Geoscience Reference
In-Depth Information
the key to the modeling is a transition probability matrix. For full details, see
the work of MacKenzie et al
.
(2003, 2006, 2009).
A transition probability matrix simply defines the probability of a unit
being in a particular category in season
t
+ 1 conditional on the category of
the unit in season
t
. This conditional approach creates the dependence struc-
ture of the model; technically, the modeling is assuming a first-order Markov
process. If the probability of a unit changing from state
m
in season
t
to state
n
in
t
+ 1 is φ
t
mn
[,]
, then the transition probability matrix for a situation with
three possible categories, for example, would look like the following:
[0,0]
[0,1]
[0,2]
φ
φ
φ
t
t
t
φ=
[1,0]
[1,1]
[1,2]
,
φ
φ
φ
t
t
t
t
[2,0]
[2,1]
[2,2]
φ
φ
φ
t
t
t
where the rows relate to the category of the unit at time
t
and columns denote
the category at
t
+ 1. Note that each row has to sum to 1.0; therefore, two of
the three probabilities in each row would be estimated and the third deter-
mined by subtraction. In this form, the parameterization is multinomial,
meaning that the two estimated probabilities are independent and uncon-
strained (except for the fact their sum must be less than 1.0.
Sometimes, the multinomial parameterization can have numerical difficul-
ties, particularly when covariates are to be included, and the interpretation
of covariates effects can also be unclear. When it makes sense, an alternative
parameterization is to use a conditional binomial approach. For example, if
ψ
+
[]
is defined as the probability of the species being present at the unit at
time
t
+ 1 conditional on the unit existing in category
m
at time
t
, and
m
1
t
R
t
m
[]
is defined as the probability of reproduction occurring at a unit at time
t
+ 1
given the species was present at the unit at
t
+ 1, and the unit was in category
m
at time
t
, then the transition probability matrix would look like this:
+
(
)
[0]
[0]
[0]
[0]
[0]
1
−ψ
ψ−
1
R
ψ
R
t
+
1
t
+
1
t
+
1
t
+
1
t
+
1
(
)
[1]
[1]
[1]
[1
[1]
φ=
1
−ψ
ψ−
1
R
ψ
R
t
t
+
1
t
+
1
t
+
1
t
+
1
t
+
1
(
)
[2]
[2]
[1]
[2]
[2]
1
−ψ
ψ−
1
R
ψ
R
t
+
1
t
+
1
t
+
1
t
+
1
t
+
1
Essentially, this parameterization is analogous to flipping a series of coins to
determine which category a unit is in rather than rolling a die.
Note also that biologically interesting questions can be considered with
these modeling approaches. For example, does the probability of reproduc-
tion occurring at a unit depend on whether reproduction occurred at the
unit previously? If so, that would imply
[1]
[2]
[1]
[2]
RR
t
≠
; if not, then
RR
t
=
. Such
+
1
t
+
1
+
1
t
+
1