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process has spent in that state. This requires modeling the regime transition as a semi-
Markov process [3].
To model this we modify the Markov transition matrix,
T
predict
, to be a weighted sum
of two matrices, the steady state matrix
T
steady
and the change matrix
T
change
.
T
steady
is
the
M
M
identity matrix, where
M
is the number of regimes.
T
change
is the Markov
transition matrix, which is computed off-line as described earlier.
×
T
predict
(
r
t
+1
|
r
t
)=(1
−
ω
(
.
))
T
steady
+
ω
(
.
)
T
change
(
r
t
+1
|
r
t
)
(8)
where
ω
(
.
) represents the probability of a regime change, and
r
t
represents the current
regime. To compute the value of
ω
(
.
), we need to introduce a few variables. We define
Δt
as the time since the last regime transition at
t
0
:
Δt
=
t
t
0
. We model the time
τ
i
spent in regime
R
i
before the transition to regime
R
j
occurs as a random variable with
distribution
F
ij
.
τ
i
is estimated from historical data. We hypothesized that the probabil-
ity density of
τ
i
is dependent on the current regime,
R
i
,i.e.
p
(
τ
i
|
−
R
i
). We computed the
frequency of all values of
τ
i
in ascending order and fitted different distributions. The
Gamma distribution,
g
(
t
;
α, λ
) is a reasonable fit to the data.
The probability of a regime transition
ω
(
r, Δt
) from the current regime,
r
, with
respect to the time
Δt
that has elapsed since the last regime transition,
t
0
, is given by:
ω
(
r
=
R
i
,Δt
)=
Δt
p
(
Δt
|
r
=
R
i
)d
Δt
(9)
0
where
p
(
Δt
r
=
R
i
)=
g
(
Δt
;
α
i
,λ
i
). Equation 10 describes a recursive computation
for predicting the posterior distribution of regimes at time
t
+
n
days into the future,
where
k
=
n
+1, for the semi-Markov process.
|
P
(
r
t
+
k
|{
np
t
0
,...,
np
t−
1
}
)=
(10)
r
t
+
k−
···
k
P
(
r
t−
1
|{
np
t
0
,...,
np
t−
1
}
·
T
predict
(
r
t
+
j
|
r
t
+
j−
1
,Δt
+
j
−
)
1)
r
t−
1
j
=1
The second measure of success, correctness of prediction of the time of regime change,
which we obtained using the semi-Markov model, is shown in Table 1.
6
Related Work
Marketing research methods have been developed to understand the conditions for
growth in performance and the role that marketing actions can play to improve sales.
For instance, in [4], an analysis is presented on how in mature economic markets strate-
gic windows of change alternate with long periods of stability.
Model selection is the task of choosing a model of optimal complexity for the given
data. A good overview of concepts, theory and model selection methods is given in [5].
Much work has focused on models for rational decision-making in autonomous
agents. Ng and Russel [6] show that an agent's decisions can be viewed as a set of
linear constraints on the space of possible utility (reward) functions. However, the sim-
ple reward structure they used in their experiments will not scale to what is needed to
predict prices in TAC SCM.
