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