Quantile approximation for robust statistical estimation and kappa-enclosing problems
Given a set P of n points in R-d, a fundamental problem in computational geometry is concerned with finding the smallest shape of some type that encloses all the points of P. Well-known instances of this problem include finding the smallest enclosing box, minimum volume ball, and minimum volume annulus. In this paper we consider the following variant: Given a set of n points in R-d, find the smallest shape in question that contains at least Ic points or a certain quantile of the data. This type of problem is known as a Ic-enclosing problem. We present a simple algorithmic framework for computing quantile approximations for the minimum strip, ellipsoid, and annulus containing a given quantile of the points. The algorithms run in O(n log n) time.
robust estimation lms regression minimum enclosing disk minimum volume ball/ellipsoid/annulus estimator