Convex Hull for Probabilistic Points

We analyze the correctness of an O(n log n) time divide-and-conquer algorithm for the convex hull problem when each input point is a location determined by a normal distribution. We show that the algorithm finds the convex hull of such probabilistic points to precision within some expected correctness determined by a user-given confidence value phi. In order to precisely explain how correct the resulting structure is, we introduce a new certificate error model for calculating and understanding approximate geometric error based on the fundamental properties of a geometric structure. We show that this new error model implies correctness under a robust statistical error model, in which each point lies within the hull with probability at least φ, for the convex hull problem

Computing Shapley Values in the Plane

We consider the problem of computing Shapley values for points in the plane, where each point is interpreted as a player, and the value of a coalition is defined by the area of usual geometric objects, such as the convex hull or the minimum axis-parallel bounding box. For sets of n points in the plane, we show how to compute in roughly O(n^{3/2}) time the Shapley values for the area of the minimum axis-parallel bounding box and the area of the union of the rectangles spanned by the origin and the input points. When the points form an increasing or decreasing chain, the running time can be improved to near-linear. In all these cases, we use linearity of the Shapley values and algebraic methods. We also show that Shapley values for the area of the convex hull or the minimum enclosing disk can be computed in O(n^2) and O(n^3) time, respectively. These problems are closely related to the model of stochastic point sets considered in computational geometry, but here we have to consider random insertion orders of the points instead of a probabilistic existence of points

Polyhedral Predictive Regions For Power System Applications

Despite substantial improvement in the development of forecasting approaches, conditional and dynamic uncertainty estimates ought to be accommodated in decision-making in power system operation and market, in order to yield either cost-optimal decisions in expectation, or decision with probabilistic guarantees. The representation of uncertainty serves as an interface between forecasting and decision-making problems, with different approaches handling various objects and their parameterization as input. Following substantial developments based on scenario-based stochastic methods, robust and chance-constrained optimization approaches have gained increasing attention. These often rely on polyhedra as a representation of the convex envelope of uncertainty. In the work, we aim to bridge the gap between the probabilistic forecasting literature and such optimization approaches by generating forecasts in the form of polyhedra with probabilistic guarantees. For that, we see polyhedra as parameterized objects under alternative definitions (under $L_1$ and $L_\infty$ norms), the parameters of which may be modelled and predicted. We additionally discuss assessing the predictive skill of such multivariate probabilistic forecasts. An application and related empirical investigation results allow us to verify probabilistic calibration and predictive skills of our polyhedra.Comment: 8 page

Convex Hulls under Uncertainty

We study the convex-hull problem in a probabilistic setting, motivated by the need to handle data uncertainty inherent in many applications, including sensor databases, location-based services and computer vision. In our framework, the uncertainty of each input site is described by a probability distribution over a finite number of possible locations including a \emph{null} location to account for non-existence of the point. Our results include both exact and approximation algorithms for computing the probability of a query point lying inside the convex hull of the input, time-space tradeoffs for the membership queries, a connection between Tukey depth and membership queries, as well as a new notion of \some-hull that may be a useful representation of uncertain hulls

The Mahler conjecture in two dimensions via the probabilistic method

The "Mahler volume" is, intuitively speaking, a measure of how "round" a centrally symmetric convex body is. In one direction this intuition is given weight by a result of Santalo, who in the 1940s showed that the Mahler volume is maximized, in a given dimension, by the unit sphere and its linear images, and only these. A counterpart to this result in the opposite direction is proposed by a conjecture, formulated by Kurt Mahler in the 1930s and still open in dimensions 4 and greater, asserting that the Mahler volume should be minimized by a cuboid. In this article we present a seemingly new proof of the 2-dimensional case of this conjecture via the probabilistic method. The central idea is to show that either deleting a random pair of edges from a centrally symmetric convex polygon, or deleting a random pair of vertices, reduces the Mahler volume with positive probability.Comment: 9 pages, 6 figures. Minor corrections from original version. To appear in Amer. Math. Monthl