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

(Tsiamyrtzis and Karlis, 2004). The main reason for this is the unmanageable
probability function, which causes inferential procedures to be somewhat problematic
and difficult. For example, consider the estimation of maximum likelihood (ML)
estimates. To estimate the likelihood, one has to calculate the probability function at all
the observations. The probability function can be calculated via recurrence
relationships, otherwise exhausting summations are needed (Tsiamyrtzis and Karlis,
2004). The efficient algorithm must be used to do the calculation of probabilities
(especially for higher dimensions) to save time. Applying purely the recurrence
relationships without trying to use them in an optimal way can be difficult and timeconsuming
(Tsiamyrtzis and Karlis, 2004). For further motivation, consider a problem
with, say three-dimensional data. For example, the data may represent the three
different weed species counts in a field. If the number of locations is not very large, this
implies that it will result many cells with zero frequency, so the calculation of the entire
three-dimensional space of all combinations for the number of count (plants) of the
three weed species is a very bad approach. For instance, if the maximum number of
counts for each species is denoted as a
i
, then to create the entire probability table, one
has to calculate
3
∏ ( ai
+ 1) different probabilities using normal strategy. Clearly, the
i=
1
calculation of these probabilities is awkward and time-consuming. It is especially usual
if one can only calculate the non-zero frequency cells, which contribute to the
likelihood. Tsiamyrtzis and Karlis (2004) proposed efficient strategies for calculating
the multivariate Poisson probabilities based on the existing recurrence relationships.
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