  Eigenvector spatial filtering (ESF) is a spatial modeling approach, which has
been applied in urban and regional studies, ecological studies, and so on.
However, it is computationally demanding, and may not be suitable for large
data modeling. The objective of this study is developing fast ESF and random
effects ESF (RE-ESF), which are capable of handling very large samples. To
achieve it, we accelerate eigen-decomposition and parameter estimation, which
make ESF and RE-ESF slow. The former is accelerated by utilizing the Nystrom
extension, whereas the latter is by small matrix tricks. The resulting fast ESF
and fast RE-ESF are compared with non-approximated ESF and RE-ESF in Monte
Carlo simulation experiments. The result shows that, while ESF and RE-ESF are
slow for several thousand samples, fast ESF and RE-ESF require only several
seconds for the samples. They also suggest that the proposed approaches
effectively remove positive spatial dependence in the residuals with very small
approximation errors when the number of eigenvectors considered is 200 or more.
Note that these approaches cannot deal with negative spatial dependence. The
proposed approaches are implemented in an R package "spmoran."
