Statistics Character and Complexity in Nonlinear Systems
Yagang Zhang, Zengping Wang · Machine Learning · 2010
The statistic character and complexity in nonlinear systems have been clarified in this chapter. These stochastic symbolic sequences bear three characters. In two kinds of typical nonlinear systems-unimodal surjective map and Lorenz type maps nonlinear systems, the distributions of frequency, inter-occurrence times, first passage time and visitation density in unimodal surjective map and Lorenz type maps are discussed carefully. These two kinds of nonlinear systems have same distributions, which have also been explained in theory, and the catholicity of the statistic character has been elicited. In the 4-symbolic dynamics, the distribution of frequency, inter-occurrence times and the alignment of two random sequences have been amplified in detail. By using transfer probability of Markov chain (MC), we have obtained analytic expressions of generating functions in four probabilities stochastic wander model, which can be applied to all 4-symbolic systems. So, a perfect symbolic platform has been set up for our utilizing statistic character, in fact, it is a stochastic signal platform of symbolic simulation. The 4-symbolic sequences have natural relations with bioinformatics sequences, in the field of application, we hope to afford a symbolic platform which satisfies these statistic character and study some properties of DNA sequences (Hao, 2000; Hao et al., 2000; Bershadskii, 2001; Grimm & Rupprecht, 1997; Allegrini et al., 1996; Natalia & Avy, 2005; Elena et al., 2005), 20 amino acids symbolic sequences of protein structure, and the time series that can be symbolic in finance market et al, which are part of our future work. The symbolic platform provides a set of effective