Page 138 - Statistics and Data Analysis in Geology

Spat ia I An a I ysis

distance between nearest neighbors, a, is equal to the value of 8 from a random
pattern of points of the same density. The test is
-
z=- d-8 (5.28)
Se
This is the form of the nearest-neighbor test that is commonly presented, but un-
- fortunately it has a serious defect for most practical purposes. The expected value
6 assumes that edge effects are not present, which means that the observed pattern
of points must extend to infinity in all directions if a and 8 are to be validly com-
pared. Since the map does not extend indefinitely, the nearest neighbors of points
near the edges must lie within the body of the map, and so d is biased toward a
greater value (Upton and Fingleton, 1985). There are several corrections for this
problem. If data are available beyond the limits of the area being analyzed, the
map can be surrounded by a guard region. Then, nearest-neighbor distances be-
tween points inside the map and points in the guard region can be included in the
calculation of d. Alternatively, we can consider our map to be drawn not on a flat
plane but on a torus. In this case, in the right map edge would be adjacent to the
left edge and the top adjacent to the bottom. The nearest neighbor of a point along
the right edge of the map might lie just inside the left edge (this concept should
be familiar to anyone who has contoured point densities on stereonets). Another
way of regarding this particular correction is to imagine that the pattern of points
repeats in all directions, like floor tiles. Any point lying adjacent to an edge of
the map has {he opportunity to find a point across the edge that may be a closer
neighbor than the nearest point within the map.
A third correction involves adjusting d so that the boundary effects are in-
cluded in its expected value. Using numerical simulation, Donnelly (1978) found
these alternative expressions for the theoretical mean nearest-neighbor distance
and its sampling variance:

(5.29)

and
JA
58 N 0.070- A + 0.035Pp
n2 (5.30)
In these approximations, p is the perimeter of the rectangular map. Note that if
the map has no edges, as when it is considered to be drawn on a torus, p is zero
and these equations are identical to equations (5.24) and (5.26):
The expected and observed mean nearest-neighbor distances can be used to
construct an index to the spatial pattern. The ratio
-
R=- d (5.31)
6
is the nearest-neighbor statistic and ranges from 0.0 for a distribution where all
points coincide and are separated by distances of zero, to 1.0 for a random dis-
tribution of points, to a maximum value of 2.15. The latter value characterizes
a distribution in which the mean distance to the nearest neighbor is maximized.
The distribution has the form of a regular hexagonal pattern where every point

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