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approximation (A) sub time series. Kisi [
9
] constructed a new series by adding some
relevant D values and constructing an effective series and used that along with
approximation component as an input of ANN. This may lead to loss in information
associate with individual data series and a better way is splitting the original series
in low resolution level and use those data series directly as input of ANN. So, the
present value of runoff has been estimated using the three resolution levels of
antecedent runoff and rainfall information (i.e. runoff and rainfall time series of
2-day mode D
q
1
D
i
p
3
and
approximate mode, where q denotes runoff, p denotes rainfall and i and j denotes
number of antecedent data sets of rainfall and runoff respectively). The GT is used
to identify the input structure (3 previous runoff information, one previous and
present rainfall information) used for the modelling. DWT is used to decompose the
input data into three wavelet decomposition levels (2
, 4-day mode D
q
2
, 8-day mode D
q
3
D
i
p
1
D
i
p
2
;
;
;
-
4
-
8). The three detailed
coef
first approximate series of the original data of runoff and
rainfall used in this study are presented in Fig.
6.25
a, b. The neuro-wavelet model
performance indices are summarised in the Table
6.5
.
The observed and estimated NW model values during the training and validation
phase is shown in the Fig.
26
a, b in terms of scatter plots. The MBE values are quite
small in the case of the NW model in comparison to the LLR, NNARX and ANFIS
models. The model ef
cient series and the
ciency of the NW model is higher than most of the models
used in this comparative study (nearly 5 % higher than ANFIS and more than 10 %
higher in the case of the NNARX model). In terms of MBE and S
d
, the performance
of the NW model outperforms all other tested models in both the training and
validation phases with a slight disparity with SVM models in the validation phase
(in terms of ef
ciency).
6.5.5 Rainfall-Runoff Modelling with SVM, W-ANFIS
and W-SVM Models
For SVM modelling, we have used C ++ based LIBSVM with
ν
-SV and
ε
-SV
regressions. Normalization of input vectors and proper identi
cation different
parameters are very important in SVM modelling. Some of the iterative analysis
had shown the importance of scaling the input vectors using data from the Brue
catchment for a concrete authenticity. The study also performed the analysis on the
Brue catchment daily data to see the performance of different kernel functions on
different regressions. The variations of modelling responses for different kernel
functions (Linear, Polynomial, Radial and Sigmoid) with different SVMs are shown
in Fig.
6.27
in terms of mean squared errors. This analysis on daily data from the
Brue catchment was performed after
fixing the parameters to default values (degree
in kernel function is set as 3, coef0 in kernel function is set as zero, cache memory
size is set as 40 MB, tolerance of termination criterion is set as a default value of
0.001). The analysis has shown the same results as that we found on the data from
