Agriculture Reference
In-Depth Information
"
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y
z
ðÞʱ þ
X
j
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y
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t
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Pr
y
z
ðÞy
z
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ð
1
:
55
Þ
Given the dependences that are assumed by the auto-logistic model, the resulting
likelihood does not have a closed form (Besag
1974
).
To solve this problem, Besag (
1975
) suggested estimating model parameters
using a maximum pseudolikelihood procedure. Here, the term pseudolikelihood
indicates the use of the product of the conditional probabilities given in Eq. (
1.55
).
Many other statistical models for spatial data have been proposed, for both
continuous and qualitative variables. See Cressie (
1993
) for detailed descriptions.
1.4.3.3 Spatial Point Pattern Analysis
, where z ¼ z
1
;
d
Consider a stochastic process
y ðÞ:
z
2 D
ℝ
ð
z
2
; ...;
z
n
Þ
are
random points where the events of interest have occurred.
The properties of a spatial point process can be described in terms of the
intensity function
(z), which represents the expected number of points in a small
area around z. A common statistical model used for random point patterns is the
homogeneous Poisson process, which is also called the complete spatial random-
ness (CSR) process. It has two important properties (Diggle
2003
):
1. The number of events in any region,
A
, follows the Poisson distribution with a
mean of
ʻ
|
A
|, where |
A
| is the area of
A
.
2. The position of any
n
points in
A
is an independent sample from the uniform
distribution on
A
.
ʻ
This process constitutes the reference statistical model for the analysis of
univariate point patterns (Diggle
2003
). Following this law, points are generated
in the study area subject to two conditions: (i) the region of interest is homoge-
neous, and (ii) there are no attractive or inhibitory interactions between the points.
In other words, (i) means that the density of points is constant (homogeneous) over
the study area. For a random sample of sub-regions, the frequency distribution of
the number of points in each region will follow a Poisson distribution, where the
expected
number of points in any sub-region is the same. Condition (ii) states that
the location of one point in space does not affect the probabilities of nearby points.
The CSR pattern is used as a benchmark to identify two broad classes of patterns
that constitute violations of the fundamental assumptions in (i) and (ii). These
non-casual patterns are defined as the uniform and clustered patterns. In the uniform
pattern, every point is regularly distanced from all of its neighbors. In the clustered
pattern, many points are concentrated close together, and there are large areas that
contain very few, if any, points (see Fig.
1.3
).
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