Prediction and Optimization of Heavy Clay Products Quality - Advanced Materials for Agriculture, Food and Environmental Safety - page 92

where x and y represent the number of input and output neurons, respec-
tively, n is the number of neurons in the hidden layer and m is the number
of weights (connections between layers) in the neural network. m can be
taken as the number of training exemplars divided by 10. Some sugges-
tions regarding the number of hidden neurons are as follows: this number
should be between the sizes of the input and output layers, it should be 2/3
the size of the input layer, plus the size of the output layer, or less than twice
the size of the input layer [16].
In this work, the ANN procedure of StatSot Statistica was used to
model the ANN, and the number of hidden neurons varied from n = 5 to
13 (proposed by the program used). h ere were x = 10 inputs, y = 10 out-
puts, and m = 115 to 283 weight coei cients (depending on n ). Broyden-
Fletcher-Goldfarb-Shanno (BFGS) algorithm, implemented in StatSot
Statistica's evaluation routine, was used for ANN modeling. h e informa-
tion is transferred, between the layers through a “transfer” or “activation”
function. h is function is typically nonlinear for hidden layers and linear
for the output layer. Most common nonlinear activation functions, used
in StatSot Statistica ANN calculation, are logistic, sigmoid, hyperbolic,
and tangent functions (also exponential, sine, sot max, Gausian). In most
applications, hyperbolic tangent function behaves better as compared to
the other functions [13].
Coei cients associated with the hidden layer (both weights and biases)
are grouped in matrices W 1 and B 1 . Similarly, coei cients associated with
the output layer are grouped in matrices W 2 and B 2 . If Y is the matrix of
the output variables, f 1 and f 2 are transfer functions in the hidden and
output layers, respectively, and X is the matrix of input variables, it is
possible to represent the neural network, by using matrix notation, as
follows [17]:

Y = f 1 ( W 2 ⋅ f 2 ( W 1 ⋅ X + B 1 ) + B 2 ) (4.4)

Weights (elements of matrices W 1 and W 2 ) are determined during the
training step, which updates them using optimization procedures to mini-
mize the error function between network and experimental outputs [15,
16], evaluated according to the sum of squares (SOS) and BFGS algorithm,
used to speed up and stabilize convergence [18].

4.2.4.1 Training, Testing and System Implementation

At er dei ning the architecture of ANN, the training step is initiated.
h e training process was repeated several times in order to get the best