NOTEBOOK FOR SPATIAL DATA ANALYSIS
Part I. Spatial Point Pattern Analysis
______________________________________________________________________________________
________________________________________________________________________
ESE 502
I.21
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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentNOTEBOOK FOR SPATIAL DATA ANALYSIS
Part I. Spatial Point Pattern Analysis
______________________________________________________________________________________
________________________________________________________________________
ESE 502
I.22
Tony E. Smith
(2)
Pr(  )
( )/ ( )
Pr(
 )
Pr(
 ) Pr(  )
Pr(
 )
Pr(  )
( )/ ( )
CS
aC aS
CR
RS
R
Sa
R
a
S
=⋅⇒
=
=
()
Pr(  )
aC
aR
⇒=
and hence that the square in (2) can be replace by any
bounded region
,
R
, in the plane.
This fundamental proportionality result, which we designate as the
Spatial Laplace
Principle
, forms the basis for all models of spatial randomness.
In probability terms, this principle induces a
uniform probability distribution
on
R
,
describing the location of a single random point. With respect to any given cell,
⊂
, it
convenient to characterize this event as a
Bernoulli
(
binary
)
random variable
,
X C
,
where
() 1
XC
=
if the point is located in
C
and
( )
0
=
otherwise. In these terms, it
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 ese502
 Probability theory, spatial data, Spatial data analysis, Tony E. Smith, I. Spatial Point, Spatial Point Pattern

Click to edit the document details