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Spectral Methods

Spatial and Temporal Modeling and Analysis

  1. Montserrat Fuentes

Published Online: 15 JAN 2013

DOI: 10.1002/9780470057339.vnn060

Encyclopedia of Environmetrics

Encyclopedia of Environmetrics

How to Cite

Fuentes, M. 2013. Spectral Methods. Encyclopedia of Environmetrics. 5.

Author Information

  1. North Carolina State University, Raleigh, NC, USA

Publication History

  1. Published Online: 15 JAN 2013

Abstract

Spectral methods are a powerful tool for characterizing and modeling spatial dependence, while offering significant computational advantages. Using the spectral representation of a spatial process, we can easily construct valid (positive definite) covariance functions and introduce new models to characterize complex dependence structures of spatial processes. Likelihood approaches for very large spatial datasets are often very difficult, if not infeasible, to implement owing to computational limitations. Even when we can assume normality, exact calculations of the likelihood for a Gaussian spatial process observed at n locations requires O(n3) operations. The spectral version of the Gaussian log-likelihood for gridded data requires O(nlog2n) operations and does not involve calculating determinants.

Keywords:

  • covariance;
  • isotropy;
  • Fourier basis;
  • Fourier transform;
  • stationary processes;
  • spectral density;
  • spectral representation