时间序列分维的改进 GP 算法

The Novel GP Algorithm of Fractal Dimension of TimeSeries

在拓扑等价的意义上，证明了系统单变量时间序列混沌吸引子的分维与度量无关，改进了计算分维的ＧＰ算法（ＮＧＰ），给出了递推ＧＰ算法，并利用此算法计算了Ｈｅｎｏｎ吸引子和Ｌｏｒｅｎｚ吸引子的分维，通过比较发现运算速度显著提高，增强了算法的实用性。

The problems of detecting deterministic chaos in experimental data, and of studying the fractal characterization of chaotic attractor, have stimulated the development of new methods for analyzing time series. Especially the numerical algorithm of fractal dimension of time series is paid much attention. A real physical system may involve many variables but only one or more of them can be detected by modern data collecting equipment. In recent years, the technique of reconstruction of state space is applied to time series analyzing and processing. Its significance is that one can obtain the topological characteristics such as time series fractal dimension. Grassberger and Procaccia put forward the GP algorithm of fractal dimension. But because of the adoption of 2norm, the GP algorithm contains many duplicated calculations and needs large number of samples. This restricts the effectiveness and practicability of GP algorithm to a certain extent. In this paper, under the meaning of topological equivalence, the theorem that the fractal dimensional of reconstructed chaotic attractor is independent of the norm of R 2 is put forward and proven. Based on this theorem, the novel GP algorithm is proposed. As a result, this algorithm leads to better accuracy of fractal dimension, low requirement of hard ware, and more practicability. The novel GP algorithm is applied to Henon map and Lorenz system as examples respectively and is proven to be effective and practical. Fractal dimension describes not only the dynamic characteristic about the system but also the number of variables about the stable attractor. The novel GP algorithm provides a new concept of time series analyzing and processing. This method raises highly the ability to understand the complex system by physics experiment. On the other hand, the technique is simple and easy. The integration of the traditional time series analysis methods with the modern chaotic and fractal theory has shown attractive promise.