Andrei Rogers and Norbert G. Gomar
Statistical
inference
in Quadrat Analysis
The growing recognition of the need for establishing a systematic
and quantitative means for describing and analyzing, the spatial
dispersion of activities in urban areas has generated an interest in the
potential applicability of cell-counting or
quadrat
methods in urban
analysis
[6,8,16].
These exploratory efforts to apply quadrat methods
in analyses of urban spatial structure have justifiably emphasized
probability theory and
its use in model formulation. This paper
strives to complement such efforts by focusing on the other, equally
important, half of quadrat analysis: statistical inference and its use
in model verification.
In quadrat analysis, we may distinguish between efforts which
focus on the development of stochastic models of point distributions
and efforts which deal with the problem of inference regarding the
model that may be said to “account” for an observed point distribu-
tion. Efforts of the first kind address questions such
as:
given that 150
points are randomly placed inside
a
grid
of
100
squares, what
is the
expected number of unoccupied squares, squares containing one point,
and so on? Efforts of the second kind consider questions such
given that in a grid of
100
squares, superimposed over
a point pattern,
35
squares are empty,
31)
squares contain a single point, and so on,
what is the likelihood that the stochastic model that generated this
point distribution was a random spatial process? This paper deals
with the latter class of problems and, in particular, focuses on the use
of the chi-square goodness of
fit test to measure the adequacy of fit,
to an observed frequency distribution, that
is provided by alternative
quadrat models of point dispersion.
Let
XI ,x2
,
- -
.
,
xN
denote independent observations on a random
variable which has an unknown distribution function
F(x).
The
problem of testing whether
F(x)
=
Fo(z),
where
Fo(x)
is some par-
Andrei Rogers is associate professm of city planning with
the
Depart-
ment
City and Regional Planning, at the University of California
at
Berkeley. Nmbert
G. Gomar
is associated
with
the Institute of Trans-
portation and TraB Engineering at the University
California at
Berkeley.