Unsupervised Bump Hunting Using Principal Components

Unsupervised Bump Hunting Using Principal Components Daniel A D ıaz-Pach on Jean-Eudes Dazard † J. Sunil Rao‡ March 14, 2018 Abstract Principal Components Analysis is a widely used technique for dimension reduction and characterization of variability in multivariate populations. Our interest lies in studying when and why the rotation to principal components can be used effectively within a response-predictor set relationship in the context of mode hunting. Specifically focusing on the Patient Rule Induction Method (PRIM), we first develop a fast version of this algorithm (fastPRIM) under normality which facilitates the theoretical studies to follow. Using basic geometrical arguments, we then demonstrate how the PC rotation of the predictor space alone can in fact generate improved mode estimators. Simulation results are used to illustrate our findings. Key words: Algorithms, Bump hunting, Computationally intensive methods, Mode hunting, Principal components. 1 Introduction The PRIM algorithm for bump hunting was first developed by Friedman and Fisher (1999). It is an intuitively useful computational algorithm for the detection of local maxima (or minima) on target functions. Roughly speaking, PRIM peels the (conditional) distribution of a response from the outside in, leaving at the end rectangular boxes which are supposed to contain a bump (see the formal description in Algorithm 1) at page 5. However, some shortcomings against this procedure have also appeared in the literature when several dimensions are under consideration. For instance, as Polonik and Wang (2010) explained it, the method could fail when there are two or more modes in high-dimensional settings.