A Bound on the Sum of Weighted Pairwise Distances of Points Constrained to Balls

We consider the problem of choosing Euclidean points to maximize the sum of their weighted pairwise distances, when each point is constrained to a ball centered at the origin. We derive a dual minimization problem and show strong duality holds (i.e., the resulting upper bound is tight) when some locally optimal configuration of points is affinely independent. We sketch a polynomial time algorithm for finding a near-optimal set of points.Comment: Cornell ORIE Tech Repor

Optimally fast incremental Manhattan plane embedding and planar tight span construction

We describe a data structure, a rectangular complex, that can be used to represent hyperconvex metric spaces that have the same topology (although not necessarily the same distance function) as subsets of the plane. We show how to use this data structure to construct the tight span of a metric space given as an n x n distance matrix, when the tight span is homeomorphic to a subset of the plane, in time O(n^2), and to add a single point to a planar tight span in time O(n). As an application of this construction, we show how to test whether a given finite metric space embeds isometrically into the Manhattan plane in time O(n^2), and add a single point to the space and re-test whether it has such an embedding in time O(n).Comment: 39 pages, 15 figure

Score, Pseudo-Score and Residual Diagnostics for Spatial Point Process Models

We develop new tools for formal inference and informal model validation in the analysis of spatial point pattern data. The score test is generalized to a "pseudo-score" test derived from Besag's pseudo-likelihood, and to a class of diagnostics based on point process residuals. The results lend theoretical support to the established practice of using functional summary statistics, such as Ripley's $K$-function, when testing for complete spatial randomness; and they provide new tools such as the compensator of the $K$-function for testing other fitted models. The results also support localization methods such as the scan statistic and smoothed residual plots. Software for computing the diagnostics is provided.Comment: Published in at http://dx.doi.org/10.1214/11-STS367 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org