ESE 502
Tony E. Smith
________________________________________________________________________
1
DISCRETE SPATIAL AUTOREGRESSIVE MODELS
The standard logistic, binomial and Poisson regression models of discrete counting data
have natural spatial generalizations in a manner similar to the conditional autoregressive
(CAR) model. We consider each in turn.
1. Autologistic Model
Suppose one considers the voting outcomes for Pennsylvania counties during a given
Presidential election. Let
1
i
v
=
if county
i
voted Democratic and
0
i
v
=
otherwise. To
model this voting behavior, one may consider a number of county covariates,
1
(
,..,
)
i
i
ik
x
x
x
=
, say with
1
i
x
=
“per capita income” in county
i
,
2
i
x
=
“average years of
education” for voters in county
i
, and so on. A
standard logistic model
of voting behavior
would then take the form:
(1)
Pr(
1)
1
i
i
i
v
µ
µ
=
=
+
where
(2)
(
)
1
exp
k
i
j
ij
j
x
µ
α
β
=
=
+
∑
However, it may well be that there are voting similarities among adjacent counties. Hence,
if we let
1
ij
w
=
if county
j
is a contiguous
neighbor
of county
i
(with
0
ii
w
=
) and if we
let
1
1
1
(
,..,
,
,..,
)
i
i
i
n
v
v
v
v
v
−
−
+
=
denote the voting outcomes of all counties other than
i
, then a
natural spatial generalization of this model would be to hypothesize that the
conditional
distribution
of
i
v
given the voting outcomes,
i
v
−
, of all other counties depends only on the
voting behavior of its neighbors, say,
(3)
(
)
Pr(
1
)
1
(
)
i
i
i
i
i
i
v
v
v
v
µ
µ
−
−
−
=
=
+
with
(4)
(
)
1
(
)
exp
k
i
i
j
ij
ij
j
j
j i
v
x
w v
µ
α
β
ρ
−
=
≠
=
+
+
∑
∑
To estimate this model, we start with the standard logistic model that has a likelihood
function of the form:
(5)
1
1
(
,
 )
Pr(
1) [1
Pr(
1)]
i
i
n
v
v
i
i
i
L
v
v
v
α β
−
=
=
=
−
=
∏
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ESE 502
Tony E. Smith
________________________________________________________________________
2
{
}
1
log
(
,
 )
logPr(
1)
(1
)log[1
Pr(
1)]
n
i
i
i
i
i
L
v
v
v
v
v
α β
=
⇒
=
=
+
−
−
=
∑
where
1
(
,..,
)
n
v
v
v
=
and where Pr(
1)
i
v
=
is given by (1) and (2). This function is quite easy
to maximize, and yields well behaved maximumlikelihood estimates of
α
and
β
.
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 Fall '08
 ese502
 Conditional Probability, Probability theory, Maximum likelihood, Likelihood function, Tony E. Smith

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