# localmoran: Local Moran's I statistic In r-spatial/spdep: Spatial Dependence: Weighting Schemes, Statistics

## Description

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.

## Usage

 1 2 3 4 5 6 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) 

## Arguments

 x a numeric vector the same length as the neighbours list in listw listw a listw object created for example by nb2listw 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 na.fail), can also be na.omit or na.exclude - in these cases the weights list will be subsetted to remove NAs in the data. It may be necessary to set zero.policy to TRUE because this subsetting may create no-neighbour observations. Note that only weights lists created without using the glist argument to nb2listw may be subsetted. If na.pass is used, zero is substituted for NA values in calculating the spatial lag. (Note that na.exclude will only work properly starting from R 1.9.0, na.omit and na.exclude assign the wrong classes in 1.8.*) 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 p.adjustSP. Note that the number of multiple tests for each region is only taken as the number of neighbours + 1 for each region, rather than the total number of regions. 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 get.spChkOption() 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 rank() and punif() for [0, 1] p-values; if FALSE folded [0, 0.5] p-values.

## Details

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.

## Value

 Ii local moran statistic E.Ii expectation of local moran statistic; for localmoran_permthe permutation sample means Var.Ii variance of local moran statistic; for localmoran_permthe permutation sample standard deviations Z.Ii standard deviate of local moran statistic; for localmoran_perm based on permutation sample means and standard deviations Pr() p-value of local moran statistic using pnorm(); for localmoran_perm using standard deviatse based on permutation sample means and standard deviations Pr() Sim For localmoran_perm, rank() and punif() of observed statistic rank for [0, 1] p-values, or if rank=FALSE, the simulation folded [0, 0.5] range ranked p-value (based on https://github.com/pysal/esda/blob/4a63e0b5df1e754b17b5f1205b8cadcbecc5e061/esda/crand.py#L211-L213) Skewness For localmoran_perm, the output of e1071::skewness() for the permutation samples underlying the standard deviates Kurtosis For localmoran_perm, the output of e1071::kurtosis() for the permutation samples underlying the standard deviates

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.

## Note

Conditional permutations added for comparative purposes; permutations are over the whole data vector omitting the observation itself.

## Author(s)

Roger Bivand Roger.Bivand@nhh.no

## References

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

localG
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 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]), check.names=FALSE)