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Considerable interest in studying the chaotic behavior of natural, physical, and socio-economic systems have led to the development of many different methods for identification and prediction of chaos. An important commonality among almost all of these methods is the concept of phase space reconstruction. Other than this, the methods largely have different bases and approaches and often aim to identify different measures of chaos. All these methods have been successfully applied in many different scientific fields. This chapter describes some of the most popular methods for chaos identification and prediction, especially those that have found applications in hydrology. These methods include: phase space reconstruction, correlation dimension method, false nearest neighbor method, Lyapunov exponent method, Kolmogorov entropy method, surrogate data method, Poincare maps, close returns plot, and nonlinear local approximation prediction method. To put the utility of these methods in a proper perspective in the identification of chaos, the superiority of two of these methods (phase space reconstruction and correlation dimension) over two commonly used linear tools for system identification (autocorrelation function and power spectrum) is also demonstrated. Further, as the correlation dimension method has been the most widely used method for chaos identification, it is discussed in far more detail.

In this paper, phase space reconstruction from stereo-electroencephalography data of ten patients with focal epilepsy forms a series of graphs. Those obtained graphs reflect the transition characteristics of brain dynamical system from pre-seizure to seizure of epilepsy. Interestingly, it is found that the rank of Laplacian matrix of these graphs has a sharp decrease when a seizure is close to happen, which thus might be viewed as a new potential biomarker in epilepsy. In addition, the reliability of this method is numerically verified with a coupled mass neural model. In particular, our simulation suggests that this potential biomarker can play the roles of predictive effect or delayed awareness, depending on the bias current of the Gaussian noise. These results may give new insights into the seizure detection.

Despite the tremendous growth in the applications of chaos theory in hydrology, there have been lingering criticisms. These criticisms have been based on the fundamental assumptions involved in the development of methods for identification and prediction of chaos (e.g. infinite and noise-free time series, lack of clear-cut guidelines on the selection of parameters involved) and/or the limitations of hydrologic data (e.g. short and noisy data, presence of zeros). A number of issues have been raised in this regard, but some have attracted far more attention than the others. This chapter presents a review of studies that have addressed such issues in chaos studies in hydrology. The review mainly focuses on four major issues: selection of an optimum delay time for phase space reconstruction, minimum data size for correlation dimension estimation, effects of data noise, and influence of the presence of zeros in data. Examples are also provided to illustrate how these issues have been addressed to gain more confidence in the applications of the methods and in the interpretation of the outcomes.

My purpose in this paper is to formulate a general completeness problem for statistical theories, and to show that the quantum theory is complete. It follows as a corollary to the completeness theorem for quantum mechanics that a phase space reconstruction of the quantum statistics is excluded, i.e., there are no ‘hidden variables’ underlying the quantum statistics.