localmoran.bi {bispdep}

BiLISA - Bivariate Local Indicators of Spatial Association Moran's Ixy statistic

Description

The bivariate 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)_{xy} = \frac{(x_i-\bar{x})}{{\sum_{k=1}^{n}(x_k-\bar{x})^2}/(n-1)}{\sum_{j=1}^{n}w_{ij}(y_j-\bar{y})}

, and its expectation and variance were given in Anselin (1995), but those from Sokal et al. (1998) are implemented here.

Usage

localmoran.bi(varX,varY,listw,zero.policy=NULL,na.action=na.fail,conditional=TRUE,
              alternative="two.sided",mlvar=TRUE,spChk=NULL,adjust.x=FALSE)

Arguments

varX

a numeric vector the same length as the neighbours list in listw and y, with the values of the primary variable x

varY

a numeric vector the same length as the neighbours list in listw and x, with the values of the secundary variable y

listw

a listw object created for example by nb2listw with spatial weights

zero.policy

default NULL, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE assign NA and stop with error for any empty neighbours sets.

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. Similar meaning and values than parameter na.action of localmoran

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

type of specifying the alternative hypothesis test, must be one of greater, less or two.sided (default).

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

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

Iyxi

local moran statistic

E.Iyxi

expectation of local moran statistic; for localmoran_permthe permutation sample means

Var.Iyxi

variance of local moran statistic; for localmoran_permthe permutation sample standard deviations

Z.Iyxi

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 using alternative=

Pr(folded) Sim

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. For p-value adjustment, use p.adjust() or p.adjustSP() on the output vector.

References

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; Czaplewski, R.L., Reich, R.M. 1993. Expected value and variance of Moran's bivariate spatial autocorrelation statistic for a permutation test, Research paper RM-309, Fort Collins, CO U.S. Department of Agriculture, Forest Service, Rocky Mountain Forest and Range Experiment Station, p, 13.; Anselin, Luc, Ibnu Syabri, and Oleg Smirnov. 2002. “Visualizing Multivariate Spatial Correlation with Dynamically Linked Windows.” In New Tools for Spatial Data Analysis: Proceedings of the Specialist Meeting, edited by Luc Anselin and Sergio Rey. University of California, Santa Barbara: Center for Spatially Integrated Social Science (CSISS)

See Also

localG

Examples

library(spdep)
# load columbus data
data(columbus)
data(oldcol)
columbus <- st_read(system.file("shapes/columbus.shp", package="spData")[1], quiet=TRUE)
col_nbq <- poly2nb(columbus)
a.lw <- nb2listw(col_nbq, style="W")
localmoran.bi(columbus$CRIME, columbus$HOVAL, a.lw, zero.policy=TRUE,
              alternative="two.sided")