Abstract
Let P be a set of points in R^d, and let M be a function that maps any subset of P to a positive real number. We examine the problem of computing the exact mean and variance of M when a subset of points in P is selected according to a well-defined random distribution. We consider two distributions; in the first distribution (which we call the Bernoulli distribution), each point p in P is included in the random subset independently, with probability pi(p). In the second distribution (the fixed-size distribution), a subset of exactly s points is selected uniformly at random among all possible subsets of s points in P. This problem is a crucial part of modern ecological analyses; each point in P represents a species in d-dimensional trait space, and the goal is to compute the statistics of a geometric measure on this trait space, when subsets of species are selected under random processes. We present efficient exact algorithms for computing the mean and variance of several geometric measures when point sets are selected under one of the described random distributions. More specifically, we provide algorithms for the following measures: the bounding box volume, the convex hull volume, the mean pairwise distance (MPD), the squared Euclidean distance from the centroid, and the diameter of the minimum enclosing disk. We also describe an efficient (1-e)-approximation algorithm for computing the mean and variance of the mean pairwise distance. We implemented three of our algorithms and we show that our implementations can provide major speedups compared to the existing imprecise approaches.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.