spdep (version 0.5-3)

spautolm: Spatial conditional and simultaneous autoregression model estimation

Description

Function taking family and weights arguments for spatial autoregression model estimation by Maximum Likelihood, using full matrix methods, not suited to large data sets with thousands of observations. With one of the sparse matrix methods, larger numbers of observations can be handled, but the interval= argument should be set. The implementation is GLS using the single spatial coefficient value, here termed lambda, found by line search using optimize to maximise the log likelihood.

Usage

spautolm(formula, data = list(), listw, weights,
 na.action, family = "SAR", method="full", verbose = NULL, interval=NULL,
 zero.policy = NULL, tol.solve=.Machine$double.eps, llprof=NULL, control=list())

Value

• A list object of class spautolm:
• fit
  a list, with items: [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
• lambda
  ML autoregressive coefficient
• LL
  log likelihood for fitted model
• LL0
  log likelihood for model with lambda=0
• call
  the call used to create this object
• parameters
  number of parameters estimated
• aliased
  if not NULL, details of aliased variables
• method
  Jacobian method chosen
• zero.policy
  zero.policy used
• weights
  case weights used
• interval
  the line search interval used
• timings
  processing timings
• na.action
  (possibly) named vector of excluded or omitted observations if non-default na.action argument used
• llprof
  if not NULL, a list with components lambda and ll of equal length

Control arguments

• tol.opt

{the desired accuracy of the optimization - passed to optimize() (default=.Machine$double.eps^(2/3)} Imult{default 2; used for preparing the Cholesky decompositions for updating in the Jacobian function} super{default FALSE using a simplicial decomposition for the sparse Cholesky decomposition, if TRUE, use a supernodal decomposition} cheb_q{default 5; highest power of the approximating polynomial for the Chebyshev approximation} MC_p{default 16; number of random variates} MC_m{default 30; number of products of random variates matrix and spatial weights matrix}

Details

This implementation is based on lm.gls and errorsarlm. In particular, the function does not (yet) prevent asymmetric spatial weights being used with "CAR" family models. It appears that both numerical issues (convergence in particular) and uncertainties about the exact spatial weights matrix used make it difficult to reproduce Cressie and Chan's 1989 results, also given in Cressie 1993.

Note that the fitted() function for the output object assumes that the response variable may be reconstructed as the sum of the trend, the signal, and the noise (residuals). Since the values of the response variable are known, their spatial lags are used to calculate signal components (Cressie 1993, p. 564). This differs from other software, including GeoDa, which does not use knowledge of the response variable in making predictions for the fitting data.

References

Cliff, A. D., Ord, J. K. 1981 Spatial processes, Pion; Ord, J. K. 1975 Estimation methods for models of spatial interaction, Journal of the American Statistical Association, 70, 120-126; Waller, L. A., Gotway, C. A. 2004 Applied spatial statistics for public health, Wiley, Hoboken, NJ, 325-380; Cressie, N. A. C. 1993 Statistics for spatial data, Wiley, New York, 548-568; Ripley, B. D. 1981 Spatial statistics, Wiley, New York, 88-95.

See Also

optimize, errorsarlm

Examples

example(NY_data)
lm0 <- lm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata)
summary(lm0)
lm0w <- lm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata, weights=POP8)
summary(lm0w)
esar0 <- errorsarlm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY)
summary(esar0)
system.time(esar1f <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, family="SAR", method="full", verbose=TRUE))
summary(esar1f)
system.time(esar1M <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, family="SAR", method="Matrix", verbose=TRUE))
summary(esar1M)
system.time(esar1M <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, family="SAR", method="Matrix", verbose=TRUE,
 control=list(super=TRUE)))
summary(esar1M)
system.time(esar1s <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, family="SAR", method="spam", verbose=TRUE))
summary(esar1s)
esar1wf <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, weights=POP8, family="SAR", method="full")
summary(esar1wf)
system.time(esar1wM <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, weights=POP8, family="SAR", method="Matrix"))
summary(esar1wM)
esar1ws <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, weights=POP8, family="SAR", method="spam")
summary(esar1ws)
esar1wlu <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, weights=POP8, family="SAR", method="LU")
summary(esar1wlu)
esar1wch <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, weights=POP8, family="SAR", method="Chebyshev")
summary(esar1wch)
ecar1f <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, family="CAR", method="full")
summary(ecar1f)
system.time(ecar1M <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, family="CAR", method="Matrix"))
summary(ecar1M)
ecar1s <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, family="CAR", method="spam")
summary(ecar1s)
ecar1wf <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, weights=nydata$POP8, family="CAR", method="full")
summary(ecar1wf)
system.time(ecar1wM <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, weights=POP8, family="CAR", method="Matrix"))
summary(ecar1wM)
ecar1ws <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, weights=POP8, family="CAR", method="spam")
summary(ecar1ws)
example(nc.sids)
ft.SID74 <- sqrt(1000)*(sqrt(nc.sids$SID74/nc.sids$BIR74) + sqrt((nc.sids$SID74+1)/nc.sids$BIR74))
lm_nc <- lm(ft.SID74 ~ 1)
sids.nhbr30 <- dnearneigh(cbind(nc.sids$east, nc.sids$north), 0, 30, row.names=row.names(nc.sids))
sids.nhbr30.dist <- nbdists(sids.nhbr30, cbind(nc.sids$east, nc.sids$north))
sids.nhbr <- listw2sn(nb2listw(sids.nhbr30, glist=sids.nhbr30.dist, style="B", zero.policy=TRUE))
dij <- sids.nhbr[,3]
n <- nc.sids$BIR74
el1 <- min(dij)/dij
el2 <- sqrt(n[sids.nhbr$to]/n[sids.nhbr$from])
sids.nhbr$weights <- el1*el2
sids.nhbr.listw <- sn2listw(sids.nhbr)
both <- factor(paste(nc.sids$L_id, nc.sids$M_id, sep=":"))
ft.NWBIR74 <- sqrt(1000)*(sqrt(nc.sids$NWBIR74/nc.sids$BIR74) + sqrt((nc.sids$NWBIR74+1)/nc.sids$BIR74))
mdata <- data.frame(both, ft.NWBIR74, ft.SID74, BIR74=nc.sids$BIR74)
outl <- which.max(rstandard(lm_nc))
as.character(nc.sids$names[outl])
mdata.4 <- mdata[-outl,]
W <- listw2mat(sids.nhbr.listw)
W.4 <- W[-outl, -outl]
sids.nhbr.listw.4 <- mat2listw(W.4)
esarI <- errorsarlm(ft.SID74 ~ 1, data=mdata, listw=sids.nhbr.listw,
 zero.policy=TRUE)
summary(esarI)
esarIa <- spautolm(ft.SID74 ~ 1, data=mdata, listw=sids.nhbr.listw,
 family="SAR")
summary(esarIa)
esarIV <- errorsarlm(ft.SID74 ~ ft.NWBIR74, data=mdata, listw=sids.nhbr.listw,
 zero.policy=TRUE)
summary(esarIV)
esarIVa <- spautolm(ft.SID74 ~ ft.NWBIR74, data=mdata, listw=sids.nhbr.listw,
 family="SAR")
summary(esarIVa)
esarIaw <- spautolm(ft.SID74 ~ 1, data=mdata, listw=sids.nhbr.listw,
 weights=BIR74, family="SAR")
summary(esarIaw)
esarIIaw <- spautolm(ft.SID74 ~ both - 1, data=mdata, listw=sids.nhbr.listw,
 weights=BIR74, family="SAR")
summary(esarIIaw)
esarIVaw <- spautolm(ft.SID74 ~ ft.NWBIR74, data=mdata,
 listw=sids.nhbr.listw, weights=BIR74, family="SAR")
summary(esarIVaw)
ecarIaw <- spautolm(ft.SID74 ~ 1, data=mdata.4, listw=sids.nhbr.listw.4,
 weights=BIR74, family="CAR")
summary(ecarIaw)
ecarIIaw <- spautolm(ft.SID74 ~ both - 1, data=mdata.4,
 listw=sids.nhbr.listw.4, weights=BIR74, family="CAR")
summary(ecarIIaw)
ecarIVaw <- spautolm(ft.SID74 ~ ft.NWBIR74, data=mdata.4,
 listw=sids.nhbr.listw.4, weights=BIR74, family="CAR")
summary(ecarIVaw)
nc.sids$fitIV <- append(fitted.values(ecarIVaw), NA, outl-1)
spplot(nc.sids, c("fitIV"), cuts=12) # Cressie 1993, p. 565
data(oldcol)
COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 nb2listw(COL.nb, style="W"))
summary(COL.errW.eig)
COL.errW.sar <- spautolm(CRIME ~ INC + HOVAL, data=COL.OLD,
 nb2listw(COL.nb, style="W"))
summary(COL.errW.sar)
data(boston)
gp1 <- spautolm(log(CMEDV) ~ CRIM + ZN + INDUS + CHAS + I(NOX^2)
 + I(RM^2) + AGE + log(DIS) + log(RAD) + TAX + PTRATIO + B + log(LSTAT), 
 data=boston.c, nb2listw(boston.soi), family="SMA")
summary(gp1)