Clusters of high-dimensional interval data and related Boolean functions of events in Euclidean space

Abstract

Clustering interval data has been studied for decades. High-dimensional interval data can be expressed in terms of hyperrectangles in $$\mathbb {R}^d$$ (or d-orthotopes) in case of real-valued d-attributes data. This paper investigates such high-dimensional interval data: the Cartesian product of intervals, or a vector of interval. For the efficient computation of related Boolean functions, some interesting aspects have been discovered using vertices and edges of the graph, generated from given events. We also study the lower and upper-bounded orthants in $$\mathbb {R}^d$$ as events for which we show the existence of a polynomial-time algorithm to calculate the probability of the union of such events. This efficient algorithm has been discovered by constructing a suitable partial order relation based on a recursive projection onto lower-dimensional spaces. Illustrative real-life applications are presented.