新的独立性度量及其在混沌信号分析中的应用

发现独立性可用一种简便的数格子方法估计,据此提出了“占格率”这一全新的独立性度量。接着又提出了一类无穷多种独立性度量－准熵,它们将变量经分布函数变换后量化,再用凸函数对其联合概率的均匀性进行度量,占格率的数学期望也是一种准熵,此外,还提出了广义互信息及其递归算法,将传统的互信息推广到一类无穷多种独立性度量,上述新度量对连续变量的分布不作先验假设,而是从样本中得出分布函数值的无偏估计,因此适合于根据观测值估计任意连续分布变量的独立性,本文还将新度量用于混沌信号分析,表明它们都是在经典的互信息之外的奇异吸引子重构时延的好的选取准则,并且指出Fraser和Swinney提出的著名的互信息递归算法是本文提出的广义互信息递归算法的一个特例。

New independence measures and its application to chaotic signal analysis

Independence can be estimated conveniently by "counting grid boxes" and thus a novel independence measure named "grid occupancy (GO)" is proposed. Then a class of infinitely many independence measures named quasi entropy (QE) is put forward. QE uses convex functions to evaluate the uniformity of the joint probability obtained by transforming and then quantizing variables through their distribution functions respectively. Interestingly, the expectation of GO is also a kind of QE. Moreover, the generalized mutual information (GMI) and a recursive algorithm for computing GMI are proposed, by which the traditional mutual information (MI) is generalized to a class of infinitely many independence measures. These new measures do not make priori assumptions on the distributions of continuous variables. Instead, they use unbiased estimates of the values of distribution functions obtained from the samples. Therefore, they are well suited for estimating from observations the independence of variables with arbitrary continuous distributions. These new measures are also applied to chaotic signal analysis. They are shown to be good criteria beyond the classical MI for choosing delay in strange attractor reconstruction. It is also noted that the well known recursive algorithm for computing MI by Fraser and Swinney is a special case of the recursive algorithm for computing GMI proposed in this paper.