Geology Reference
In-Depth Information
employs gradient descent and gradient descent with momentum which are often too
slow for practical problems because they require small learning rates for stable
learning. Algorithms such as Conjugate Gradient, quasi-Newton, and Leven-
berg
Marquardt (LM) are considered to be some of the faster algorithms, which all
make use of standard numerical optimization techniques. Architecture of the model
including number of hidden layers is also a very important factor.
This study has used a three-layer feed-forward neural network (one input layer,
one hidden layer, and one output layer) which is the most commonly used topology
in hydrology. This topology has proved its ability in modeling many real-world
functional problems. The selection of hidden neurons is the tricky part in ANN
modeling, as it relates to the complexity of the system being modeled. In this study
we have used 15 hidden neurons which were identi
-
ed through a trial and error
method. We have applied the Gamma Test to identify the input space and training
data length for ANN modeling [
1
] (the equations of the Gamma Test procedure is
given in Chap.
3
).
We have tabulated the Gamma static values corresponding to all data sets in our
input space and selected the
first four inputs with minimum Gamma static value.
The embedding 000001 model (i.e., wind speed data as input and
ow data as
output) was identi
ed as the best structure in comparison to other models with
single inputs for daily discharge modeling in the Beas catchment. The humidity
data is also considered as the second most effective input data series because of its
low noise level (
value), the rapid fall off of the M-test error graph, relatively low
V-ratio value (indicating the existence of a reasonably accurate and smooth model),
and the regression line
Г
fit with slope A = 1.3882 (low enough as a simple nonlinear
model with less complexity). The relative importance of six input data sets in
modeling are wind speed followed by humidity, minimum temperature, precipita-
tion, maximum temperature, and solar radiation. As per cross correlation and
Gamma Test method, we have used four inputs for modeling and those inputs are
daily values of wind speed, humidity, minimum temperature, and precipitation. To
discover the suitable length of data for training the model, we have employed the
M-test associated with Gamma Test software. The M-test (a repeat execution of the
Gamma Test with a different number of input data lengths) was performed on four
input space (mentioned earlier). The results obtained from the M-test with four
inputs (110011 Model) are shown in Fig.
8.12
.
The M-test produced an asymptotic convergence of the Gamma statistics to a
value of 0.0805 at around 566 data points, then increases. Gamma static value gave
a clear indication that it is quite adequate to construct a nonlinear predictive model
using around 566 data points with reasonable accuracy. So, in this study, the
training data length is selected as 566 data points. The ANN model is calibrated
using this input space and data length of 566; the modeling results are shown in
Table
8.1
. The line plots during ANN training and validation are shown in
Figs.
8.13
and
8.14
for training and validation phases.
