Intrinsic Gaussian processes on complex constrained domains. (19th April 2019)

• Record Type: Journal Article
• Authors: Niu, Mu; Cheung, Pokman; Lin, Lizhen; Dai, Zhenwen; Lawrence, Neil; Dunson, David
• Is Part Of: Journal of the Royal Statistical Society. Volume 81:Number 3(2019)
• Page Start: 603
• Page End: 627
• Publication Date: 2019-04-19
• Subjects: Brownian motion -- Constrained domain -- Gaussian process -- Heat kernel -- Intrinsic covariance kernel -- Manifold; Statistics -- Periodicals; Great Britain -- Statistics -- Periodicals; 519.2
• DOI: 10.1111/rssb.12320
• Languages: English
• ISSNs: 1369-7412

Abstract

Summary: We propose a class of intrinsic Gaussian processes (GPs) for interpolation, regression and classification on manifolds with a primary focus on complex constrained domains or irregularly shaped spaces arising as subsets or submanifolds of R, R 2, R 3 and beyond. For example, intrinsic GPs can accommodate spatial domains arising as complex subsets of Euclidean space. Intrinsic GPs respect the potentially complex boundary or interior conditions as well as the intrinsic geometry of the spaces. The key novelty of the approach proposed is to utilize the relationship between heat kernels and the transition density of Brownian motion on manifolds for constructing and approximating valid and computationally feasible covariance kernels. This enables intrinsic GPs to be practically applied in great generality, whereas existing approaches for smoothing on constrained domains are limited to simple special cases. The broad utilities of the intrinsic GP approach are illustrated through simulation studies and data examples.