Method Of Curve Fitting Based On Depth Functions

The concept of location depth was introduced as a way to extend the univariate notion of ranking to a bivariate configuration of data points. It has used successfully for robust estimation, hypothesis testing, and graphical display. The depth contour is the boundaries of regions of all points with equal depth, all contours form a collection of nested polygons, and the center of the deepest contour is called the tukey median. The only available implemented algorithms for the depth contours and the tukey median are slow, which limits their usefulness. In this paper we describe an algorithm which computes all bivariate depth contours in O(n~2) time and space, using topological sweep of the dual arrangement of lines. Once these contours are known, the location depth of any point can be computed in O(log~2n) time.so this algorithm solved the problem of computing speed, and make the depth function used more conveniently in practice. Some numerical examples are given here to show the algorithm's validity and effectivity. At the end of this paper, we illustrate the application of depth contours.