Fast Computation of 2-Dimensional Depth Contours

"One person’s noise is another person’s signal." For many applications, including the detection of credit card frauds and the monitoring of criminal activities in electronic commerce, an important knowledge discovery problem is the detection of exceptional/outlying events. In computational statistics, a depth-based approach detects outlying data points in a 2-D dataset by, based on some definition of depth, organizing the data points in layers, with the expectation that shallow layers are more likely to contain outlying points than are the deep layers. One robust notion of depth, called depth contours, was introduced by Tukey. ISODEPTH, developed by Ruts and Rousseeuw, is an algorithm that computes 2-D depth contours. In this paper, we give a fast algorithm, FDC, which computes the first k 2-D depth contours by restricting the computation to a small selected subset of data points, instead of examining all data points. Consequently, FDC scales up much better than ISODEPTH. Also, while ISODEPTH relies on the non-existence of collinear points, FDC is robust against collinear points.