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(34)
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where zi,p is the activation value of input unit zi , and vji is the weight between input zi and hidden unit yj An alternative to the above formulation of the net input signal for PUs is to include a distortion factor within the product [406], such as
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where zI+1,p = 1 for all patterns; vj,I+1 represents the distortion factor The purpose of the distortion factor is to dynamically shape the activation function during training to more closely t the shape of the true function represented by the training data If zi,p < 0, then zi,p can be written as the complex number zi,p = 2 |zi,p | ( = 1) that, substituted in (34), yields netyj,p = e
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31 Neural Network Types Let c = 0 + = a + b be a complex number representing Then, ln c = ln re = ln r + + 2 k where r = a2 + b2 = 1
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Considering only the main argument, arg(c), k = 0 which implies that 2 k = 0 Furthermore, = for = (0, 1) Therefore, = , which simpli es equation (310) 2 2 to ln c = , and consequently, 2 (38) ln 2 = Substitution of (38) in (36) gives netyj,p = e = e
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vji ) + sin
vji
(39)
Leaving out the imaginary part ([222] show that the added complexity of including the imaginary part does not help with increasing performance),
netyj,p = e Now, let
vji ln |zi,p |
cos
(310)
j,p j,p with Ii = and zi,p = 0 Then,
i=1 I
vji ln |zi,p | vji Ii
(311)
(312)
0 if zi,p > 0 1 if zi,p < 0
(313)
netyj,p = e j,p cos( j,p ) The output value for each output unit is then calculated as ok,p = fok
(314)
wkj fyj (e j,p cos( j,p ))
(315)
Note that a bias is now included for each output unit
3 Supervised Learning Neural Networks
Simple Recurrent Neural Networks
Simple recurrent neural networks (SRNN) have feedback connections which add the ability to also learn the temporal characteristics of the data set Several di erent types of SRNNs have been developed, of which the Elman and Jordan SRNNs are simple extensions of FFNNs
Context layer
Figure 33 Elman Simple Recurrent Neural Network The Elman SRNN [236], as illustrated in Figure 33, makes a copy of the hidden layer, which is referred to as the context layer The purpose of the context layer is to store the previous state of the hidden layer, ie the state of the hidden layer at the previous pattern presentation The context layer serves as an extension of the input layer, feeding signals representing previous network states, to the hidden layer The input vector is therefore z = (z1 , , zI+1 , zI+2 , , zI+1+J )
actual inputs context units
(316)
Context units zI+2 , , zI+1+J are fully interconnected with all hidden units The connections from each hidden unit yj (for j = 1, , J) to its corresponding context
31 Neural Network Types
unit zI+1+j have a weight of 1 Hence, the activation value yj is simply copied to zI+1+j It is, however, possible to have weights not equal to 1, in which case the in uence of previous states is weighted Determining such weights adds additional complexity to the training step Each output unit s activation is then calculated as ok,p = fok
I+1+J
vji zi,p ) (317)
wkj fyj (
j=1 i=1
where (zI+2,p , , zI+1+J,p ) = (y1,p (t 1), , yJ,p (t 1))
State layer
Figure 34 Jordan Simple Recurrent Neural Network Jordan SRNNs [428], on the other hand, make a copy of the output layer instead of the hidden layer The copy of the output layer, referred to as the state layer, extends the input layer to z = (z1 , , zI+1 , zI+2 , , zI+1+K ) (318)