Karhunen-Loeve Expansion and Optimal Low-Rank Model for Spatial Processes

Hao Zhang

Department of Statistics

Purdue University

Karhunen-Loeve expansion has been well-studied in theory and applied in mathematics, physics, engineering, hydrology and other disciplines. It has found recent applications in the analysis of massive spatial data when the exact likelihood function becomes numerically intractable due to the large covariance matrix. Karhunen-Loeve expansion results in an optimal approximation to the underlying spatial process by a low-rank model, and makes the analysis of massive spatial data feasible. However, we find that some well-adopted algorithms for calculating the eigenvalues and eigenfunctions do not work satisfactorily for spatial processes due to either the speed or the precision. I will introduce some new algorithms and compare their performances. I will also show some examples where we used the KL expansion for statistical estimation and prediction.