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spdep (version 0.5-74)

spautolm: Spatial conditional and simultaneous autoregression model estimation

Description

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.

Usage

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())
## S3 method for class 'spautolm':
summary(object, correlation = FALSE, adj.se=FALSE,
 Nagelkerke=FALSE, ...)

Arguments

formula
a symbolic description of the model to be fit. The details of model specification are given for lm()
data
an optional data frame containing the variables in the model. By default the variables are taken from the environment which the function is called.
listw
a listw object created for example by nb2listw
weights
an optional vector of weights to be used in the fitting process
na.action
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
family
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"
method
character string: default "eigen" for use of dense matrices, "Matrix_J" or "spam" for sparse matrices (restricted to spatial weights symmetric or similar to symmetric) using methods in either the Matrix or spam packa
verbose
default NULL, use global option value; if TRUE, reports function values during optimization.
trs,
default NULL, if given, a vector of powered spatial weights matrix traces output by trW; when given, used in some Jacobian methods
interval
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 appro
zero.policy
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
tol.solve
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:
llprof
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
control
list of extra control arguments - see section below
object
spautolm object from spautolm
correlation
logical; if 'TRUE', the correlation matrix of the estimated parameters is returned and printed (default=FALSE)
adj.se
if TRUE, adjust the coefficient standard errors for the number of fitted coefficients
Nagelkerke
if TRUE, the Nagelkerke pseudo R-squared is reported
...
further arguments passed to or from other methods

Value

  • A list object of class spautolm:
  • fita list, with items: [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
  • lambdaML autoregressive coefficient
  • LLlog likelihood for fitted model
  • LL0log likelihood for model with lambda=0
  • callthe call used to create this object
  • parametersnumber of parameters estimated
  • aliasedif not NULL, details of aliased variables
  • methodJacobian method chosen
  • familyfamily chosen
  • zero.policyzero.policy used
  • weightscase weights used
  • intervalthe line search interval used
  • timingsprocessing timings
  • na.action(possibly) named vector of excluded or omitted observations if non-default na.action argument used
  • llprofif not NULL, a list with components lambda and ll of equal length
  • lambda.seNumerical Hessian-based standard error of lambda
  • fdHessNumerical Hessian-based variance-covariance matrix
  • Xcovariates used in model fitting
  • Yresponse used in model fitting
  • weightsweights used in model fitting

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

See Also

optimize, errorsarlm, do_ldet

Examples

Run this code
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="eigen", 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="eigen")
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="eigen")
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$POP8, family="CAR", method="eigen")
summary(ecar1wf)
system.time(ecar1wM <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, weights=POP8, family="CAR", method="Matrix"))
summary(ecar1wM)
ecar1ws <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, weights=POP8, family="CAR", method="spam")
summary(ecar1ws)
example(nc.sids)
ft.SID74 <- sqrt(1000)*(sqrt(nc.sids$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)

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