NOTEBOOK FOR SPATIAL DATA ANALYSIS
Part I. Spatial Point Pattern Analysis
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ESE 502
I.2-1
Tony E. Smith
2. Models of Spatial Randomness
As with most all statistical analyses, cluster analysis of point patterns begins by asking:
What would point patterns look like if points were
randomly distributed
?
This requires a
statistical model of randomly located points.
2.1 Spatial Laplace Principle
To develop such a model, we begin by considering a square region,
S
, on the plane and
divide it in half, as shown on the left in Figure 1 below:
The
Laplace Principle
of probability theory asserts that if there is no information to
indicate that either of two events is more likely, then they should be treated as equally
likely, i.e., as having the
same probability
of occuring.
1
Hence by applying this principle
to the case of a randomly located point in square,
S
, there is no reason to believe that this
point is more likely to appear in either left half or the (identical) right half. So these two
(mutually exclusive and collectively exhaustive) events should have the same probability,
1/2, as shown in the figure. But if these halves are in turn divided into equal quarters,
then the same argument shows that each of these four “occupancy” events should have
probability 1/4. If we continue in this way, then the square can be divided into a large
number of
n
grid cells, each with the same probability,
1
n
, of containing the point. Now
for any subregion (or
cell
),
CS
⊂
, the probability that
C
will contain this point is at
least as large as the sum of probabilities of all grid cells inside
C
, and similarly is no
greater that the sum of probabilities of all cells that intersect
C
. Hence by allowing
n
to
become arbitrarily large, it is evident that these two sums will converge to the same limit
– namely the fractional area of
S
inside
C
. Hence the probability,
Pr( | )
that a
random point in
S
lies in any cell
⊂
is
proportional to the area of
C
.
2
(1)
()
Pr( | )
aC
aS
=
Finally, since this must hold for any pair of nested regions
CRS
⊂⊂
it follows that
1
This is also known as Laplace’s “Principle of Insufficient Reason”.
2
This argument in fact simply repeats the construction of area itself in terms of Riemann sums (as for
example in Bartle (1975, section 24).
1/4
1/4
1/4
1/4
1/2
1/2
•
•
•
C
S
Fig. 1. Spatial Laplace Principle