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

1
1
− d
− 1
2
−1
2
−1
−1
g(
λ | µ , Σ)
= (2π
) ( λ1,...,
λd
) | Σ | exp{ − (logλ
− µ ) Σ (logλ
− µ )} (7.10)
2
The multivariate Poisson-log normal distribution denoted by PΛ d ( µ, Σ)
is the
Λ d ( µ , Σ)
mixture of independent Po λ ) distributions (i =1,…, d ) with probability
density function
where
d
∫∏
d
R i=
1
+
( i
p( y | µ, Σ)
= f ( y | λ ) g(
λ | µ, Σ)
dλ
; ( y 1
,..., = 0,1,...)
(7.11)
d
R +
denotes the positive orthant of d-dimensional real space
i
i
y d
d
R .
It is not easy to simplify the multiple integral (7.11), but its moments can be easily
obtained through conditional expectation results and standard properties of the Poisson
and log normal distributions. The expectation, variance, covariance, and correlation for
the multivariate Poisson-log normal model are given below (Aitchison and Ho, 1989).
Let σ
ij
denotes the (i, j) element of Σ . Then
1
E( Y i
) = exp( µ
i
+ σ
ii
) = α
i
. (7.12)
2
2
Var(
) α + α {exp( σ ) −1}
. (7.13)
Y i
=
i i
ii
) =
iα
j{exp(
σ
ii
Cov( Y i
, Y j
α ) −1}
. (7.14)
Corr(
Y i
, Y j
) =
exp( σ ) −1
[{exp( σ ) −1+
α }{exp( σ ) −1+
α }]
ii
−1
i
ij
jj
−1
j
1
2
. (7.15)
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