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The corresponding recommendations by GT and entropy theory were 1,056 and
1,040 data points. However the AIC and BIC values were minimum when we used
1,750 data points with AIC and BIC values of 11420.54 and 5081.889 respectively.
Figure
6.12
shows that there is more than one local minimum in the curve and GT
and entropy results are close to the
first depression in the
gure. The AIC and BIC
analysis did not perfectly matching with previous
findings of GT. Entropy Theory
and traditional approaches. This mismatching result indicates the need for further
exploration on data selection approaches.
6.5 Data Based Rainfall: Runoff Modelling
In previous sections we have adopted many novel data selection approaches to
identify the most in
cient data length for a reliable
smooth model. In this section we perform further data based modelling concen-
trating on the recommendations of the Gamma Test.
uencing data series and suf
6.5.1 Modelling with ARX, ARMAX and ANN
The ARX and ARMAX models can be viewed as a simpler version of the ANN
model with a linear threshold function as the transfer function and no hidden layer.
In general, the modeller needs to identify the (unknown) number of past outputs,
inputs, and error terms (in case of ARMAX) to perform the modelling. The normal
procedure adopted is to make different models with different combination of past
inputs, outputs and error terms and, later, to evaluate the ef
ciency of such models
using some residual statistics like RMSE or information criteria like AIC, and BIC.
However, in this study we stick to the suggestions of GT (i.e. three antecedent
information of runoff and two antecedent information of rainfall). Later the GT
suggestion was veri
ed by making different models with different combinations of
the inputs. In this section we show the results obtained from ARX (3, 2) and
ARMAX (3, 2, 2) in comparison with predictions made by three ANN models with
different training algorithm (Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm,
the Conjugate Gradient Algorithm and the Levenberg
Marquardt training algo-
-
rithm). The statistical performance of the identi
ed models for the 1,056 training
data and remaining validation data are summarised in Table
6.4
. The order selection
adopted for ARMAX model is shown in Fig.
6.13
. The Y-axis in the
gure shows
the unexplained output variance (in %) which is an indicator of model prediction
error for a speci
figure that, after
the seventh parameter, there is little difference in the variance. However, the best
c number of parameters. It has identi
ed from the
t
was observed when the total number of parameters was 20. The AIC criterion has
identi
ed the best order as seven (n
a
=3n
b
= 2 and n
k
= 2) for the ARMAX model.
