resf_vc: Spatially varying coefficient modeling with/without automatic coefficient selection

Description

This function estimates a spatially varying coefficient model based on the random effects eigenvector spatial filtering (RE-ESF) approach. Spatially varying coefficients are selected to stablize the estimates.

Usage

resf_vc(y, x=NULL, xgroup = NULL, xconst=NULL, meig, method="reml",
        penalty="bic", allsvc=FALSE, maxiter=30, sizelimit=2000 )

Arguments

Vector of explained variables (N x 1)

Matrix of explanatory variables whose coefficients are allowed to vary across geographical space (N x K). Default is NULL

xgroup

Matrix of group indexes. The indeces may be group IDs (numbers) or group names (N x K_group). Default is NULL

xconst

Matrix of explanatory variables whose coefficients are assumed constant across space (N x K_const). Default is NULL

meig

Moran's eigenvectors and eigenvalues. Output from meigen or meigen_f

method

Estimation method. Restricted maximum likelihood method ("reml") and maximum likelihood method ("ml") are available. Default is "reml"

penalty

Penalty to select varying coefficients and stablize the estimates. The current options are "bic" for the Baysian information criterion-type penalty (N x log(K)) and "aic" for the Akaike information criterion (2K) (see Muller et al., 2013). Default is "bic"

allsvc

If it is TRUE, SVCs are selected using the penalty. If it is FALSE, all the coefficients on the explanatory variables in x are SVCs. Default is FALSE

maxiter

Maximum number of iterations. Default is 30

sizelimit

Maximum size of matrix being inverted. Roughly speaking, this value contraints K + KL, where K and L are the numbers of the explanatory variables and eigen-pairs, respectively. If K + KL exceeds sizelimit, L is reduced to fullfill the contraint. Default is 2000

Value

Matrix with columns for the estimated coefficients on xconst, their standard errors, t-values, and p-values (K_const x 4)

b_g

List of K_group matrices with columns for the estimated group effects, their standard errors, and t-values

Matrix of estimated variance parameters (2 x K). The (1, k)-th element denotes the standard error of the k-th SVCs, while the (2, k)-th element denotes the Moran's I value of the SVCs. 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

s_g

Vector of estimated standard errors of the group effects

Vector whose elements are residual standard error (resid_SE), adjusted conditional R2 (adjR2(cond)), restricted log-likelihood (rlogLik), Akaike information criterion (AIC), and Bayesian information criterion (BIC). When method = "ml", restricted log-likelihood (rlogLik) is replaced with log-likelihood (logLik)

b_vc

Matrix of estimated spatially varying coefficients (SVCs) on x (N x K)

bse_vc

Matrix of estimated standard errors for the SVCs (N x k)

t_vc

Matrix of estimated t-values for the SVCs (N x K)

p_vc

Matrix of estimated p-values for the SVCs (N x K)

pred

Vector of predicted values (N x 1)

resid

Vector of residuals (N x 1)

Vector indicating whether spatial variations are found or not from each coefficients (K x 1)

other

List of other outcomes, which are internally used

References

Muller, S., Scealy, J.L., and Welsh, A.H. (2013) Model selection in linear mixed models. Statistical Science, 28 (2), 136-167.

Murakami, D., Yoshida, T., Seya, H., Griffith, D.A., and Yamagata, Y. (2017) A Moran coefficient-based mixed effects approach to investigate spatially varying relationships. Spatial Statistics, 19, 68-89.

Murakami, D., and Griffith, D.A. (2019). Spatially varying coefficient modeling for large datasets: Eliminating N from spatial regressions. Spatial Statistics, 30, 39-64.

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

Examples

Run this code

# NOT RUN {
require(spdep)
data(boston)
y	<- boston.c[, "CMEDV"]
x       <- boston.c[,c("ZN", "LSTAT")]
xconst  <- boston.c[,c("CRIM", "NOX", "AGE", "DIS", "RAD", "TAX", "PTRATIO", "B", "RM")]
xgroup  <- boston.c[,"TOWN"]
coords  <- boston.c[,c("LAT","LON")]
meig 	  <- meigen(coords=coords)
# meig_f 	<- meigen_f(coords=coords)  ## for fast computation


######## Without the penalty-based SVC selection (just like the usual GWR)
res	    <- resf_vc(y=y,x=x,xconst=xconst,meig=meig,allsvc=TRUE)

######## With the penalty-based SVC selection
# res2	<- resf_vc(y=y,x=x,xconst=xconst,meig=meig)

######## SVC modeling with group effects
# res3	<- resf_vc(y=y,x=x,xconst=xconst,xgroup=xgroup,meig=meig)

res$b
res$s
res$e

res$vc
res$b_vc[1:10,]
res$bse_vc[1:10,]
res$t_vc[1:10,]
res$p_vc[1:10,]

###Plot the first SVC (INDUS)
#
#require(ggplot2)
#ggplot(mapping = aes(x = coords$LON, y = coords$LAT)) +
#  geom_point(aes(colour = res$b_vc[,2])) +
#  scale_color_gradientn(colours=rev(rainbow(4)))
# }