Page 60 - Neural Network Modeling and Identification of Dynamical Systems

48 2. DYNAMIC NEURAL NETWORKS: STRUCTURES AND TRAINING METHODS

FIGURE 2.20 Neuron as a module converting n-dimen-
sional input vector into m-dimensional output vector. From
[109], used with permission from Moscow Aviation Institute.

where N L is the number of layers into which the
set of ANN elements is divided; p, q, r are the
indices used to number the arbitrary (“current”)
ANN layer.
In the list (2.14) L (0) is the input (zero) layer,
the purpose of which is to “distribute” the input
data to the neuron elements, which perform the
(1)
primary data processing. Layers L ,...,L (N L )
FIGURE 2.21 The primitive mappings of which the neu-
ensure the processing of the inputs of the ANN ron consists. From [109], used with permission from Moscow
into the outputs. Aviation Institute.
Suppose that in the ANN there are N L layers
L (p) ,p = 0,1,...,N L . 1) set of input mappings f i (x (in) ):
The layer L (p) has N (p) elements of the neu- i
L
(p) (in)
ron elements S , i.e., f i : R → R; u i = f i (x ), i = 1,...,n;
j i
(2.17)
(p)
(p)
L (p) ={S }, j = 1,...,N . (2.16)
j L
2) aggregating mapping (“input star”) ϕ(u 1 ,...,
(p) (p) (p,q) u n ):
The element S has N inputs x and
j j i,j
(p) (p,q) n
M outputs x . ϕ : R → R; v = ϕ(u 1 ,...,u n ); (2.18)
j j,k
(p)
The connections of the element S
j 3) converter (activation function) (v):
with other elements of the network can be
represented as a set of tuples showing where ψ : R → R; y = ψ(v); (2.19)
(p)
the outputs of the element S are trans-
j (m)
ferred. 4) output mapping (“output star”) E :
Thus, a single neuron as a module of the (m) m (m) (out)
E : R → R ; E (y) ={x },
ANN (Fig. 2.20) is a mapping of the n- j
(in)
dimensional input vector x (in) = (x (in) ,...,x n ) j = 1,...,m,
1
into the m-dimensional output vector x (out) = x (out) = y, ∀j ∈{j = 1,...,m}.
(out) (out) (out) (in) j
(x ,...,x m ), i.e., x = (x ).
1 (2.20)
The mapping is formed as a composi-
tion of the following primitive mappings The relations (2.20) are interpreted as follows:
(Fig. 2.21): mapping E (m) (y) generates as a result an m-