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

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.