Rejoinder: Discussion of Bump Hunting in High Dimensional Data

We thank all of the discussants for taking the time to contribute, and for both their compliments and helpful criticism. Before responding in detail, we address two general concerns that ran throughout many of the discussions. 1 Computation Several discussants raised the issue of the computational requirements of the peeling algorithm (Section 7). They were concerned about its ability to scale to large (in memory) data sets. Here we provide some insight in the form of a worst case analysis. Let be the minimum fraction of observations permitted to be trimmed from each box in the peeling sequence. The current version of the software (Friedman 1998) has been modiied to incorporate such a limit for both real{valued and categorical variables. For the latter, this implies that more than one value may need to be trimmed from the current box at some iterations. This requires no additional computation since the values can be considered in the order of their individual output means within each current box. Initially, before any peeling trajectories are constructed, the values of each real{valued variable are sorted in ascending order. The corresponding order information is stored as a linked list so that individual deletions can be performed later in constant time. This requires an initial computation proportional to n R N log N, where n R is the number of real{valued variables and N is the training sample size. An upper bound L on the number of boxes in each trajectory is given by L = log 0 log(1 ?) where 0 is the speciied minimum box size (7.4). Here it is assumed that the minimum fraction of observations is peeled at each iteration of the trajectory. In this case, the number of observations in the current box at the lth step is N l = N(1 ?) l : For each categorical input variable the computation required to compute the peeling criterion (14.2) is proportional to N l. For real{valued variables it is N l since the linked lists can be used, along with updating formulae for the means, so that only the ?fraction of the observations at the extremes need be considered. The total computation bound for producing a peeling trajectory in this case is proportional to