C++ Neural Networks and Fuzzy Logic: Preface

Y are an exemplar pair (in bipolar versions), you take the product X

T

 Y and add it to similar products from

other exemplar pairs, to get a weight matrix W. Some of the elements of the matrix W may be negative

numbers. In the unipolar context you do not have negative values and positive values at the same time. Only

one of them is allowed. Suppose you do not want any negative numbers; then one way of remedying the

situation is by adding a large enough constant to each element of the matrix. You cannot choose to add to only

the negative numbers that show up in the matrix. Let us look at an example.

Suppose you choose two pairs of vectors as possible exemplars. Let them be,

C++ Neural Networks and Fuzzy Logic:Preface

Additional Issues

176

---

X

1

 = (1, 0, 0, 1), Y

1

= (0, 1, 1)

and

X

2

 = (0, 1, 1, 0), Y

2

 = (1, 0, 1)

These you change into bipolar components and get, respectively, (1, –1, –1, 1), (–1, 1, 1), (–1, 1, 1, –1), and

(1, –1, 1). The calculation of W for the BAM was done as follows.

     1 [−1 1 1]   −1 [1 −1 1]  −1  1  1    −1  1 −1   −2  2 0

W = −1       + 1         =  1 −1 −1+1 −1  1 =  2 −2 0

    −1             1            1 −1 −1     1 −1  1    2 −2 0

     1            −1           −1  1  1    −1  1 −1   −2  2 0

and

                  −2    2    2   −2

W

T

 =    2   −2   −2    2

                   0    0    0    0

You see some negative numbers in the matrix W. If you add the constant m, m = 2, to all the elements in the

matrix, you get the following matrix, denoted by W

~

.

                   0    4    2

W

~

 =    4    0    2

                   4    0    2

                   0    4    2

You need a modification to the thresholding function as well. Earlier you had the following function.

      1  if y

j

 > 0   1   if  x

i

 > 0

      b

j

|

t+1

 = b

j

|

t

    if  y

j

 = 0   and      a

i

|

t+1

 = a

i

|

t

     if  x

i

 = 0

      0  if  y

j

 < 0  0   if  x

i

 < 0

Now you need to change the right−hand sides of the conditions from 0 to the product of m and sum of the

inputs of the neurons, that is to m times the sum of the inputs. For brevity, let us use S

i

 for the sum of the

inputs. This is not a constant, and its value depends on what values you get for neuron inputs in any one

direction. Then the thresholding function can be given as follows:

      1   if y

j

 > m S

i

         1  if  x

i

 > m S

i

b

j

|

t+1

 = b

j

|

t

      if  y

j

 = m S

i

       and      a

i

|

t+1

 = a

i

|

t

     if  x

i

 = m S

i

       0  if  y

j

 < m S

i

                         0  if  x

i

 < m S

i

For example, the vector X