The Wavelet-Galerkin method on global random processes

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Romesh Ruwan Thanuja Athuruliye Liyana Arachchige (Creator)
The University of North Carolina at Greensboro (UNCG )
Web Site:
Haimeng Zhang

Abstract: One of the main usages of covariance function or kernel is to capture the spatial or temporal dependency of a random process or random field. Covariance functions have been widely used in many areas such as environmental statistics, economics, machine learning, atmospheric sciences, imaging analysis, etc. Hence, the understanding of a covariance function is crucial to the modeling, estimation, and prediction of a random process. In this dissertation, based on Mercer’s theorem, we first modify an existing algorithm that uses the Wavelet-Galerkin method to approximate real-valued stationary covariance functions. We then apply the algorithm to approximate real-valued and complex-valued nonstationary covariance functions. In particular, we demonstrate the validity of the algorithm in the approximation of complex-valued covariance functions. In the second part of this dissertation, we apply the proposed algorithm to implement the Karhunen-Loéve expansion for studying axially symmetric Gaussian random processes on the sphere. The convergence of the truncated Karhunen-Loéve expansion to approximate axially symmetric Gaussian processes is established, and an expression for L2 error bound of the above approximation is derived. Also, we propose an efficient algorithm to generate axially symmetric Gaussian data on the sphere with a given covariance structure, and we demonstrate that our method is comparable with the classical data generation method. [This abstract has been edited to remove characters that will not display in this system. Please see the PDF for the full abstract.]

Additional Information

Language: English
Date: 2021
Random process, Wavelet-Galerkin method, Covariance function
Wavelets (Mathematics)
Galerkin methods
Stochastic processes
Analysis of covariance

Email this document to