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help for spautoc
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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.

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

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