Guillaume Pouliot and Zhen Xie
This R package implements regression analysis for spatially misaligned data, where the geographic locations of the outcome variable and explanatory variables do not coincide, as developed in Pouliot (2020). We implement estimation and inference for three complementary estimators: a two-step bootstrap estimator, a minimum-distance estimator, and a one-step Quasi Maximum Likelihood estimator. These three methods trade off statistical power for weaker assumptions on the covariance structure of regression residuals.
SpReg can be installed from its GitHub repository via
devtools::install_github("zhenxie23/SpReg", ref = 'main')Example: Madsen et al. (2008)
We use the dataset of Madsen et al. (2008). There are 558 observations over an area of 400,000 squared kilometers.
library(SpReg)
rivers <- read.csv(system.file("extdata", "rivers.csv", package = "SpReg"))
rivers <- rivers[which(rivers$FOR_NLCD<100),]The simulation is implemented as follows: for each run, the 558 locations in the data set are randomly split into two equal halves. The covariate R = logit(for.nlcd) is assumed to have been observed on one half, and the outcome variable Y = log(cl) is assumed to have been observed on the other. This creates a misaligned data set. For each round of the cross-validation exercise, all estimates, along with their standard errors, are calculated. The performance of estimators can then be compared with each other.
train_set <- sample(1:558,277);
test_set <- setdiff(1:558,train_set);
rivers_train <- rivers[train_set, ];
rivers_test <- rivers[test_set, ];
DatR <- rivers_train[,c("FOR_NLCD", "LAT_DD", "LON_DD")];
DatR$X <- log((DatR$FOR_NLCD)/(100-DatR$FOR_NLCD));
sp::coordinates(DatR) <- ~LON_DD+LAT_DD;
sp::proj4string(DatR) = "+proj=longlat +datum=WGS84";
DatY <- rivers_test[, c("CL","LAT_DD","LON_DD")];
DatY$Y <- log(DatY$CL);
sp::coordinates(DatY) <- ~LON_DD+LAT_DD;
sp::proj4string(DatY) = "+proj=longlat +datum=WGS84";Estimation and Inference for Two-Step Estimators
TwoStep_Results <- TwoStepBootstrap(DatR, "X", DatY, "Y", "Exp", FALSE, FALSE,
cutoff.R = 295, cutoff.res = 40);
summary(TwoStep_Results, alpha = 0.05)
=====================================================================
Point Std.
n Est. Error [ 95% C.I. ]
=====================================================================
Krig-and-OLS
(Intercept) 281 5.168 0.132 [4.910 , 5.427]
R.hat 281 -0.414 0.060 [-0.532 , -0.297]
---------------------------------------------------------------------
Krig-and-GLS
(Intercept) 281 5.150 0.168 [4.821 , 5.480]
R.hat 281 -0.396 0.076 [-0.545 , -0.247]
---------------------------------------------------------------------
Two Step Bootstrap
(Intercept) 281 5.182 0.154 [4.880 , 5.485]
R.hat 281 -0.426 0.074 [-0.571 , -0.281]
=====================================================================Estimation and Inference for Minimum Distance Estimator
## specifying starting value
MD.Starting.Value <- c(psill = unname(TwoStep_Results$vario.par.point.est[1]),
range = unname(TwoStep_Results$vario.par.point.est[2]),
beta = unname(TwoStep_Results[["tsbs.point.est"]][2]),
beta_std = unname(TwoStep_Results[["tsbs.var.mat"]][2,2]));
MD_Results <- MinimumDistance(DatR, "X", DatY, "Y", "Exp", MD.Starting.Value,
cutoff.R = 150, cutoff.YR = 70);
summary(MD_Results, alpha = 0.05)
=====================================================================
Point Std.
n Est. Error [ 95% C.I. ]
=====================================================================
Minimum Distance
R 558 -0.344 0.056 [-0.453 , -0.234]
=====================================================================