Page 253 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

242 UNSUPERVISED LEARNING

2. Iterate:

2.1 Find, for each object z n in the training set, the most similar
(i)
neuron w .
k
ðiÞ
kðz n Þ¼ arg min kz n w k ð7:31Þ
j
j
This is called the best-matching or winning neuron for this input
vector.
2.2 Update the winning neuron and its neighbours using the update
rule:

ðiÞ
w ðiþ1Þ ¼ w ðiÞ þ h ðjkðz n Þ jjÞðz n w Þ ð7:32Þ
ðiÞ ðiÞ
j j j
2.3 Repeat 2.1 and 2.2 for all samples z n in the data set.
2.4 If the weights in the previous steps did not change significantly,
then stop. Else, increment i and go to step 2.1.

(i)
Here (i) is the learning rate and h (jk(z n ) jj) is a weighting function.
Both can depend on the iteration number i. This weighting function
weighs how much a particular neuron in the grid is updated. The term
jk(z n ) jj indicates the distance between the winning neuron k(z n ) and
neuron j, measured over the grid. The winning neuron (for which
(i)
j ¼ k(z n )) will get the maximal weight, because h () is chosen such that:
ðiÞ
ðiÞ
h ðÞ 1 and h ð0Þ¼ 1 ð7:33Þ
Thus, the winning neuron will get the largest update. This update moves
the neuron in the direction of z n by the term (z n w j ).
The other neurons in the grid will receive smaller updates. Since we
want to preserve the neighbourhood relations only locally, the further
the neuron is from the winning neuron, the smaller the update.
A commonly used weighting function which satisfies these requirements
is the Gaussian function:

!
x 2
ðiÞ
h ðxÞ¼ exp 2 ð7:34Þ

ðiÞ
For this function a suitable scale (i) over the map should be defined.
This weighting function can be used for a grid of any dimension (not just
one-dimensional), when we realize that jk(z n ) jj means in general the
distance between the winning neuron k(z n ) and neuron j over the grid.