Spatial conditional and simultaneous autoregression model estimation

Function taking family and weights arguments for spatial autoregression model estimation by Maximum Likelihood, using dense 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.

spautolm(formula, data = list(), listw, weights, na.action, family = "SAR", method="eigen", verbose = NULL, trs=NULL, interval=NULL, zero.policy = NULL, tol.solve=.Machine$double.eps, llprof=NULL, control=list()) "summary"(object, correlation = FALSE,, Nagelkerke=FALSE, ...)
a symbolic description of the model to be fit. The details of model specification are given for lm()
an optional data frame containing the variables in the model. By default the variables are taken from the environment which the function is called.
a listw object created for example by nb2listw
an optional vector of weights to be used in the fitting process
a function (default options("na.action")), can also be na.omit or na.exclude with consequences for residuals and fitted values - in these cases the weights list will be subsetted to remove NAs in the data. Note that only weights lists created without using the glist argument to nb2listw may be subsetted.
character string: either "SAR" or "CAR" for simultaneous or conditional autoregressions; "SMA" for spatial moving average added thanks to Jielai Ma - "SMA" is only implemented for method="eigen" because it necessarily involves dense matrices
character string: default "eigen" for use of dense matrices, "Matrix_J" for sparse matrices (restricted to spatial weights symmetric or similar to symmetric) using methods in the Matrix package; “Matrix” provides updating Cholesky decomposition methods. Values of method may also include "LU", which provides an alternative sparse matrix decomposition approach, and the "Chebyshev" and Monte Carlo "MC" approximate log-determinant methods.
default NULL, use global option value; if TRUE, reports function values during optimization.
default NULL, if given, a vector of powered spatial weights matrix traces output by trW; when given, used in some Jacobian methods
search interval for autoregressive parameter when not using method="eigen"; default is c(-1,0.999), optimize will reset NA/NaN to a bound and gives a warning when the interval is poorly set; method="Matrix" will attempt to search for an appropriate interval, if find\_interval=TRUE (fails on some platforms)
default NULL, use global option value; Include list of no-neighbour observations in output if TRUE --- otherwise zero.policy is handled within the listw argument
the tolerance for detecting linear dependencies in the columns of matrices to be inverted - passed to solve() (default=double precision machine tolerance). Errors in solve() may constitute indications of poorly scaled variables: if the variables have scales differing much from the autoregressive coefficient, the values in this matrix may be very different in scale, and inverting such a matrix is analytically possible by definition, but numerically unstable; rescaling the RHS variables alleviates this better than setting tol.solve to a very small value
default NULL, can either be an integer, to divide the feasible range into llprof points, or a sequence of spatial coefficient values, at which to evaluate the likelihood function
list of extra control arguments - see section below
spautolm object from spautolm
logical; if 'TRUE', the correlation matrix of the estimated parameters is returned and printed (default=FALSE)
if TRUE, adjust the coefficient standard errors for the number of fitted coefficients
if TRUE, the Nagelkerke pseudo R-squared is reported
further arguments passed to or from other methods

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.


A list object of class spautolm:


The standard errors given in Waller and Gotway (2004) are adjusted for the numbers of parameters estimated, and may be reproduced by using the additional argument in the summary method. In addition, the function returns fitted values and residuals as given by Cressie (1993) p. 564.

Control arguments


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; LeSage J and RK Pace (2009) Introduction to Spatial Econometrics. CRC Press, Boca Raton.

See Also

optimize, errorsarlm, do_ldet

  • spautolm
  • residuals.spautolm
  • deviance.spautolm
  • coef.spautolm
  • fitted.spautolm
  • print.spautolm
  • summary.spautolm
  • LR1.spautolm
  • logLik.spautolm
  • print.summary.spautolm
lm0 <- lm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata)
lm0w <- lm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata, weights=POP8)
esar0 <- errorsarlm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
system.time(esar1f <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, family="SAR", method="eigen", verbose=TRUE))
res <- summary(esar1f)
system.time(esar1M <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, family="SAR", method="Matrix", verbose=TRUE))
## Not run: 
# 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)
# esar1wf <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
#  listw=listw_NY, weights=POP8, family="SAR", method="eigen")
# summary(esar1wf)
# system.time(esar1wM <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
#  data=nydata, listw=listw_NY, weights=POP8, family="SAR", method="Matrix"))
# summary(esar1wM)
# 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="eigen")
# summary(ecar1f)
# system.time(ecar1M <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
#  data=nydata, listw=listw_NY, family="CAR", method="Matrix"))
# summary(ecar1M)
# ecar1wf <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
#  listw=listw_NY, weights=nydata$POP8, family="CAR", method="eigen")
# summary(ecar1wf)
# system.time(ecar1wM <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
#  data=nydata, listw=listw_NY, weights=POP8, family="CAR", method="Matrix"))
# summary(ecar1wM)
# example(nc.sids)
# ft.SID74 <- sqrt(1000)*(sqrt(nc.sids$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)
# ## End(Not run)
Documentation reproduced from package spdep, version 0.6-9, License: GPL (>= 2)

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