Image Processing Reference

In-Depth Information

classes with a small number of patterns is more important than major classes with a large

number of training patterns, a situation that is often the case for medical datasets where

the number of malignant cases far exceeds those of benign ones. The weight of a training

pattern x
p
from class C is specified by the inverse of the proportion of the class over the

given training patterns as

Z
·
m

1

ω
p
= ω
C
=

,

p = 1, . . . , m,

C = 1, . . . , M,

(6.17)

N
C

where ω
p
is the weight of the training pattern x
p
that is from class C, ω
C
is the weight of

patterns from the class C, m is the number of given training patterns, N
c
is the number of

patterns from the class C, and Z is a normalisation factor that makes the maximum value

of weights from the class C a unit value (i.e. max

C

ω
C
= 1).

Learning Fuzzy If-Then rules for weighted training patterns is a strategy that adjusts the

grades of certainty CF
j
and can be employed to achieve improved classification performance.

It is based on an incremental learning approach where the adjustment occurs whenever clas-

sification of training patterns is performed. When a training pattern is correctly classified

we reinforce the grade of certainty of the fuzzy rule that is used for the classification. On the

other hand, we decrease the grade of certainty of rule if a training pattern is not successfully

classified.

Let us assume that we have generated fuzzy If-Then rules by the rule-generation procedure

detailed by Eq. 6.8 and Eq. 6.9. We also assume that a fuzzy If-Then rule R
j
is used for

the classification of a training pattern x
p
. That is, R
j
has the maximum product of the

compatibility and the grade of certainty (see Eq. 6.13).

The proposed learning method

adjusts the grades of certainty of R
j
as

CF
new

j

= CF
ol
j
−η·ω
p
·CF
old

if x
p
is misclassified

(6.18)

j

and

CF
new

j

= CF
old

j

+ η·ω
p
·(1−CF
old

)

if x
p
is correctly classified

(6.19)

j

where ω
p
is the weight of the training pattern x
p
, and η (the learning rate) is a positive

constant value in the interval [0; 1].

One epoch of the proposed learning method involves examining all given training patterns.

Thus, there will be m adjustments of fuzzy If-Then rules after all m training patterns are

examined. The learning process is summarised as follows:

1. Generate fuzzy If-Then rules from m given training patterns by the procedure

described by Eq. 6.4 and Eq. 6.5.

2. Set K as K = 1.

3. Set p as p = 1.

4. Classify x
p
by using the rules generated in Step 1.

5. After x
p
is classified, adjust the grades of certainty using Eq. 6.18 or eq. 6.19.

6. If p < m let p := p + 1 and go to Step 4. Otherwise go to Step 7.

7. If K reaches a pre-specified value, stop the learning procedure.

Otherwise let

K := K + 1 and go to Step 3.

Note that K in the above learning procedure corresponds to the number of epochs.

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