R: Local spatial heteroscedasticity

LOSH {spdep} | R Documentation

## Local spatial heteroscedasticity

### 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

      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.

### See Also

`LOSH.cs`, `LOSH.mc`

### Examples

        data(boston, package="spData")
        resLOSH <- LOSH(boston.c$NOX, nb2listw(boston.soi))
        hist(resLOSH[,"Hi"])
        mean(resLOSH[,"Hi"])

[Package spdep version 1.2-2 Index]