrates concentration time series in the watersheds of the Small Carpathians are presented. In previous paper [1] to reconstruct the corresponding attractor, the time delay and embedding dimension have been needed and computed. The parameters are determined by the methods of autocorrelation function and average mutual information. Besides, there are used the advanced versions of the correlation dimension method and algorithm of false nearest neighbours. The Fourier spectrum of the con-centration of nitrates in the water catchment area Ondava: Stropkov for the period 1969-1996 is listed. Here we present new advanced data on the correlation dimension (d2), embedding dimension (dE), Kaplan-Yorke dimension (dL), average limit of predictability (Prmax) and parameter K for the nitrates concentrations in the watersheds of the Small Carpathians.

Dimension Estimation Using Weighted Correlation Dimension Method

Dimension reduction is an important tool for feature extraction and has been widely used in many fields including image processing, discrete-time systems, and fault diagnosis. As a key parameter of the dimension reduction, intrinsic dimension represents the smallest number of variables which is used to describe a complete dataset. Among all the dimension estimation methods, correlation dimension (CD) method is one of the most popular ones, which always assumes that the effect of every point on the intrinsic dimension estimation is identical. However, it is different when the distribution of a dataset is nonuniform. Intrinsic dimension estimated by the high density area is more reliable than the ones estimated by the low density or boundary area. In this paper, a novel weighted correlation dimension (WCD) approach is proposed. The vertex degree of an undirected graph is invoked to measure the contribution of each point to the intrinsic dimension estimation. In order to improve the adaptability of WCD estimation,k-means clustering algorithm is adopted to adaptively select the linear portion of the log-log sequence(log⁡δk,log⁡C(n,δk)). Various factors that affect the performance of WCD are studied. Experiments on synthetic and real datasets show the validity and the advantages of the development of technique.

The Analysis of Surface EMG Signals with the Wavelet-Based Correlation Dimension Method

Many attempts have been made to effectively improve a prosthetic system controlled by the classification of surface electromyographic (SEMG) signals. Recently, the development of methodologies to extract the effective features still remains a primary challenge. Previous studies have demonstrated that the SEMG signals have nonlinear characteristics. In this study, by combining the nonlinear time series analysis and the time-frequency domain methods, we proposed the wavelet-based correlation dimension method to extract the effective features of SEMG signals. The SEMG signals were firstly analyzed by the wavelet transform and the correlation dimension was calculated to obtain the features of the SEMG signals. Then, these features were used as the input vectors of a Gustafson-Kessel clustering classifier to discriminate four types of forearm movements. Our results showed that there are four separate clusters corresponding to different forearm movements at the third resolution level and the resulting classification accuracy was 100%, when two channels of SEMG signals were used. This indicates that the proposed approach can provide important insight into the nonlinear characteristics and the time-frequency domain features of SEMG signals and is suitable for classifying different types of forearm movements. By comparing with other existing methods, the proposed method exhibited more robustness and higher classification accuracy.

Prediction of the Reference Evapotranspiration Using a Chaotic Approach

Evapotranspiration is one of the most important hydrological variables in the context of water resources management. An attempt was made to understand and predict the dynamics of reference evapotranspiration from a nonlinear dynamical perspective in this study. The reference evapotranspiration data was calculated using the FAO Penman-Monteith equation with the observed daily meteorological data for the period 1966–2005 at four meteorological stations (i.e., Baotou, Zhangbei, Kaifeng, and Shaoguan) representing a wide range of climatic conditions of China. The correlation dimension method was employed to investigate the chaotic behavior of the reference evapotranspiration series. The existence of chaos in the reference evapotranspiration series at the four different locations was proved by the finite and low correlation dimension. A local approximation approach was employed to forecast the daily reference evapotranspiration series. Low root mean square error (RSME) and mean absolute error (MAE) (for all locations lower than 0.31 and 0.24, resp.), high correlation coefficient (CC), and modified coefficient of efficiency (for all locations larger than 0.97 and 0.8, resp.) indicate that the predicted reference evapotranspiration agrees well with the observed one. The encouraging results indicate the suitableness of chaotic approach for understanding and predicting the dynamics of the reference evapotranspiration.

Downstream prediction using a nonlinear prediction method

The estimation of river flow is significantly related to the impact of urban hydrology, as this could provide information to solve important problems, such as flooding downstream. The nonlinear prediction method has been employed for analysis of four years of daily river flow data for the Langat River at Kajang, Malaysia, which is located in a downstream area. The nonlinear prediction method involves two steps; namely, the reconstruction of phase space and prediction. The reconstruction of phase space involves reconstruction from a single variable to the m-dimensional phase space in which the dimension m is based on optimal values from two methods: the correlation dimension method (Model I) and false nearest neighbour(s) (Model II). The selection of an appropriate method for selecting a combination of preliminary parameters, such as m, is important to provide an accurate prediction. From our investigation, we gather that via manipulation of the appropriate parameters for the reconstruction of the phase space, Model II provides better prediction results. In particular, we have used Model II together with the local linear prediction method to achieve the prediction results for the downstream area with a high correlation coefficient. In summary, the results show that Langat River in Kajang is chaotic, and, therefore, predictable using the nonlinear prediction method. Thus, the analysis and prediction of river flow in this area can provide river flow information to the proper authorities for the construction of flood control, particularly for the downstream area.

Hydrologic system complexity and nonlinear dynamic concepts for a catchment classification framework

The absence of a generic modeling framework in hydrology has long been recognized. With our current practice of developing more and more complex models for specific individual situations, there is an increasing emphasis and urgency on this issue. There have been some attempts to provide guidelines for a catchment classification framework, but research in this area is still in a state of infancy. To move forward on this classification framework, identification of an appropriate basis and development of a suitable methodology for its representation are vital. The present study argues that hydrologic system complexity is an appropriate basis for this classification framework and nonlinear dynamic concepts constitute a suitable methodology. The study employs a popular nonlinear dynamic method for identification of the level of complexity of streamflow and for its classification. The correlation dimension method, which has its base on data reconstruction and nearest neighbor concepts, is applied to monthly streamflow time series from a large network of 117 gaging stations across 11 states in the western United States (US). The dimensionality of the time series forms the basis for identification of system complexity and, accordingly, streamflows are classified into four major categories: low-dimensional, medium-dimensional, high-dimensional, and unidentifiable. The dimension estimates show some "homogeneity" in flow complexity within certain regions of the western US, but there are also strong exceptions.

Structural Analysis and Building an EDM Process Model

The insufficient information about the mechanism of EDM process affects decision-making related to operation, parameter adjustment, and automatic control. Without a thorough understanding of the process dynamical features, it is difficult to determine a suitable model to describe such processes. This research employs the chaotic analytical techniques to determine the level of complexity of an EDM process; the correlation dimension method. With the understanding of the process complexity and its composition by correlation dimension analysis, a well-defined model is proposed. Finally, by using this model structure and size, an online time-varied predictive model was developed and verified by another experiment.