Method for extracting geometrical characteristics of joint probability density based on contour lines

一种基于等高线提取联合概率密度几何特征的方法

This paper presents a method for extracting geometrical features of the joint probability density function(PDF)of two-dimensional systems based on its contour lines,with particular interests given to the number and position of peaks and craters.In order to detect those two types of structures,a series of horizontal planes are applied to truncate the joint PDF with contour lines generated.Starting with the analysis of contour lines in a single plane,shape characteristics of the peak and the crater can be reflected on the contour lines in the aspects of gradient direction and inclusion relationship.Aided by the properties of PDF,the information about gradient direction and inclusion relationship of contour lines can be obtained simultaneously if the contour tree is built.According to the contour tree,the contour lines can be classified as two groups.Then the corresponding relation between contour lines in different planes is discussed.Based on the corresponding relation,clustering analysis about contour lines belonging to the same group but having different heights is performed.Two sets of contour lines are finally obtained as the simplest expression of geometrical characteristics of a joint PDF.They can be used to obtain the number and position of each peak and crater.Three oscillators of different types are chosen to test this method,which shows that this method can pave the way for numerical calculation about the stochastic P-bifurcation of multi-dimensional systems.

文提出了一种基于联合概率密度的等高线来提取其几何特征的方法.该方法主要关注peak和crater的位置和个数.为了探测到这两种结构,这里采用了一系列水平面去截取联合概率密度的等高线.从分析位于单一平面上的等高线着手,发现peak和crater的形状特征都可以各自反映在这些等高线的梯度方向和包含关系上面.根据概率密度函数的性质,如果这些等高线对应的等高线树能够被建立起来,那么它们各自的梯度方向和相互之间的包含关系都可以同时得到.所以根据等高线树就可以把这些等高线分为两组.之后讨论了位于不同平面上等高线之间的对应关系.由这些对应关系可以对属于同一组但是却具有不同高度的等高线进行聚类分析.最终,该方法得到两个包含等高线的集合作为联合概率密度几何特征的最简表达.这两个集合可以被用来进一步提取peak和crater的位置和个数.文中选择了三个不同类型的振子对这个方法进行了检验.结果表明该方法可为多维系统的随机P分岔的计算奠定基础.