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284 ENTERPRISE INFORMATION SYSTEMS VI
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APPENDIX A
In this appendix, we briefly describe the DAR-HMM according
to the formalism adopted by Rabiner in (Rabiner,
1989) to specify classical hidden Markov models:
• S � {si}, subcategories set with N = |S|;
• V � {vj}, predictable symbols set with M = |V |;
• V (i) = {v (i)
k }, set of symbols predictable in subcategory
i with M (i) = |V (i) | and V = �
i V (i) ;
• O(t) ∈ V , observed symbol at time t;
• Q(t) ∈ S, state at time t;
• πi(t) =P (Q(t) =si), probability that si is the actual
subcategory at time t;
• aii ′ = P (Q(t +1)=si|Q(t) =si ′), transition probability
from si ′ to si;
• b i k = P (O(0) = v (i)
k |Q(0) = si), probability of observing
v (i)
k from subcategory si at t =0;
• b ii′
kk ′ = P (O(t) =v(i)
k |Q(t) =si,O(t − 1) = v(i′ )
k ′ ),
probability of observing v (i)
k from the subcategory si
having just observed v (i′ )
k ′ .
DAR-HMM for symbolic prediction can thus be described
using a parameter vector λ =< Π 0 ,A,B >, where
Π 0 [N] is the vector of inital subcategory probability πi(0),
A[N][N] is the matrix with subcategory transition probabilities
aii ′, and B[N][M][M +1]is the emission matrix with
symbol probabilities b ii′
kk ′ and b i k.Givenλthe vector of parameters
describing a specific language behavior model, we
can predict the first observed symbol as the most probable
one at time t =0:
Ô(0) = arg max
v (i)
k
= argmax
v (i)
k
= argmax
v (i)
k
�
P (O(0) = v (i)
k |λ)
�
(P (O (0) |Q (0) ,λ) P (Q(0)))
�
b i �
k · πi(0) .
Then mimicking the DAR-HMM generative model, to predict
the t th symbol of a sentence we want to maximize the
symbol probability in the present (hidden) state given the
last observed symbol:
�
P O(t) =v (i)
k ,Q(t) =si|O(t − 1) = v(i′ �
)
k ′ ,λ .
Recalling that we can compute the probability of the current
(hidden) state as:
P (Q(t)) =
=
N� P (Q(t)|Q(t − 1)) P (Q(t − 1)) =
N�
i ′ =1
πi ′(t − 1)aii ′ = πi(t),
we obtain a recursive form for symbol prediction at time t:
�
b ii′
N�
�
kk ′ · πi ′(t − 1)aii ′ .
Ô(t) = arg max
v (i)
k
i ′ =1