multivariate poisson hidden markov models for analysis of spatial ...

Y and assumes a probability distribution for it. The systematic component specifies the
explanatory variables used as predictors in the model. The link function describes the
functional relationship between the systematic component and the expected value
(mean) of the random component. The generalized linear model relates a function of
that mean to the explanatory variables through a prediction equation having linear form
(Agresti, 2002). More details of the GLM and the loglinear analysis can be found in
Agresti (2002).
A generalized linear model using the log link function with a Poisson response is called
a loglinear model. The general use is modelling cell counts in contingency tables. The
models specify how the expected count depends on levels of the categorical variables
for that cell as well as associations and interactions among those variables. To calculate
the level of interdependence between two species and higher-order associations,
loglinear analysis provides a good statistical background to directly examine the higherorder
associations. Loglinear models methodology is mainly applicable when there is no
clear distinction between response and explanatory variables, for example, when all the
variables are observed simultaneously (Stokes et al., 2000). The loglinear model point
of view treats all variables as response variables, and the focus is on statistical
independence and dependence. Loglinear modelling of multi-way categorical data is
analogous to correlation analysis for normally distributed response variables and is
useful in assessing patterns of statistical dependence among the subsets of variables.
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