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

The observation sequences of three variables are denoted by Y 1
, Y 2
, Y 3
row vectors
(with uppercase letters)
Y
1
= ⎡ ⎣ y11 y12 ... y1j
... y ⎤
1n
⎦ .
Y
2
= ⎡ ⎣ y21 y22 ... y2 j
... y ⎤
2n
⎦ .
Y
3
= ⎡ ⎣ y31 y32 ... y3j
... y ⎤
3n
⎦ .
To denote the observation sets we will use column vectors
y ,...,
1
, y
2
y
n
(with lowercase
letters)
y
1
⎡ y
=
⎢
⎢
y
⎢⎣
y
11
21
31
⎤
⎥
⎥
⎥⎦
⎡ y12
⎤
y
⎢ ⎥
2
=
⎢
y22
⎥
…
⎢⎣
y ⎥
32 ⎦
⎡ y1
n ⎤
y
⎢ ⎥
n
=
⎢
y2n
⎥
.
⎢⎣
y ⎥
3n
⎦
Then Y = ⎡ ⎣ y1 y2 ... y
j
... y ⎤
n ⎦ where y
n
is a trivariate observation.
5.4.2 Estimation for the multivariate Poisson hidden Markov models (Extension of
the univariate Markov-dependent mixture model by Leroux and Puterman, 1992)
Let Y= ⎡ ⎣ y1 y2 . . . yj
. . . y ⎤
n ⎦ be the realization of a hidden Markov model
with original m state Markov Chain { i
}
S . Define Φ by ( P P ,..., , λ , λ ,..., λ )
11
,
12
P mm 1 2
m
92