Self-Organizing Maps and Their Applications in Image Processing, Information Organization, and Retrieval - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 412

We can also observe for T MMSOM in relation to T SOM and T opt that:

1. For 5, the MMSOM yields the same threshold with the SOM.

In this case, both methods attain the same probability of false clas-
sification.
2. When

= 0 .

10, MMSOM
introduces a correction to the threshold toward the direction of T opt ,
i.e.,

( m 2 − m 1 ) σ , e.g., σ = 1 for m 1 = 5 and m 2 =

< 0 . 5: T opt < T mid < T MMSOM < T SOM

(11.72)

> 0 . 5: T SOM < T MMSOM < T mid < T opt .

As a consequence, the MMSOM attains a smaller probability of false
classification than the SOM for the same .

3. When the overlap between the cluster pdfs progressively increases
(e.g., σ = 3 for m 1 = 5 and m 2 = 10), the following inequalities are

satisfied:

< 0 . 5: T opt < T mid < T SOM < T MMSOM

(11.73)

> . < < <

0 5: T MMSOM T SOM T mid T opt

which imply that the MMSOM is inferior to the SOM with respect
to the probability of false classification for the same .

The bias introduced by the SOM and the MMSOM in estimating the true
cluster means of the Gaussian mixture model (Equation 11.49) is studied in
Figure 11.6a and b. We have also included the conditional means that cor-
respond to the decision regions predicted by T opt . The bias introduced in
estimating the true mean for the dominating cluster by all methods is plotted
in Figure 11.6a. In other words, in Figure 11.6a we have plotted the following
quantities:

| w − | ≤ .

m 2 for 0 5

2

(11.74)

| w − m 1 | for > 0 . 5

1

vs. 3. The sum of biases introduced in estimating both the true

means is plotted vs.

for σ =

σ = > . < .

45 the
conditional mean for the decision region determined by T opt that corresponds
to the dominating cluster introduces the smallest bias. MMSOM introduces
smaller bias in estimating the true mean of the dominating cluster than the
SOM in all cases. For

for 3inFigure 11.6b. For 0 55 or 0

= 0 . 5, T opt and T SOM yield the same bias. On the other

hand, for 0 55, it is worth noting that MMSOM outperforms both

the choice T opt and the SOM with respect to the bias. From the preceding
analysis, we conclude that:

. 45 << 0 .

1. The conditional mean for the decision region defined by T opt enables
a more accurate representation of the patterns that belong to the
dominating cluster. Such an approach favors the majority.