aple: Approximate profile-likelihood estimator (APLE)
In spdep: Spatial Dependence: Weighting Schemes, Statistics and Models

Description Usage Arguments Details Value Author(s) References See Also Examples

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.

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

`x`	a zero-mean detrended continuous variable
`listw`	a `listw` object from for example `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)

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

A scalar APLE value.

Roger Bivand Roger.Bivand@nhh.no

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.

nb2listw, aple.mc, aple.plot

example(wheat)
nbr1 <- poly2nb(wheat, queen=FALSE)
nbrl <- nblag(nbr1, 2)
nbr12 <- nblag_cumul(nbrl)
cms0 <- with(as(wheat, "data.frame"), 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(wheat, "data.frame"),
 tapply(yield_detrend, c, median))), rep(0.0, 25),
 check.attributes=FALSE))
moran.test(wheat$yield_detrend, nb2listw(nbr12, style="W"))
aple(as.vector(scale(wheat$yield_detrend, scale=FALSE)), nb2listw(nbr12, style="W"))

errorsarlm(yield_detrend ~ 1, wheat, nb2listw(nbr12, style="W"))