Local Moran's I statistic
The local spatial statistic Moran's I is calculated for each zone based on the spatial weights object used. The values returned include a Z-value, and may be used as a diagnostic tool. The statistic is:
I_i = \frac{(x_i-\bar{x})}{{∑_{k=1}^{n}(x_k-\bar{x})^2}/(n-1)}{∑_{j=1}^{n}w_{ij}(x_j-\bar{x})}
, and its expectation and variance were given in Anselin (1995), but those from Sokal et al. (1998) are implemented here.
localmoran(x, listw, zero.policy=NULL, na.action=na.fail, conditional=TRUE,
alternative = "two.sided", p.adjust.method="none", mlvar=TRUE,
spChk=NULL, adjust.x=FALSE)
localmoran_perm(x, listw, nsim=499, zero.policy=NULL, na.action=na.fail,
alternative = "two.sided", p.adjust.method="none", mlvar=TRUE,
spChk=NULL, adjust.x=FALSE, sample_Ei=TRUE, iseed=NULL, rank=TRUE)x |
a numeric vector the same length as the neighbours list in listw |
listw |
a |
zero.policy |
default NULL, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE assign NA |
na.action |
a function (default |
conditional |
default TRUE: expectation and variance are calculated using the conditional randomization null (Sokal 1998 Eqs. A7 & A8). Elaboration of these changes available in Sauer et al. (2021). If FALSE: expectation and variance are calculated using the total randomization null (Sokal 1998 Eqs. A3 & A4). |
alternative |
a character string specifying the alternative hypothesis, must be one of greater (default), less or two.sided. |
p.adjust.method |
a character string specifying the probability value adjustment for multiple tests, default "none"; see |
mlvar |
default TRUE: values of local Moran's I are reported using the variance of the variable of interest (sum of squared deviances over n), but can be reported as the sample variance, dividing by (n-1) instead; both are used in other implementations. |
spChk |
should the data vector names be checked against the spatial objects for identity integrity, TRUE, or FALSE, default NULL to use |
adjust.x |
default FALSE, if TRUE, x values of observations with no neighbours are omitted in the mean of x |
nsim |
default 499, number of conditonal permutation simulations |
sample_Ei |
default TRUE; if conditional permutation, use the sample $E_i$ values, or the analytical values, leaving only variances calculated by simulation. |
iseed |
default NULL, used to set the seed for possible parallel RNGs |
rank |
default TRUE, use |
The values of local Moran's I are divided by the variance (or sample variance) of the variable of interest to accord with Table 1, p. 103, and formula (12), p. 99, in Anselin (1995), rather than his formula (7), p. 98. The variance of the local Moran statistic is taken from Sokal et al. (1998) p. 334, equations 4 & 5 or equations 7 & 8 located depending on user specification. By default, the implementation divides by n, not (n-1) in calculating the variance and higher moments. Conditional code contributed by Jeff Sauer and Levi Wolf.
Ii |
local moran statistic |
E.Ii |
expectation of local moran statistic; for |
Var.Ii |
variance of local moran statistic; for |
Z.Ii |
standard deviate of local moran statistic; for |
Pr() |
p-value of local moran statistic using |
Pr() Sim |
For |
Skewness |
For |
Kurtosis |
For |
In addition, an attribute data frame "quadr" with mean and median quadrant columns, and a column splitting on the demeaned variable and lagged demeaned variable at zero.
Conditional permutations added for comparative purposes; permutations are over the whole data vector omitting the observation itself.
Roger Bivand Roger.Bivand@nhh.no
Anselin, L. 1995. Local indicators of spatial association, Geographical Analysis, 27, 93–115; Getis, A. and Ord, J. K. 1996 Local spatial statistics: an overview. In P. Longley and M. Batty (eds) Spatial analysis: modelling in a GIS environment (Cambridge: Geoinformation International), 261–277; Sokal, R. R, Oden, N. L. and Thomson, B. A. 1998. Local Spatial Autocorrelation in a Biological Model. Geographical Analysis, 30. 331–354; Bivand RS, Wong DWS 2018 Comparing implementations of global and local indicators of spatial association. TEST, 27(3), 716–748 doi: 10.1007/s11749-018-0599-x; Sauer, J., Oshan, T. M., Rey, S., & Wolf, L. J. 2021. On Null Hypotheses and Heteroskedasticity. OSF Preprints. doi: 10.31219/osf.io/ugkhp
data(afcon, package="spData")
oid <- order(afcon$id)
resI <- localmoran(afcon$totcon, nb2listw(paper.nb))
printCoefmat(data.frame(resI[oid,], row.names=afcon$name[oid]),
check.names=FALSE)
hist(resI[,5])
mean(resI[,1])
sum(resI[,1])/Szero(nb2listw(paper.nb))
moran.test(afcon$totcon, nb2listw(paper.nb))
# note equality for mean() only when the sum of weights equals
# the number of observations (thanks to Juergen Symanzik)
resI <- localmoran(afcon$totcon, nb2listw(paper.nb),
p.adjust.method="bonferroni")
printCoefmat(data.frame(resI[oid,], row.names=afcon$name[oid]),
check.names=FALSE)
hist(resI[,5])
totcon <-afcon$totcon
is.na(totcon) <- sample(1:length(totcon), 5)
totcon
resI.na <- localmoran(totcon, nb2listw(paper.nb), na.action=na.exclude,
zero.policy=TRUE)
if (class(attr(resI.na, "na.action")) == "exclude") {
print(data.frame(resI.na[oid,], row.names=afcon$name[oid]), digits=2)
} else print(resI.na, digits=2)
resG <- localG(afcon$totcon, nb2listw(include.self(paper.nb)))
print(data.frame(resG[oid], row.names=afcon$name[oid]), digits=2)
set.seed(1)
resI_p <- localmoran_perm(afcon$totcon, nb2listw(paper.nb))
printCoefmat(data.frame(resI_p[oid,], row.names=afcon$name[oid]),
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