An Empirical Bayes Approach for Distributed Estimation of Spatial Fields

TL;DR

This paper introduces a distributed estimation method for spatial fields using an Empirical Bayes approach with Gaussian Processes, accommodating heterogeneity and leveraging sparsity for efficient computation.

Contribution

It presents a novel distributed estimator that handles heterogeneous sensor data and combines physics-based and data-driven models within an Empirical Bayes framework.

Findings

Effective estimation of temperature fields using PDE models.

Successful data-driven inference with spline parametrization.

Distributed algorithm performs well with sparse covariance structures.

Abstract

In this paper we consider a network of spatially distributed sensors which collect measurement samples of a spatial field, and aim at estimating in a distributed way (without any central coordinator) the entire field by suitably fusing all network data. We propose a general probabilistic model that can handle both partial knowledge of the physics generating the spatial field as well as a purely data-driven inference. Specifically, we adopt an Empirical Bayes approach in which the spatial field is modeled as a Gaussian Process, whose mean function is described by means of parametrized equations. We characterize the Empirical Bayes estimator when nodes are heterogeneous, i.e., perform a different number of measurements. Moreover, by exploiting the sparsity of both the covariance and the (parametrized) mean function of the Gaussian Process, we are able to design a distributed spatial field estimator. We corroborate the theoretical results with two numerical simulations: a stationary temperature field estimation in which the field is described by a partial differential (heat) equation, and a data driven inference in which the mean is parametrized by a cubic spline.