Dynamic Update of Half-space Depth Contours

Data depth is an approach to statistical analysis based on the geometry of the data. Half-space depth has been studied most frequently by computational geometers. The half-space depth of a point x relative to a set of points S = {X1, ..., Xn} in R d is the minimum number of points of S lying in any closed half-space determined by a line through x [2, 11]. Depth contours, enclosing regions with increasing depth, help to visualize, quantify and compare data sets. Prior work investigated combinatorial properties and algorithms for computation of depth contours for static data sets. We present a dynamic algorithm for computing the twodimensional rank-based half-space depth contours of a set of n points in O(n log n) time per operation and in O(n) overall space, an improvement over the static version of O(n) time per operation. The same algorithm can compute the half-space depth of a single point relative to a data set dynamically in O(log n) time and O(n) space. The algorithm does not compute the entire set of contours explicitly but maintains the order (ranking) of points according to their half-space depth. A constant number of contours (e.g. 10%, · · · 100%) can be constructed in O(n) time from the sorted list of the data points, ranked by depth. Our algorithm uses generalized dynamic segment trees to update the depth of every data point and is based on key characterizations of the potential changes in the depth contours upon insertions or deletions. We only consider data sets in general position.