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

5. The probability distribution of the observation symbol in state j : B = { (n)}
b ( n) = P[
v at time t | S = j],
1 ≤ j ≤ K,
1 ≤ n ≤ M ,
j
n
t
where v
n
denotes the n th observation symbol in a given state j .
b j
(n) should also satisfy the stochastic constraints
b j
( n)
≥ 0 , 1 ≤ j ≤ K,
1 ≤ n ≤ M and
b j
M
∑
n=
1
b j
( n)
= 1, 1 ≤ j ≤ K .
6. The above probability distribution is the case when the observations are discrete.
The initial state distribution π = { }
π i
, where
π = [ = ] , 1 ≤ j ≤ K .
i
PS1
i
From above definitions, it is clear that a complete specification of an HMM involves
three model parameters ( KMT , , ) and three sets of probability parameters ( ABπ , , ).
Therefore, for convenience, we can use the compact notation λ = ( A,
B,
π ) to denote
the complete set of parameters of the model throughout the thesis.
Before we go further, there are some assumptions that are made in the theory of hidden
Markov models for mathematical and computational tractability. First, it is assumed that
the next state is dependent only on the current state, which is called the Markov
assumption. That is,
PS [ = j| S = i, S = i ,..., S = i] = PS [ = j| S = i]
.
t+ 1 t t t−1 t− 1 0 0 t+
1 t t
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