Essential Statistical Concepts, Definitions, and Terminology - Sampling Spatial Units for Agricultural Surveys - page 28

methods for estimating SLM: the ML technique and the instrumental variables
(IV) method (Anselin 1988 ; Kelejian and Prucha 1998 ). In spdep the two methods
are implemented using two different functions: lagsarlm and stsls. Consid-
ering the data set LasRosas, the ML estimate for SLM in R is computed as
follows.

> library(spdep)
> SLM < - lagsarlm((YIELD~N+N2), type ¼ "lag", data ¼ LasRosas@data,
+ LasRosas_listw,tol.solve ¼ 1.0e-18)

> summary(SLM)

Call:lagsarlm(formula ¼ (YIELD ~ N + N2), data ¼ LasRosas@data, listw
¼ LasRosas_listw,

type ¼ "lag", tol.solve ¼ 1e-18)

Residuals:
Min 1Q Median 3Q Max

-23.528379 -2.570107 -0.002049 2.592077 25.259360

Type: lag
Coefficients: (asymptotic standard errors)
Estimate

Std. Error z value Pr( > |z|)

(Intercept) 4.769249730 0.861163904 5.5381 0.00000003057

N 0.111569453 0.008046086 13.8663 < 2.2e-16

N2 -0.000302859 0.000057528 - 5.2645 0.00000014056

Rho: 0.84398, LR test value: 2074.5, p-value: < 2.22e-16
Asymptotic standard error: 0.013498
z-value: 62.527, p-value: < 2.22e-16
Wald statistic: 3909.6, p-value: < 2.22e-16
Log likelihood: -5102.06 for lag model
ML residual variance (sigma squared): 17.855, (sigma: 4.2255)
Number of observations: 1738
Number of parameters estimated: 5
AIC: 10214, (AIC for lm: 12287)
LM test for residual autocorrelation
test value: 2.9824, p-value: 0.084176

ˁ 1 of Eq. ( 1.54 ) is replaced with Rho in the R output.
The data may not be continuous. If they are, for example, binary (i.e., take the value
0 if a phenomenon is absent at a geographical unit, or 1 if it is present), then we can
define an MRF that is known as an auto-logistic model.
A process

Note that the parameter

is said to obey to the auto-logistic law with the
presence of covariates (Besag 1974 ; Alf` and Postiglione 2002 ), if the conditional
density function of y can be written as

d

y ðÞ: z 2 D ℝ