Approximate profile-likelihood estimator (APLE)

The Approximate profile-likelihood estimator (APLE) of the simultaneous autoregressive model's spatial dependence parameter was introduced in Li et al. (2007). It employs a correction term using the eigenvalues of the spatial weights matrix, and consequently should not be used for large numbers of observations. It also requires that the variable has a mean of zero, and it is assumed that it has been detrended. The spatial weights object is assumed to be row-standardised, that is using default style="W" in nb2listw.

Usage

aple(x, listw, override_similarity_check=FALSE, useTrace=TRUE)

Arguments

x: a zero-mean detrended continuous variable
listw: a listw object from for example spdep::nb2listw
override_similarity_check: default FALSE, if TRUE - typically for row-standardised weights with asymmetric underlying general weights - similarity is not checked
useTrace: default TRUE, use trace of sparse matrix W %*% W (Li et al. (2010)), if FALSE, use crossproduct of eigenvalues of W as in Li et al. (2007)

Details

This implementation has been checked with Hongfei Li's own implementation using her data; her help was very valuable.

Value

A scalar APLE value.

References

Li, H, Calder, C. A. and Cressie N. A. C. (2007) Beyond Moran's I: testing for spatial dependence based on the spatial autoregressive model. Geographical Analysis 39, 357-375; Li, H, Calder, C. A. and Cressie N. A. C. (2012) One-step estimation of spatial dependence parameters: Properties and extensions of the APLE statistic, Journal of Multivariate Analysis 105, 68-84.

Author

Roger Bivand Roger.Bivand@nhh.no

Examples

wheat <- st_read(system.file("shapes/wheat.gpkg", package="spData")[1], quiet=TRUE)
library(spdep)
#> 
#> Attaching package: ‘spdep’
#> The following objects are masked from ‘package:spatialreg’:
#> 
#>     get.ClusterOption, get.VerboseOption, get.ZeroPolicyOption,
#>     get.coresOption, get.mcOption, set.ClusterOption,
#>     set.VerboseOption, set.ZeroPolicyOption, set.coresOption,
#>     set.mcOption
nbr1 <- spdep::poly2nb(wheat, queen=FALSE)
nbrl <- spdep::nblag(nbr1, 2)
#> Warning: lag 2 neighbour object has 2 sub-graphs
nbr12 <- spdep::nblag_cumul(nbrl)
cms0 <- with(as.data.frame(wheat), tapply(yield, c, median))
cms1 <- c(model.matrix(~ factor(c) -1, data=wheat) %*% cms0)
wheat$yield_detrend <- wheat$yield - cms1
isTRUE(all.equal(c(with(as.data.frame(wheat),
 tapply(yield_detrend, c, median))), rep(0.0, 25),
 check.attributes=FALSE))
#> [1] TRUE
spdep::moran.test(wheat$yield_detrend, spdep::nb2listw(nbr12, style="W"))
#> 
#> 	Moran I test under randomisation
#> 
#> data:  wheat$yield_detrend  
#> weights: spdep::nb2listw(nbr12, style = "W")    
#> 
#> Moran I statistic standard deviate = 10.305, p-value < 2.2e-16
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>      0.1935469610     -0.0020040080      0.0003600869 
#> 
aple(as.vector(scale(wheat$yield_detrend, scale=FALSE)), spdep::nb2listw(nbr12, style="W"))
#> [1] 0.6601805
# \dontrun{
errorsarlm(yield_detrend ~ 1, wheat, spdep::nb2listw(nbr12, style="W"))
#> 
#> Call:
#> errorsarlm(formula = yield_detrend ~ 1, data = wheat, listw = spdep::nb2listw(nbr12, 
#>     style = "W"))
#> Type: error 
#> 
#> Coefficients:
#>      lambda (Intercept) 
#>  0.60189687 -0.00251772 
#> 
#> Log likelihood: -192.9519 
# }