nach oben

Journal of Geographical Systems

Erschienen in:

01.10.2015 | Original Article

Random effects specifications in eigenvector spatial filtering: a simulation study

verfasst von: Daisuke Murakami, Daniel A. Griffith

Erschienen in: Journal of Geographical Systems | Ausgabe 4/2015

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

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.

Vorheriger Artikel Software for Bayesian cross section and panel spatial model comparison

Nächster Artikel Optimizing distance-based methods for large data sets

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

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.

\( {\hat{\mathbf{\beta }}} \) in Eq. (5) is identical to the generalized least squared estimator (Henderson 1975).

Models with known mean structure are widely applied in geostatistics (e.g., Cressie and Wikle 2011).

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.

When M = I–11 ′/N, RE-ESF approximates the geostatistical model with known constant mean.

The asymmetric matrix W has real eigenvalues and eigenvectors (see Griffith 2000).

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 \( k = \frac{N}{{\sum\nolimits_{l} {\lambda_{l}^{\alpha } } }} \).

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.

Anselin L, Rey S (1991) Properties of tests for spatial dependence in linear regression models. Geogr Anal 23(2):112–131CrossRef

Arbia G, Fingleton B (2008) New spatial econometric techniques and applications in Regional Science. Pap Reg Sci 87(3):311–317CrossRef

Bates DM (2010) lme4: Mixed-effects modeling with R. http://lme4.r-forge.r-project.org/book

Blancheta FG, Legendre P, Borcard D (2008) Modelling directional spatial process in ecological data. Ecol Model 215(4):325–336CrossRef

Borcard D, Legendre P (2002) All-scale spatial analysis of ecological data by means of principal coordinates of neighbor matrices. Ecol Model 153(1–2):51–68CrossRef

Chun Y (2008) Modeling network autocorrelation within migration flows by eigenvector spatial filtering. J Geogr Syst 10(4):317–344CrossRef

Chun Y (2014) Analyzing space–time crime incidents using eigenvector spatial filtering: an application to vehicle burglary. Geogr Anal 46(2):165–184CrossRef

Chun Y, Griffith DA (2011) Modeling network autocorrelation in space–time migration flow data: an eigenvector spatial filtering approach. Ann Assoc Am Geogr 101(3):523–536CrossRef

Chun Y, Griffith DA (2013) Spatial statistics and geostatistics: theory and applications for geographic information science and technology. SAGE, London

Chun Y, Griffith DA (2014) A quality assessment of eigenvector spatial filtering based parameter estimates for the normal probability model. Spat Stat 10:1–11MathSciNetCrossRef

Cressie N (1993) Statistics for spatial data. John Wiley & Sons, New York

Cressie N, Wikle CK (2011) Statistics for spatio-temporal data. Wiley, New YorkMATH

Cuaresma CJ, Feldkircher M (2013) Spatial filtering, model uncertainty and the speed of income convergence in Europe. J Appl Econom 28(4):720–741CrossRef

Dray S, Legendre P, Peres-Neto PR (2006) Spatial modeling: a comprehensive framework for principal coordinate analysis of neighbor matrices (PCNM). Ecol Model 196(3–4):483–493CrossRef

Drineas P, Mahoney MW (2005) On the Nyström method for approximating a gram matrix for improved kernel-based learning. J Mach Learn Res 6:2153–2175MathSciNetMATH

Elhorst JP (2010) Applied spatial econometrics: raising the bar. Spat Econ Anal 5(1):9–28CrossRef

Fischer MM, Griffith DA (2008) Modeling spatial autocorrelation in spatial interaction data: an application to patent citation data in European Union. J Reg Sci 48(5):969–989CrossRef

Franzese RJ, Hays JC (2014) Testing for spatial-autoregressive lag versus (unobserved) spatially correlated error-components. Benjamin F. Shambaugh conference: New frontiers in the study of policy diffusion, University of Iowa, Iowa City

Getis A (1990) Screening for spatial dependence in regression analysis. Paper Reg Sci Assoc 69(1):69–81CrossRef

Getis A, Griffith DA (2002) Comparative spatial filtering in regression analysis. Geogr Anal 34(2):130–140CrossRef

Getis A, Ord K (1992) The analysis of spatial association by use of distance statistics. Geogr Anal 24(3):189–206CrossRef

Griffith DA (1996) Spatial autocorrelation and eigenfunctions of the geographic weights matrix accompanying georeferenced data. Can Geogr 40(4):351–367CrossRef

Griffith DA (2000) Eigenfunction properties and approximations of selected incidence matrices employed in spatial analyses. Linear Algebr Appl 321(1–3):95–112MathSciNetCrossRefMATH

Griffith DA (2002) A spatial filtering specification of the auto-Poisson model. Stat Prob Letters 58(3):245–251MathSciNetCrossRefMATH

Griffith DA (2003) Spatial autocorrelation and spatial filtering: Gaining understanding through theory and scientific visualization. Springer, BerlinCrossRef

Griffith DA (2004a) A spatial filtering specification for the autologistic model. Environ Plan 36(10):1791–1811CrossRef

Griffith DA (2004b) Distributional properties of georeferenced random variables based on the eigenfunction spatial filter. J Geogr Syst 6(3):263–288MathSciNetCrossRef

Griffith DA (2006) Hidden negative spatial autocorrelation. J Geogr Syst 8(4):335–355CrossRef

Griffith DA (2008) Spatial-filtering-based contributions to a critique of geographically weighted regression (GWR). Environ Plan 40(11):2751–2769CrossRef

Griffith DA (2009) Modeling spatial autocorrelation in spatial interaction data: empirical evidence from 2002 Germany journey-to-work flows. J Geogr Syst 11(2):117–140CrossRef

Griffith DA (2010) Spatial filtering. In: Fischer MM, Getis A (eds) Handbook of applied spatial analysis. Springer, Berlin, pp 301–318CrossRef

Griffith DA (2011) Visualizing analytical spatial autocorrelation components latent in spatial interaction data: an eigenvector spatial filter approach. Comput Environ Urban Syst 35(2):140–149MathSciNetCrossRef

Griffith DA, Chun Y (2013) Spatial autocorrelation and spatial filtering. In: Fischer MM, Nijkamp P (eds) Handbook of regional science. Springer, Berlin, pp 1477–1507

Griffith DA, Csillag F (1993) Exploring relationship between semi-variogram and spatial autoregressive models. Paper Reg Sci 72(3):283–295CrossRef

Griffith DA, Fischer MM (2013) Constrained variants of the gravity model and spatial dependence: model specification and estimation issues. J Geogr Syst 15(3):291–317CrossRef

Griffith DA, Layne L (1997) Uncovering relationships between geo-statistical and spatial autoregressive models. In: American Statistical Association (ed) Proceedings of the section on statistics and the environment 1997, American Statistical Association, Alexandria, Virginia, pp 91–96

Griffith DA, Paelinck JHP (2011) Non–standard spatial statistics and spatial econometrics. Springer, BerlinCrossRef

Griffith DA, Peres-Neto PR (2006) Spatial modeling in ecology: the flexibility of eigenfunction spatial analyses in exploiting relative location information. Ecol 87(10):2603–2613CrossRef

Grimpe C, Patuelli R (2009) Regional knowledge production in nanomaterials: a spatial filtering approach. Ann Reg Sci 46(3):519–541CrossRef

Haskard KA, Lark RM (2009) Modelling non-stationary variance of soil properties by tempering an empirical spectrum. Geoderma 153(1–2):18–28CrossRef

Helbich M, Arsanjani JJ (2014) Spatial eigenvector filtering for spatiotemporal crime mapping and spatial crime analysis. Cartogr Geogr Inf Sci 42(2):134–148CrossRef

Henderson CR (1975) Best linear unbiased estimation and prediction under a selection model. Biometrics 31(2):423–447CrossRefPubMedMATH

Hodges JS (2013) Richly parameterized linear models: additive, time series, and spatial models using random effects. CRC Press, Boca Raton

Hodges JS, Reich BJ (2010) Adding spatially-correlated errors can mess up the fixed effect you love. Am Stat 64(4):325–334MathSciNetCrossRefMATH

Hughes J, Haran M (2013) Dimension reduction and alleviation of confounding for spatial generalized linear mixed models. J R Stat Soc B Stat Methodol 75(1):139–159MathSciNetCrossRef

Jacob GB, Muturi EJ, Caamano EX, Gunter JT et al (2008) Hydrological modeling of geophysical parameters of arboviral and protozoan disease vectors in Internally Displaced People camps in Gulu, Uganda. Int J Health Geogr 7:11PubMedCentralCrossRefPubMed

Legendre P, Legendre L (2012) Numerical Ecology, 3rd edn. Elsevier Science BV, Amsterdam

LeSage JP, Pace RK (2009) Introduction to spatial econometrics. Chapman & Hall/CRC, Boca RatonCrossRefMATH

LeSage JP, Pace RK (2014) The biggest myth in spatial econometrics. Econom 2(4):217–249

LeSage JP, Parent O (2007) Bayesian model averaging for spatial econometric models. Geogr Anal 39(3):241–267CrossRef

Moniruzzaman M, Paez A (2012) A model-based approach to select case sites for walkability audits. Health Place 18(6):1323–1334CrossRefPubMed

Moran PAP (1950) Notes on continuous stochastic phenomena. Biom 37(1/2):17–23MATH

Pace RK, LeSage JP (2010) Omitted variable biases of OLS and spatial lag models. In: Páez A, Gallo J, Buliung RN, Dall’erba S (eds) Progress in spatial analysis. Springer, Berlin, pp 17–28CrossRef

Pace RK, LeSage JP, Zhu S (2013) Interpretation and computation of estimates from regression models using spatial filtering. Spat Econ Anal 8(3):352–369CrossRef

Paciorek CJ (2010) The importance of scale for spatial-confounding bias and precision of spatial regression estimators. Stat Sci 25(1):107–125PubMedCentralMathSciNetCrossRefPubMed

Patuelli R, Griffith DA, Tiefelsdorf M, Nijkamp P (2011) Spatial filtering and eigenvector stability: space-time models for German unemployment data. Int Reg Sci Rev 34(2):253–280CrossRef

Peres-Neto PR, Legendre P, Dray S, Borcard D (2006) Variation partitioning of species data matrices: estimation and comparison of fractions. Ecol Soc Am 87(10):2614–2625

Pintore A, Holmes CC (2004) Non-stationary covariance functions via spatially adaptive spectra. University of Oxford, Oxford

Scherngell T, Lata R (2013) Towards an integrated European Research Area? Findings from eigenvector spatially filtered spatial interaction models using European Framework Programme data. Pap Reg Sci 92(3):555–577

Seya H, Tsutsumi M, Yamagata Y (2014a) Weighted-average least squares applied to spatial econometric models: a Monte Carlo investigation. Geogr Anal 46(2):126–147CrossRef

Seya H, Murakami D, Tsutsumi M, Yamagata Y (2014b) Application of LASSO to the eigenvector selection problem in eigenvector-based spatial filtering. Geogr Anal. doi:10.1111/gean.12054

Thayn JB, Simanis JM (2013) Accounting for spatial autocorrelation in linear regression models using spatial filtering with eigenvectors. Ann Assoc Am Geogr 103(1):47–66CrossRef

Tiefelsdorf M, Griffith DA (2007) Semiparametric filtering of spatial autocorrelation: the eigenvector approach. Environ Plan A 39(5):1193–1221CrossRef

Wang Y, Kockelman KM, Wang X (2013) Understanding spatial filtering for analysis of land use-transport data. J Transp Geogr 31:123–131CrossRef

Titel: Random effects specifications in eigenvector spatial filtering: a simulation study
verfasst von: Daisuke Murakami
Daniel A. Griffith
Publikationsdatum: 01.10.2015
Verlag: Springer Berlin Heidelberg
Erschienen in: Journal of Geographical Systems / Ausgabe 4/2015
Print ISSN: 1435-5930
Elektronische ISSN: 1435-5949
DOI: https://doi.org/10.1007/s10109-015-0213-7

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner