Information Technology Reference
In-Depth Information
Using Bayes' rule we determine the posterior probability:
c
i
)
P
(
c
i
)
i
=1
p
(np
p
(np
|
P
(
c
i
|
∀
i
=1
,
···
,N
np) =
(2)
|
c
i
)
P
(
c
i
)
We then define the N-dimensional vector, whose components are the posterior proba-
bilities from the GMM,
η
(np) = [
P
(
c
1
|
np)
,P
(
c
2
|
np)
,...,P
(
c
N
|
np)]
(3)
and for each normalized price np
j
we compute
η
(np
j
) which is
η
evaluated at the np
j
price. We cluster these collections of vectors using k-means. The center of each cluster
corresponds to regime
R
k
for
k
=1
,
,M
,where
M
is the number of regimes.
Figure 3 shows the cluster centers, which correspond to regimes, for the low market
segment. The figure shows only some of sample points for better visualization.
···
P(c|R
1
)
P(c|R
2
)
P(c|R
3
)
1
0.8
0.6
0.4
0
0.2
0.2
0.4
0
0.6
0
0.2
0.8
0.4
0.6
0.8
P(c
1
|np)
1
1
P(c
2
|np)
Fig. 3.
K-means clustering applied to the posterior probability
P
(
c|
np) in the low market segment
We distinguish three regimes, namely over-supply (
R
1
), balanced (
R
2
), and scarcity
(
R
3
). Regime
R
1
represents a situation where there is a glut in the market, i.e. an over-
supply situation, which depresses prices. Regime
R
2
represents a balanced market sit-
uation, where most of the demand is satisfied. In regime
R
2
the agent has a range
of options of price vs sales volume. Regime
R
3
represents a situation where there is
scarcity of products in the market, which increases prices. In this case the agent should
price close to the customer reserve price - the maximum price a customer is willing to
pay.
The number of regimes was selected a priori, after examining the data and looking
at economic analyses of market situations. Both the computation of the GMM and k-
means clustering were tried with different initial conditions, but consistently converged