# LOSH: Local spatial heteroscedasticity In r-spatial/spdep: Spatial Dependence: Weighting Schemes, Statistics

## Description

Local spatial heteroscedasticity is calculated for each location based on the spatial weights object used. The statistic is:

H_i = \frac{∑_j^n w_{ij} \cdot |e_j|^a}{h_1 \cdot ∑_j^n w_{ij}}

with

e_j = x_j - \bar{x}_j

and

\bar{x}_j = \frac{∑_k^n w_{jk} \cdot x_k}{∑_k^n w_{jk}}

Its expectation and variance are given in Ord & Getis (2012). The exponent a allows for investigating different types of mean dispersal.

## Usage

 1  LOSH(x, listw, a=2, var_hi=TRUE, zero.policy=NULL, na.action=na.fail, spChk=NULL) 

## Arguments

 x a numeric vector of the same length as the neighbours list in listw listw a listw object created for example by nb2listw a the exponent applied to the local residuals; the default value of 2 leads to a measure of heterogeneity in the spatial variance var_hi default TRUE, the moments and the test statistics are calculated for each location; if FALSE, only the plain LOSH measures, \bar{x}_i and e_i are calculated 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.*) spChk should the data vector names be checked against the spatial objects for identity integrity, TRUE, or FALSE, default NULL to use get.spChkOption()

## Details

In addition to the LOSH measure, the values returned include local spatially weighted mean values \bar{x}_i and local residuals e_i estimated about these means. These values facilitate the interpretation of LOSH values. Further, if specified through var_hi, the statistical moments and the test statistics as proposed by Ord & Getis (2012) are also calculated and returned.

## Value

 Hi LOSH statistic E.Hi (optional) expectation of LOSH Var.Hi (optional) variance of LOSH Z.Hi (optional) the approximately Chi-square distributed test statistics x_bar_i local spatially weighted mean values ei residuals about local spatially weighted mean values

## Author(s)

René Westerholt rene.westerholt@tu-dortmund.de

## References

Ord, J. K., & Getis, A. 2012. Local spatial heteroscedasticity (LOSH), The Annals of Regional Science, 48 (2), 529–539.

LOSH.cs, LOSH.mc
 1 2 3 4 5  data(boston, package="spData") resLOSH <- LOSH(boston.c\$NOX, nb2listw(boston.soi)) hist(resLOSH[,"Hi"]) mean(resLOSH[,"Hi"])