esf function

Spatial regression with eigenvector spatial filtering

This function estimates the linear eigenvector spatial filtering (ESF) model. The eigenvectors are selected by a forward stepwise method.

esf( y, x = NULL, vif = NULL, meig, fn = "r2" )

Arguments

• y: Vector of explained variables (N x 1)
• x: Matrix of explanatory variables (N x K). Default is NULL
• vif: Maximum acceptable value of the variance inflation factor (VIF) (scalar). For example, if vif = 10, eigenvectors are selected so that the maximum VIF value among explanatory variables and eigenvectors is equal to or less than 10. Default is NULL
• meig: Moran eigenvectors and eigenvalues. Output from meigen or meigen_f
• fn: Objective function for the stepwise eigenvector selection. The adjusted R2 ("r2"), AIC ("aic"), or BIC ("bic") are available. Alternatively, all the eigenvectors in meig are used without the stepwise selection if fn = "all". This is acceptable for large samples (see Murakami and Griffith, 2019). Default is "r2"

Returns

• b: Matrix with columns for the estimated coefficients on x, their standard errors, t-values, and p-values (K x 4)

• s: Vector of statistics for the estimated spatial component (2 x 1). The first element is the standard deviation and the second element is the Moran's I value of the estimated spatially dependent component. The Moran's I value is scaled to take a value between 0 (no spatial dependence) and 1 (the maximum possible spatial dependence). Based on Griffith (2003), the scaled Moran'I value is interpretable as follows: 0.25-0.50:weak; 0.50-0.70:moderate; 0.70-0.90:strong; 0.90-1.00:marked

• r: Matrix with columns for the estimated coefficients on Moran's eigenvectors, their standard errors, t-values, and p-values (L x 4)

• vif: Vector of variance inflation factors of the explanatory variables (N x 1)

• e: Vector whose elements are residual standard error (resid_SE), adjusted R2 (adjR2), log-likelihood (logLik), AIC, and BIC

• sf: Vector of estimated spatial dependent component (E$\gamma$) (N x 1)

• pred: Vector of predicted values (N x 1)

• resid: Vector of residuals (N x 1)

• other: List of other outputs, which are internally used

References

Griffith, D. A. (2003). Spatial autocorrelation and spatial filtering: gaining understanding through theory and scientific visualization. Springer Science & Business Media.

Tiefelsdorf, M., and Griffith, D. A. (2007). Semiparametric filtering of spatial autocorrelation: the eigenvector approach. Environment and Planning A, 39 (5), 1193-1221.

Murakami, D. and Griffith, D.A. (2019) Eigenvector spatial filtering for large data sets: fixed and random effects approaches. Geographical Analysis, 51 (1), 23-49.

Author(s)

Daisuke Murakami

See Also

resf

Examples

require(spdep)
data(boston)
y       <- boston.c[, "CMEDV" ]
x       <- boston.c[,c("CRIM","ZN","INDUS", "CHAS", "NOX","RM", "AGE")]
coords  <- boston.c[,c("LON", "LAT")]

#########Distance-based ESF
meig    <- meigen(coords=coords)
esfD    <- esf(y=y,x=x,meig=meig, vif=5)
esfD

#########Fast approximation
meig_f<- meigen_f(coords=coords)
esfD    <- esf(y=y,x=x,meig=meig_f, vif=10, fn="all")
esfD

############################Not run
#########Topoligy-based ESF (it is commonly used in regional science)
#
#cknn   <- knearneigh(coordinates(coords), k=4) #4-nearest neighbors
#cmat   <- nb2mat(knn2nb(cknn), style="B")
#meig <- meigen(cmat=cmat, threshold=0.25)
#esfT   <- esf(y=y,x=x,meig=meig)
#esfT