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Journal of Geographical Systems
Published in: Journal of Geographical Systems 4/2015

01-10-2015 | Original Article

Random effects specifications in eigenvector spatial filtering: a simulation study

Authors: Daisuke Murakami, Daniel A. Griffith

Published in: Journal of Geographical Systems | Issue 4/2015

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Abstract

Eigenvector spatial filtering (ESF) is becoming a popular way to address spatial dependence. Recently, a random effects specification of ESF (RE-ESF) is receiving considerable attention because of its usefulness for spatial dependence analysis considering spatial confounding. The objective of this study was to analyze theoretical properties of RE-ESF and extend it to overcome some of its disadvantages. We first compare the properties of RE-ESF and ESF with geostatistical and spatial econometric models. There, we suggest two major disadvantages of RE-ESF: it is specific to its selected spatial connectivity structure, and while the current form of RE-ESF eliminates the spatial dependence component confounding with explanatory variables to stabilize the parameter estimation, the elimination can yield biased estimates. RE-ESF is extended to cope with these two problems. A computationally efficient residual maximum likelihood estimation is developed for the extended model. Effectiveness of the extended RE-ESF is examined by a comparative Monte Carlo simulation. The main findings of this simulation are as follows: Our extension successfully reduces errors in parameter estimates; in many cases, parameter estimates of our RE-ESF are more accurate than other ESF models; the elimination of the spatial component confounding with explanatory variables results in biased parameter estimates; efficiency of an accuracy maximization-based conventional ESF is comparable to RE-ESF in many cases.
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Footnotes
1
The expectation of MC, E[MC], equals \( - \frac{1}{N - 1} \) when M = I–11′/N, whereas it equals \( - \frac{N}{{{\mathbf{1^{\prime}C1}}}}\frac{{tr[({\mathbf{X^{\prime}X}})^{ - 1} {\mathbf{X^{\prime}CX}}]}}{N - K - 1} \) when M = I–X(X′X) -1 X′. E[MC] < MC, MC < E[MC], and MC = E[MC] imply positive, negative, and no spatial dependence, respectively.
 
2
\( {\hat{\mathbf{\beta }}} \) in Eq. (5) is identical to the generalized least squared estimator (Henderson 1975).
 
3
Models with known mean structure are widely applied in geostatistics (e.g., Cressie and Wikle 2011).
 
4
Although the covariance matrix of ω usually is defined by σ NS 2 I + σ γ 2 (I + C), where σ NS 2 is a variance parameter, because it can be expanded as follows: σ NS 2 I + σ γ 2 (I + C) = (σ NS 2 +σ 2γ)I + σ γ 2 C = σ 2 I + σ γ 2 C, the assumption of zero diagonal elements is consistent with the usual assumption.
 
5
When M = I–11 ′/N, RE-ESF approximates the geostatistical model with known constant mean.
 
6
The asymmetric matrix W has real eigenvalues and eigenvectors (see Griffith 2000).
 
7
Expanding the variance of Eγ in Eq. (22) as \( {\text{Var}}[{\mathbf{E\gamma }}] = \frac{{{\mathbf{\gamma^{\prime}E^{\prime}E\gamma }}}}{N} = \frac{{{\mathbf{\gamma^{\prime}\gamma }}}}{N} = \frac{1}{N}\sum\nolimits_{l} {{\text{diag}}[\sigma_{\gamma }^{2} k{\varvec{\Lambda}}(\alpha )]_{l} } = k\frac{{\sigma_{\gamma }^{2} }}{N}\sum\nolimits_{l} {\lambda_{l}^{\alpha } } \), where \( {\text{diag}}[ \cdot ]_{l} \) returns the lth diagonal of the matrix \( \cdot \). This equation shows that Var[Eγ] equals σ γ 2 when \( k = \frac{N}{{\sum\nolimits_{l} {\lambda_{l}^{\alpha } } }} \).
 
8
Equation (5) is expanded using X ′ E = 0, which M = I–X(X ′ X) -1 X ′ implies, as \( \left[ {\begin{array}{*{20}c} {{\hat{\mathbf{\beta }}}} \\ {{\hat{\mathbf{\gamma }}}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{X^{\prime}X}}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{I}} + \frac{{\sigma_{{}}^{2} }}{{\sigma_{\gamma }^{2} }}{\varvec{\Lambda}}^{ - 1} } \\ \end{array} } \right]^{ - 1} \left[ {\begin{array}{*{20}c} {{\mathbf{X^{\prime}y}}} \\ {{\mathbf{E^{\prime}y}}} \\ \end{array} } \right] \) \( = \left[ {\begin{array}{*{20}c} {({\mathbf{X^{\prime}X}})^{ - 1} {\mathbf{X^{\prime}y}}} \\ {\left( {{\mathbf{I}} + \frac{{\sigma_{{}}^{2} }}{{\sigma_{\gamma }^{2} }}{\varvec{\Lambda}}^{ - 1} } \right)^{ - 1} {\mathbf{E^{\prime}y}}} \\ \end{array} } \right] \). Thus, \( {\hat{\mathbf{\beta }}} \) equals the estimate of the OLS estimate.
 
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Metadata
Title
Random effects specifications in eigenvector spatial filtering: a simulation study
Authors
Daisuke Murakami
Daniel A. Griffith
Publication date
01-10-2015
Publisher
Springer Berlin Heidelberg
Published in
Journal of Geographical Systems / Issue 4/2015
Print ISSN: 1435-5930
Electronic ISSN: 1435-5949
DOI
https://doi.org/10.1007/s10109-015-0213-7

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