Design of monitoring networks using k-determinantal point processes

In this paper, we introduce a design strategy based on k‐determinantal point processes (k‐DPPs). The k‐DPP design is a flexible design scheme that is able to yield spatially balanced designs while also imposing diversity of the selected locations based on extra sources of information known to be related to the underlying process of interest. The methodology is able to handle both the designing and the redesigning of a monitoring network. In particular, we discuss how a k‐DPP design can be used as a randomized alternative for the space‐filling designs when the objective is to provide a good spatial coverage of the region of interest. Furthermore, we discuss how the k‐DPP optimal design objective is remarkably similar to that of entropy design for Gaussian fields. Because the optimization for entropy designs is a NP‐hard problem, we explore an approximate solution based on a k‐DPP sampling design strategy. Through a case study of augmentation of a network for monitoring temperatures, we illustrate how a k‐DPP sampling design strategy can yield an approximation for the entropy solution.