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

eta[T,]=matrix(1,nrow=1,ncol=N)
for (t in (T-1):1){
beta[t,]=(beta[t+1,]*dens[t+1,])%*%t(TRANS)
beta[t,]=beta[t,]/scale[,t]
}
####M-step, reestmation of the transition matrix
##compute unnormalized transition probabilities (this is indeed still the end of the E-
#step, which explains that TRANS appears on the right-hand side below)
#[Refer to equation (5.24)]
P
n
∑
jk
i=
2
jk
= n m
∑∑
i= 2 l=
1
vˆ
( i)
. (5.24)
vˆ
( i)
jl
TRANS=TRANS*(t(alpha[1:(T-1),])%*%(dens[2:T,]*beta[2:T,]))
###Normalization of the transition matrix
oness=matrix(1,nrow=1,ncol=N)
sumtrans=matrix(0,nrow=N,ncol=1,byrow=T)
for (n in 1:N){
sumtrans[n,]=sum(TRANS[n,])
}
sumtrans=sumtrans%*%oness
TRANS=TRANS/sumtrans
####M-step, reestimation of the rates
###Compute a posteriori probabilities and store them in matrix beta to save space
#[Refer to equation (5.25) and (5.26)]
#The forward and backward variables
α ( i) = P[ y , y ,..., y , S = j]
and
#
j 1 2 n i
( i) = P[ y ,..., y | S = j]
(5.25)
# β
j i+
1 n i
#which yield the quantities of interest by
α
j() i β
j() i α
j() i β
j()
i
# uˆ j
() i = =
m
∑αl
( n)
α () i β () i
l
∑
j=
1
j
j
and
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