Self-Organizing Maps and Unsupervised Classification - Neural Networks: Methodology and Applications - page 414

• Minimization phase. During that phase, the allocation function is kept

constant, and E ( W,σ,χ ) is minimized with respect to W and σ .
The parameters W and σ are updated as in the batch version of the SOM al-
gorithm by canceling the partial derivatives of the cost function E ( W t ,σ t ,χ t ).
To solve the equation, an iterative procedure is used as in [Duda et al. 1973],
assuming that for i th iteration the initial values of the parameters are close
to the optimal values. The update relations are the following:

f r z i , w t− 1

N

z i K δ r,χ t− 1 ( z i )

,σ t− 1
r

r

P χ t− 1 ( z i ) ( z i )

i =1

w r =

K δ r,χ t− 1 ( z i ) f r z i , w t− 1

r
P χ t− 1 ( z i ) ( z i )

N

,σ t− 1

r

i =1

K δ r,χ t− 1 ( z i ) f r z i , w t− 1

r
P χ t− 1 ( z i ) ( z i )

N

w t− 1

r − z i

,σ t− 1

2

r

σ r 2 =

i =1

.

K δ r,χ t− 1 ( z i ) f r z i , w t− 1

r
P χ t− 1 ( z i ) ( z i )

n N

,σ t− 1

r

i =1

In both above relations, the parameters at iteration t are expressed as func-
tions of the parameters at iteration t− 1.
Since the model is complex, an appropriate initialization is desirable. Since
PRSOM can be considered as extensions of SOM, one can first perform a SOM
estimation of the reference vector set W in order to initialize the mean vector
set of PRSOM.
Thus, the PRSOM training algorithm can be summarized as follows:

PRSOM Algorithm with Constant Temperature T

1. Initialization : t = 0. The initial values W 0 of the references are computed
using a SOM training algorithm, the σ 0
r

is computed by the mean of the

(Sect. 7.2.1). The initial allocation function χ 0 local inertia I r is derived

from the update relation

z i K δ r,χ t− 1 ( z i ) f r z i , w t− 1

r
P χ t− 1 ( z i ) ( z i )

N

,σ t− 1

r

w r =

i =1

,

K δ r,χ t− 1 ( z i ) f r z i , w t− 1

N

,σ t− 1
r r

P χ t− 1 ( z i ) ( z i )

i =1

f r z i , w t− 1

r
P χ t− 1 ( z i ) ( z i )

N

w t− 1

z i K δ r,χ t− 1 ( z i )

,σ t− 1

2

r

r −

σ r 2 =

i =1

.

K δ r,χ t− 1 ( z i ) f r z i , w t− 1

r
P χ t− 1 ( z i ) ( z i )

n N

,σ t− 1

r

i =1