Figure 6 Local authorities within Klang River Basin.
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where x is the output from the previous neuron, w is the weight of the output and k is the slope steepness of the sig- moid function. Extra neuron Xo is added in the input layer and hidden layer with the output value of 1. This input is called bias and its function is to stabilize computed output between 0 and 1. It does not have a link from the previous neuron. The network architecture of the MLP-NN is shown in Fig. 1. It contains three layers: input, hidden and output layers. Each layer consists of one or more neurons and there are two types of them. First, there are passive neurons that consider the input and output data. Another type is an active neuron that computes data input using Activation Transfer Function (ATF) and produces an output. The most common use of ATF in the hidden and output neuron is the sigmoid func- tion (Zhang et al., 1998; Fernando et al., 2000). The input into an active neuron is a summation of the previous neu- ron’s weighted outputs and the output is a result of applying the sigmoid function to the input. The process is shown in Fig. | and the equations for the input and output are Fig. 2. Architecture of radial basis function neural network. Fig. 1. MLP-NN model architecture. A. El-Shafie et al.: Dynamic versus static neural network model for rainfall forecasting chosen as a sigmoid activation function of neuron i: The output of the IDNN, assuming that it has one output neu- ron j, a single hidden layer with m hidden neurons, and one input variable as shown in Fig. 4, is given by Fig. 5. IDNN model architecture. Fig. 4. Input delay neural network architecture and single neuron calculations. Fig. 7. Monthly actual rainfall records on Klang River on for period 1987-2008. Fig. 8. Weekly actual rainfall records on Klang River on for period 1997-2008. Fig. 9. Neural network model architecture utilized for rainfall fore- casting. where 7, =number of peak rainfall greater than one-third of the mean peak rainfall observed; 7; =number of low rain- fall lower than one-third of the mean low rainfall observed. Coulibaly (2001) reported that both PFC and LFC provide better performance indicators for assessment of the forecast- ing model performance for the extreme rainfall events. As the model can provide low PFC or LFC, the model represents better fit. Table 1. Maximum RE% and correlation coefficient associated with the output of MLP-NN model on a weekly/monthly basis for years 2007 and 2008. model-based MLP-NN method, Table 1 illustrates the perfor- mance measures for both scenarios. It can be observed that scenario B outperformed scenario A and provides a better ac- curacy level. It might be due to that scenario A includes the rainfall record of the same month of the previous year, which might not correlate and relate to rainfall values of the same month in the current year, and thus, the model scenario A provides relatively poor forecasting accuracy. Fig. 11. Training curve for monthly basis using the MLP-NN model. Table 2. Maximum RE% and correlation coefficient associated with the output of ANN and ARMA models on a weekly/monthly basis for years 2007 and 2008. ment of the spread of the RBFNN model configuration is (as described in Sect. 2.1.2) the only parameter to be obtained. The spread (step size) is achieved by trial and error as well. The optimal values of the spread were found to be equal to 0.07 for the monthly and 0.03 for the weekly model. remarkable improvements in the correlation coefficient at both the weekly and monthly basis. Fig. 12. Neural network performance “RMSE and maximum error%” utilizing different architecture, (a) one hidden layer (b) and (c) two hidden layers. Fig. 13. Performance of the MLP-NN model for monthly and weekly rainfall forecasting during training and testing stages. Table 3. RMSE and maximum RE for monthly and weekly rainfall forecasting model utilizing IDNN. for other river basins that might require introducing a higher order level. value of RMSE is of the IDNN model in the weekly rainfall forecasting model with only 7.3mm. It could be remarked that, generally, the performance for the weekly rainfall fore- casting is better than monthly rainfall forecasting. This is due to the inadequacy of historical data records on the monthly basis, which is 261 records, while for weekly, historical 621 rainfall records; thus, the model could capture most of tem- poral dimension of rainfall pattern and be able to provide lower forecasting error. In addition, it is obvious that the optimal results were received when using the IDNN method. Furthermore, with one-time step input delay, the IDNN is sufficient to achieve a significant level of accuracy. With re- spect to this observation, in Klang River Basin rainfall pat- tern, it might be the temporal dimension feature of rainfall that is in second order level. However, it could be inadequate Fig. 14. Performance of the RBFNN model for monthly and weekly rainfall forecasting during training and testing stages. Fig. 15. Performance of the IDNN model for monthly and weekly rainfall forecasting during training and testing stages Fig. 16. Forecasted and actual rainfall at Klang River (weekly ba- sis). Table 5. Models performance for matching the rainfall consequences. could also be applicable to other studies in other river basins with different time step input delay, according to how far is the temporal dimension of rainfall pattern at this river basin. The results of the present study also show that the pro- posed IDNN achieves better accuracy than the MLP-NN and RBENN for the extreme rainfall pattern events; however, the IDNN is still less accurate if compared with the forecasting accuracy for non-extreme events. In addition, the proposed IDNN showed better accuracy for mimicking the consecutive rainfall pattern than the classical MLP-NN and /or RBFNN model. Table 4. IDNN performance based on the peak and low flow error criteria for monthly and weekly forecasting Fig. 17. Forecasted and actual rainfall at Klang River (monthly basis).
