Adaptive estimation of external fields in reproducing kernel Hilbert spaces

Abstract

Summary This article studies the distributed parameter system that governs adaptive estimation by mobile sensor networks of external fields in a reproducing kernel Hilbert space (RKHS). The article begins with the derivation of conditions that guarantee the well‐posedness of the ideal, infinite dimensional governing equations of evolution for the centralized estimation scheme. Subsequently, convergence of finite dimensional approximations is studied. Rates of convergence in all formulations are established using history‐dependent bases defined from translates of the RKHS kernel that are centered at sample points along the agent trajectories. Sufficient conditions are derived that ensure that the finite dimensional approximations of the ideal estimator equations converge at a rate that is bounded by the fill distance of samples in the agents' assigned subdomains. The article concludes with examples of simulations and experiments that illustrate the qualitative performance of the introduced algorithms.

Classification of Human Learning Stages via Kernel Distribution Embeddings

“…Many of these approaches are based in Gaussian mixture models [39] and mixture models more generally [40], with application to car following [41], [42], [43], [44], driver influence via economic models [45], [46], and robotic manipulation [47], [48]. We choose reproducing kernel Hilbert spaces (RKHS) based tools because a) they are amenable to non-parametric modeling, meaning that unlike many machine learning approaches such as mixture models, there are very few parameters that are required, and hence less vulnerability to the need for excessive tuning; b) they established tools for statistical inference and estimation [27], [49], [50], and are gaining traction as tools for data-driven verification [51], [52] and control [53], [54] of dynamical systems; and c) numerical methods scale with the number of samples of observed data, as opposed to dimension of the state, which makes them a promising approach for extension to run-time adaptive automation.…”