Spatial autocorrelation (Moran and Geary measures)
spautoc xvar neivar [if exp] [in range] [, weight(strvar) lmean(newvar) lmedian(newvar)]
Description
spautoc calculates Moran and Geary measures of spatial autocorrelation for a spatial variable xvar and neighbourhood information given by neivar.
Remarks For n values of a spatial variable x defined for various locations, which might be points or areas, calculate the deviations
_ z = x - x
and for pairs of locations i and j, define a weights matrix
W = ( w ) ij describing which locations are neighbours in some precise sense. For example, a weight might be assigned as 1 if i and j are contiguous areas and 0 otherwise; or it might be a function of the distance between i and j and/or the length of boundary shared by i and j.
The Moran measure of autocorrelation is
n n n n n 2 n ( SUM SUM z w z ) / ( 2 (SUM SUM w ) SUM z ) i=1 j=1 i ij j i=1 j=1 ij i=1 i
and the Geary measure of autocorrelation is
n n 2 n n n 2 (n - 1) ( SUM SUM w (z - z ) ) / ( 4 (SUM SUM w ) SUM z ) i=1 j=1 ij i j i=1 j=1 ij i=1 i
and these measures may used to test the null hypothesis of no spatial autocorrelation, using both a sampling distribution assuming that x is normally distributed and a sampling distribution assuming randomisation, that is, we treat the data as one of n! assignments of the n values to the n locations.
spautoc avoids the use of Stata's matrix language, to avoid any limit on the number of locations which that would imply, and to avoid the need to handle a matrix that would typically be very sparse. The price paid is a data structure for the neighbourhood information that is idiosyncratic by Stata standards.
Here is a toy example:
area neighbours value ---- ---------- ----- 1 2, 3 and 4 3 2 1 and 4 2 3 1 and 4 2 4 1, 2 and 3 1 This would be matched by the data
_n (obs no) x (numeric variable) nei (string variable) ----------- -------------------- --------------------- 1 3 "2 3 4" 2 2 "1 4" 3 2 "1 4" 4 1 "1 2 3"
That is, nei contains the observation numbers of the neighbours of the location in the current observation, separated by spaces. Therefore, the data must be in precisely this sort order when spautoc is called.
Note various assumptions made here:
1. The neighbourhood information can be fitted into at most a str244 variable.
2. If i neighbours j, then j also neighbours i and both facts are specified.
By default this data structure implies that those locations listed have weights in W that are 1, while all other pairs of locations are not neighbours and have weights in W that are 0.
If the weights in W are not binary (1 or 0), use the weights() option. The variable specified must be another string variable.
_n (obs no) nei (string variable) weight (string variable) ----------- --------------------- ------------------------ 1 "2 3 4" ".1234 .5678 .9012" etc.
The weights matrix need not be symmetric.
Again, the assumption here is that these weights can be fitted into at most a str244 variable.
Options
weight() specifies a string variable containing numeric weights, as explained above.
lmean() and lmedian() specify that new variables should be generated containing local means and local medians, that is, means and medians of the neighbours of each location (not including that location). Any weights specified will be used in the calculation.
Examples
. spautoc cows nei
References
Cliff, A.D. and Ord, J.K. 1973. Spatial autocorrelation. London: Pion.
Cliff, A.D. and Ord, J.K. 1981. Spatial processes: models and applications. London: Pion.
Author
Nicholas J. Cox, Durham University, U.K. n.j.cox@durham.ac.uk
Also see
On-line: help for dupneigh, neigh, numids