# Spatial Autocorrelation

# Module Description

“The first law of geography: Everything is related to everything else, but near things are more related than distant things.” Waldo R. Tobler (Tobler, 1970)

A question often asked is whether or not features with similar values are clustered, randomly distributed or dispersed. Spatial autocorrelation measures the degree of correlation on space (Cliff and Ord 1973, 1981). Tests of spatial autocorrelation examine the independency of observed value in relation to values of that variable at neighboring locations.

In this module calculates the Moran’s I Index value and both a a z-score and p-value to evaluate the significance of that Index. P-values are numerical approximations of the area under the curve for a known distribution, limited by the test statistic. We use the spdep package (Bivand, R. et al., 2018)

# Argument

`Variable`: a numeric vector the same length as the neighbours list in listw.

`zero.policy`: default NULL, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE assign NA.

`style`: Starting from a binary neighbours list, in which regions are either listed as neighbours or are absent (thus not in the set of neighbours for some definition), the function adds a weights list with values given by the coding scheme style chosen. B is the basic binary coding, W is row standardised (sums over all links to n), C is globally standardised (sums over all links to n), U is equal to C divided by the number of neighbours (sums over all links to unity), while S is the variance-stabilizing coding scheme proposed by Tiefelsdorf et al. 1999, p. 167-168 (sums over all links to n).

`nsim`: number of permutations

`alternative`: a character string specifying the alternative hypothesis, must be one of “greater” (default),“less” or “less”.

# Value

A dataframes containing the following components: Moran I test under randomisation, Monte-Carlo simulation of Moran I, Local Moran’s I statistic summary, and Monte-Carlo simulation of Local Moran’s I statistic summary. Also the value of the standard deviate of Moran’s I, p.value the p-value of the test, the value of the observed Moran’s I, its expectation and variance under the method assumption.

The Spatial Autocorrelation (Global Moran’s I) tool is an inferential statistic, which means that the results of the analysis are always interpreted within the context of its null hypothesis.

- When the p-value returned by this tool is statistically significant, you can reject the null hypothesis.

- When the p-value is not statistically significant You cannot reject the null hypothesis. It is quite possible that the spatial distribution of feature values is the result of random spatial processes.

- If the p-value is statistically significant, and the z-score is positive.You may reject the null hypothesis. The spatial distribution of high values and/or low values in the dataset is more spatially clustered than would be expected if underlying spatial processes were random.

*If p-value is statistically significant, and the z-score is negative.You may reject the null hypothesis. The spatial distribution of high values and low values in the dataset is more spatially dispersed than would be expected if underlying spatial processes were random

# Video

Video 1 : Spatial Autocorrelation

# References

Bivand, Roger S. and Wong, David W. S. (2018) Comparing implementations of global and local indicators of spatial association TEST, 27(3), 716-748. URL https://doi.org/10.1007/s11749-018-0599-x

Tiefelsdorf, M., Griffith, D. A., Boots, B. (1999). A variance-stabilizing coding scheme for spatial link matrices, Environment and Planning A, 31, pp. 165–180; Kelejian, H. H., and I. R. Prucha. 2010. Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances. Journal of Econometrics, 157: pp. 53–67.