J. Civil Eng. Urban., 10 (5): 42-52, 2020
composed of 1-5 variables. Then, the patterns will be
As shown, cohesion and internal friction angle have
the most positive effect on the safety factor; , while soil
density and the distance between anchor rods have a
negative effect . These cases were expected.
selected according to some criteria such as the coefficient
of determination, the coefficient of determination adjusted,
Mean Square Error (MSE), Mallow’s C_{p }and the best
model used by the five mean variables. (Table. 6)
As represented in Table 6, the processed models
includes cohesion; the second model includes cohesion
and the internal friction angle. The three-variable model
includes cohesion, internal friction angle and soil density;
the fourth model consists of cohesion, internal friction
angle, soil density and the distance between anchor rods.
Finally, the fifth model covers cohesion, internal friction
angle, soil density, the distance between anchor rods and
the elastic modulus of soil.
The results obtained from neural networks
The neural network has been considered to expect the
safety factor. It is supposed that there is a nonlinear and
complex relation between the safety factor and the
specifications of soil. Consequently, the neural network
has been considered to study the correlation between the
safety factor and the model parameters and to compare
the results obtained from multiple regression analysis.
Preprocessing the data
Table 6. Summary of the results obtained from all
possible regression models
Training of the neural network can be more effective
if some targets and inputs are preprocessed. Error
estimation method is used to scale the inputs whose value
of error is equal to zero. Then, the inputs and target will be
normal.
Number of
Variables
MSE
R^{2 }
Adjusted R^{2 }
Mallow ‘sCp
1
2
3
4
5
0.133
0.710
0.040
0.022
0.023
0.637
0.808
0.892
0.939
0.939
0.635
0.806
0.890
0.938
0.938
-0.45
-0.16
1.025
5.6
Making model by back propagation networks to
estimate the safety factor
0.34
In this research, to estimate the safety factor, a leading
network with a back propagation algorithm and error was
used. To train the network, the data were divided
randomly in three classes including training, validation
and test. Then, 70%, 15% and 15% of the data were
divided for training, validation and test, respectively.
Levenberg-Marquardt algorithm was used to train the
network and the root-mean-squared error was also applied
as a cost function. The network was composed of two
hidden layers and one output layer with arrangement (1,
50, and 20); the tangent sigmoid function was in the
hidden layers and linear function was in the output layer
(Figure.19). The number of the optimized layers and
neurons was obtained based on the trial and error; then the
desirable network was not unique. The results obtained
from three subsets of training, validation and test have
been illustrated in Figure 20. The correlation coefficient
between the measured and expected values was 0.998,
0.993 and 0.994 for training (Figure 20 A), validation
(Figure 20 B) and test (Figure 20 C), respectively.
Standardized regression coefficients
It is difficult to compare the regression coefficients,
because ꢅ_{j }(dependent variables coefficients) is a reflex of
the measurement units of the independent variable X_{j . }It
will be useful to use the scaled dependent and independent
variables that can lead to the regression coefficients
without unit. There are currently two scaling methods. The
first one is unit normal scaling. The second one is unit
length scaling. Then, the effect of each independent
variables can be addressed by the standardized regression
coefficients (Figure 18).
Figure 19. Schematic network including two hidden
layers and one output layer
Figure 18. The standardized regression coefficients for
the regression model with 4 variables
49