C++ Neural Networks and Fuzzy Logic: Preface

F

1

 layer calculations:

       a

i

 = I

i

 / ( 1 + A ( I

i

 + B ) + C )

       x

i

 = 1 if a

i

 > 0

            0 if a

i

 d 0

F

2

 layer calculations:

       b

j

 = £ w

ij

 x

i

, the summation being on i from 1 to m

       y

j

 = 1 if jth neuron has the largest activation value in the F

2

            layer

          = 0 if jth neuron is not the winner in F

2

 layer

Top−down inputs:

       z

i

 = £v

ji

y

j

, the summation being on j from 1 to n (You will

       notice that exactly one term is nonzero)

F

1

 layer calculations:

C++ Neural Networks and Fuzzy Logic:Preface

Algorithm for ART1 Calculations

202

---

       a

i

 = ( I

i

 + D z

i

 − B ) / ( 1 + A ( I

i

 + D z

i

 ) + C )

       x

i

 = 1 if a

i

 > 0

          = 0 if a

i

 d 0

Checking with vigilance parameter:

If ( S

x

 / S

I

 ) <£, set y

j

 = 0 for all j, including the winner r in F

2

 layer, and consider the jth neuron inactive (this

step is reset, skip remaining steps).

If ( S

x

 / S

I

 ) e £, then continue.

Modifying top−down and bottom−up connection weight for winner r:

       v

ir

  = ( L / ( S

x

 + L −1 ) if x

i

 = 1

            = 0 if x

i

 = 0

       w

ri

  = 1 if x

i

 = 1

            = 0 if x

i

 = 0

Having finished with the current input pattern, we repeat these steps with a new input pattern. We lose the

index r given to one neuron as a winner and treat all neurons in the F

2

 layer with their original indices

(subscripts).

The above presentation of the algorithm is hoped to make all the steps as clear as possible. The process is

rather involved. To recapitulate, first an input vector is presented to the F

1

 layer neurons, their activations are

determined, and then the threshold function is used. The outputs of the F

1

 layer neurons constitute the inputs

to the F

2

 layer neurons, from which a winner is designated on the basis of the largest activation. The winner

only is allowed to be active, meaning that the output is 1 for the winner and 0 for all the rest. The equations

implicitly incorporate the use of the 2/3 rule that we mentioned earlier, and they also incorporate the way the

gain control is used. The gain control is designed to have a value 1 in the phase of determining the activations

of the neurons in the F

2

 layer and 0 if either there is no input vector or output from the F

2

 layer is propagated

to the F

1

 layer.

Other Models

Extensions of an ART1 model, which is for binary patterns, are ART2 and ART3. Of these, ART2 model

categorizes and stores analog−valued patterns, as well as binary patterns, while ART3 addresses

computational problems of hierarchies.