Published online 17 June 2009 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/hyp.7352
Auto-configuring radial basis function networks for chaotic
time series and flood forecasting
Li-Chiu Chang,
1Fi-John Chang
2* and Yuan-Peng Wang
21Department of Water Resources and Environmental Engineering, Tamkang University, Taipei, Taiwan, ROC 2Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei, Taiwan, ROC
Abstract:
The learning strategy of the radial basis function network (RBFN) commonly uses a hybrid learning process to identify the structure and then proceed to search the model parameters, which is a time-consuming procedure. We proposed an evolutionary way to automatically configure the structure of RBFN and search the optimal parameters of the network. The strategy can effectively identify an appropriate structure of the network by the orthogonal least squares algorithm and then systematically search the optimal locations of centres and the widths of their corresponding kernel function by the genetic algorithm. The proposed strategy of auto-configuring RBFN is first testified in predicting the future values of the chaotic Mackey-Glass time series. The results demonstrate the superiority, on both effectiveness and efficiency, of the proposed strategy in predicting the chaotic time series. We then further investigate the model’s suitability and reliability in flood forecast. The Lan-Young River in north-east Taiwan is used as a case study, where the hourly river flow of 23 flood events caused by typhoons or storms is used to train and validate the neural networks. The back propagation neural network (BPNN) is also performed for the purpose of comparison. The results demonstrate that the proposed RBFN has much better performance than the BPNN. The RBFN not only provides an efficient way to model the rainfall-runoff process but also gives reliable and precise one-hour and two-hour ahead flood forecasts. Copyright 2009 John Wiley & Sons, Ltd.
KEY WORDS radial basis function network; genetic algorithm; Mackey-Glass time series; flood forecast
Received 19 September 2008; Accepted 5 April 2009
INTRODUCTION
Accurate time series forecasting is very important in hydrology, especially for flood defense and water resources management. One of the most widely used methods in time series forecasting is the model described by Box and Jenkins (1976). This approach is both simple (parsimony) and yields rational results. It gives simplistic equations that only use several previous items of infor-mation to forecast the future. As computer power has increased, other approaches have attempted to find subtle trends and complex relationships in the data. The arti-ficial neural networks (ANNs) have been accepted as a potentially useful tool for modelling complex non-linear systems and widely used for prediction. In the hydro-logical forecasting context, ANNs have also proven to be an efficient alternative to traditional methods for rain-fall forecasting (Hsu et al., 1997; Chiang et al., 2007; Kumarasiri and Sonnadara, 2008), streamflow forecast-ing (Chang et al., 2004, 2007; Chiang et al., 2004; Jeong and Kim, 2005; Tayfur et al., 2007; Kisi, 2008; Lin and Chen, 2008), evapotranspiration simulation (Kisi, 2007; Jain et al., 2008), groundwater prediction (Krishna et al., 2008), reservoir operation (Aminian and Ameri, 2005; Chang et al., 2005; Kuo et al., 2006) and prediction of
* Correspondence to: Fi-John Chang, Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei, Taiwan, ROC. E-mail: changfj@ntu.edu.tw
water quality (Maier et al., 2004; Sahoo et al., 2006; Chaves and Kojiri, 2007).
Of the many types of neural networks, one of the most widely used is the radial basis function neural (RBFN) network. One reason is that it can approximate any con-tinuous function to any prescribed degree of accuracy by given sufficient size of the network (Poggio and Girosi, 1990; Ham and Kostanic, 2001). Moreover, train-ing RBFN is usually faster than that of multilayer per-ceptron networks. The faster learning speed comes from the fact that RBFN has just two layers and each layer can be determined sequentially. The parameters of the hidden neurons are determined by clustering, which is relatively fast, while the output layer weights are determined by a linear least-square algorithm, which is simply solving a linear system. Even though RBFNs popularity is continu-ously increasing, it is not straightforward to construct an optimal architecture of RBFN to solve a given problem. To continuously increase their popularity and gain far-ther acceptance, the development of methodologies that can automatically configure RBF networks using a set of input-output patterns is a key element.
The standard RBF training procedures are usually divided into two stages: (i) The centres of hidden nodes are determined first in a self-organizing manner and (ii) the weights that connect the hidden layer with output layer are then searched. This is a time-consuming proce-dure, since it requires the examination of many different network structures by using a trial and error procedure.
Another drawback is that the centres of hidden nodes are determined by using a clustering approach that is separate from the searching objective. To overcome the time-consuming trial and error procedure, various methods have been proposed to determine the appropriate struc-ture of the network, such as the orthogonal least squares (OLS) algorithm (Chen et al., 1991), constructive meth-ods, where the structure of the network is incrementally built (Fritzke, 1994) and pruning methods that start with an initial selection of a large number of hidden units and then reduce the units as the algorithm proceeds (Musavi
et al., 1992). These methods could substantially decrease
the total required training time, but still suffer from the fact that the formulation of the hidden layer is based on local, rather than global, search methods. Therefore, they cannot guarantee the selection of the optimal number of hidden nodes.
An interesting alternative for solving this problem can be found by the recently developed evolutionary compu-tation research area that involves a number of innovative optimization strategies such as genetic algorithms (GAs). Based on the principles of natural selection and evolution, GAs have been successfully utilized for the selection of the optimal structure of ANNs, such as feedforward neu-ral networks (Whitely et al., 1990; Arifovic and Gencay, 2001; Boozarjomehry and Svrcek, 2001). Surprisingly, not many applications of GAs have been reported for the development of training algorithms for RBF net-works (Billings and Zheng, 1995; Kuncheva, 1997). In this study, we propose a novel evolving strategy: Using the OLS algorithm to determine the appropriate structure of the network, then implementing the GA to directly search the optimizing locations of centres and the widths of their kernel function and using the least square errors (LSE) method to identify the weights of output layer.
The rest of the article first briefly presents the RBF network architecture and training problem, the imple-mented GA algorithm and auto-configuring the network for a theoretical chaotic series and flood forecasting in the Lan-Yang River. The results of the proposed method are shown and compared with the back propagation neural network (BPNN) for the case of flood forecasting. The concluding remarks are then given.
RBFN THEORY AND CONFIGURATION The RBF network was introduced into the neural network literature by Broomhead and Lowe (1988). The RBF net-work is motivated by the locally tuned response observed in biologic neurons. The architecture of the RBFN con-sists of three layers: (i) an input layer, (ii) a single hidden layer and (iii) an output layer. The input layer is only used to connect the network to its environment. The nodes in the hidden layer are associated with centres, which are vectors with dimension equal to the number of inputs. The output of the RBFN can be calculated
according to following equation: yiDfx D N kD1 wikkjjx ckjj2, i D1, 2, . . . , m 1 where x 2 <nð1 is an input vector, kÐ is a function,
jj Ð jj2 denotes the Euclidean norm, wik are the weights
in the output layer, N is the number of neurons in the hidden layer and ck2 <nð1 are the RBF centres in the
input vector space. The centres of ck are defined points
that are assumed to perform an adequate sampling of the input vector space. The functional form of kÐ is
chosen as the Gaussian function or Bell function. The generalized Gaussian functions and bell functions are defined in Equations (2) and (3).
jjx cjj Dexp jjx cjj 2 22 2 jjx cjj D 1 1 C jjx cjj a 2b 3
where is the standard deviation and c is the centre of the Gaussian function. All parameters of bell functions (i.e. a, b and c) have their own physical meaning: c is the centre of the corresponding membership function, a is the distance between the centre point and the crossover point (where the value of function is 0Ð5) and b2a is the slope at the crossover point.
Training a RBF network involves selecting the appro-priate number of basis functions and their parameters and estimating the weights and biases. A two-stage procedure has been commonly used for training RBFNs. In the first stage, the number of hidden units (nodes) and parameters governing them are determined by clustering. The second stage involves estimating the weights in the output layer, locations of the RBFN centres and the width of their cor-responding kernel functions. Most of the standard RBF training methods require a designer to fix the structure of the network and then proceed with the calculation of the model parameters. A major challenge in the design of the RBFN is the selection of the centres. Based on the Gram-Schmidt orthogonalization procedure, the OLS method offers a systematic way for centre selection and can significantly reduce the size of the RBFN. A detailed description of the algorithm can be found in the ANN text books, such as Ham and Kostanic (2001).
Here, we propose an automatically evolving strategy: (1) using the OLS algorithm to determine the appropriate structure of the network, (2) implementing the GA to search the optimizing locations of centres and the widths of their kernel function and then (3) using the LSE method to identify the weights of output layer. Figure 1 shows the flowchart of proposed methodology for auto-configuring RBFN. A detailed presentation of the OLS method can be found in the textbook by Ham and Kostanic (2001) and a brief description of the GA and its implementation is given in the following sections.
OLS procedure
Determine the number of centers and initialize
their positions
Setting tolerance parameter
r
Prepare training data set
Pick up a new center to reduce the maximum error ratio
< r ?
No
Determine the number of centers
Combination of GA procedure and LSE
Training RBFNs
Generate 30% of chromosomes closed to
the centers obtained from OLS Initialize chromosome Randomly generate The rest of 70% chromosomes The GA operations of selection, reproduction, crossover and mutation Encode variables and
predetermine the parameters of GA
Obtain the weights of the output layer by
LSE
No
Stop criteria?
Stop
Figure 1. The flowchart of self-configuring radial basis function networks GA
The GA is similar to Darwinian natural selection, and its primary monograph is Holland’s adaptation in nat-ural and artificial systems in 1975 (Holland, 1975). To evaluate the suitability of the derived solution, an objec-tive function is required. The evolution starts from a set of coded solutions and proceeds from generation to generation through genetic operations: reproduction, crossover and mutation. The reproduction process copies parent chromosomes into a tentative new population. The probability of selected chromosomes for the next gener-ation is directly proportional to its fitness value. Roulette
Wheel selection is the most common method and is used in our study. The crossover recombines two parent chro-mosomes to produce offspring new chrochro-mosomes for the next generation. To sustain genetic diversity in the pop-ulation, mutation is also made occasionally with small probability. Through the GAs, an optimal solution can be found and represented by the final winner of the processes.
Implemented GA to RBF network
In the present study, GAs are developed to optimize the centre locations and width of the hidden nodes (kernel functions). If the number of hidden nodes becomes large,
a huge number of parameters must be estimated. For example, if the optimum number of hidden node is 22 (in the case of chaotic series), 132 variables (22 nodes ð 4 input dimensions in each node C 22 nodes’ ð 2 (coefficients of kernel function a, b)) must be identified. To make a global search in feasible solution domain more effectively and efficiently, we recommend the following:
(1) Choose 30% of initial solutions closed to the results obtained from OLS, and randomly generate the rest of the 70% initial solutions.
(2) Due to the long chromosomes (132 variables), multi-crossover points should be adopted (5 to 12 multi-crossover points are recommended; we used 11 crossover points.)
(3) The GA parameters used in this study are population (1500), crossover rate (0Ð8), mutation rate (0Ð1) and generation (100).
TRAINING AND TESTING THE RBFN BY A CHAOTIC TIME SERIES
The prediction of future values of the chaotic Mackey-Glass time series generated by Equation (4) is recog-nized as a benchmark problem that has been used and reported by a number of researchers for comparing the learning and generalization ability of different neural networks (Moody and Darken, 1989; Jang, 1993). The Mackey-Glass chaotic time series is described by the following:
dxt
dt D
0Ð2xt
1 C x10t 0Ð1xt 4 When > 17, the equation shows chaotic behaviour. The goal of this task is to use known values of the time series up to the point x D t to predict the value at some point in the future, x D t C 6 in our case. To demonstrate the goodness of fit of the proposed methods and make the comparison with earlier work fair (Jang, 1993; Gholipour et al., 2006), we predicted the xt C 6 using the input variables xt 18, xt 12, xt 6 and xt, respectively. To obtain the Mackey-Glass time series xt, we also used the fourth-order Runge-Kutta method to find the numerical solutions of Equation (4). We extracted 1000 (t D 118 to 1117) input-output data pairs. The first 500 pairs were used to train the networks, and the rest of the 500 pairs were then used to test the
trained network. The root mean square error (RMSE) is used as the criterion for determining the structure of the built network and for selecting the units in the hidden layer.
Results
The size of network can effectively and efficiently be identified by the OLS algorithm based on a designed error tolerance parameter , which is important for balancing the accuracy and complexity of the network. To demon-strate the influence of the parameter, three different small values of tolerance parameter (i.e. 105, 106, 107) were given. We found (1) the hidden node is 22, if D 105,
(2) the hidden node is 31, if D 106and (3) the hidden
node is 63, if D 107. It appears that the number of
hid-den nodes would increase as the error tolerance parameter decreased. The results of implementing the constructed RBFN by fusing OLS with GA(s) are shown in Table I. It appears that (1) the values of RMSE would decrease as the number of hidden nodes increases; (2) the con-structed network could closely fit the actual values and give very small values of RMSE in all sets, i.e. train-ing and testtrain-ing sets; and (3) implementtrain-ing the GA into the OLS (i.e. OLS C GA) can significantly improve the accuracy of the RBFN. Using the RBFN with 63 hidden nodes, the time series of real and estimated values and the residual time series in two different stages, i.e. train-ing and testtrain-ing, are presented in Figure 2. The desired and predicted values for training and testing data sets are essentially the same, where their differences (errors) can only be seen on a finer scale (Figure 2). As the num-ber of hidden nodes is 63, the values of RMSE in both training and testing sets are smaller than those of the adaptive-network-based fuzzy inference systems model (Jang, 1993). The results demonstrate that the RBFN can be effectively constructed and the network can pro-vide reliable and accurate prediction. Considering that the values of RMSE in two stages are all very small, we conclude that the constructed RBFN has captured the essential components of the underlying dynamics and can be implemented to configure the non-linear dynamic sys-tem such as flood forecast.
STUDY WATERSHED AND IMPLEMENTING THE PROPOSED RBFN FOR FLOOD FORECAST To illustrate the practical application of the proposed model, the Lan-Yang River, northeastern Taiwan, is used Table I. The root mean square error values of radial basis function networks (RBFN; orthogonal least squares OLS C genetic
algorithms GA) for forecasting a chaotic time series
Method # data RBFN (22) RBFN (31) RBFN (63) ANFISa
OLS OLS C GA OLS OLS C GA OLS OLS C GA
Training (500) 0Ð00923 0Ð00473 0Ð00604 0Ð00390 0Ð00275 0Ð00143 0Ð0016
Testing (500) 0Ð00933 0Ð00475 0Ð00635 0Ð00387 0Ð00249 0Ð00140 0Ð0015
Training data set
Testing data set
Figure 2. Desired and predicted chaotic time series by radial basis function networks
as a case study. The river originates on Nanhu Mountain at 3535 m above sea level and gradually descends from the west to the east in 73-km long running through to the Pacific Ocean. The steep Lan-Yang River sweeps masses of sandstone downstream, forming the alluvial plain with a drainage area of 980 km2. The average
annual temperature is 21Ð8°C, has mean annual rainfall of 3170 mm that results in luxuriant vegetation and
beautiful scenery. About 40% of the runoff is derived from summer rain related to typhoons. There is no dam on the entire river. The river provides the Lan-Yang Plain with water for irrigation and household use, and its estuary is one of most important coastal wetlands in Taiwan. The estuary is one of the nation’s three major fishing grounds and attracts more than 230 kinds of birds.
The available data
There are four rainfall stations upstream of the water-shed and a water-level gauge station at the Lan-Yang bridge, which are all equipped with automatic rainfall amount (or water level) recorders and transmitted by wireless (Figure 3). The hourly rainfall and water level of the gauge stations are used. These are 23 typhoon (or heavy rainfall) events from the past 21 years (1981 to 2001), published by the Water Resources Agency, Tai-wan. A total of 1632 data sets are obtained. The 23 events data are divided into three independent subsets: (i) the training, (ii) validation and (iii) the testing sub-sets. The training subset includes 15 events (1104 data sets), the validation subset has 5 events (288 data sets), while the testing subset has the remaining 3 events (240 data sets). The ratio of training sets to verification sets to testing sets is approximately 4 : 1 : 1. First, the train-ing subsets are repeatedly used to build networks and to adjust the connected weights of the constructed networks. Afterwards, the validation subset is used to simulate the performance of the built models for checking its suit-ability of generalization, and the best network is selected for later use. The testing data set is then used for final evaluation of the selected network performance.
To assess the models performances, RMSE, mean absolute error (MAE) and Gbench are used. Gbench is to
compute the goodness of fit measures and shown as below. GbenchD1 n iD1 OQiQi2 n iD1 OQiQi,bench2 . 5
Here, OQi and Qirepresent the forecasted and observed
water level, respectively. The variable n means the number of data points. Qi,bench is the previous observed
value, e.g. for n-step ahead prediction Qi,benchDQin.
If Gbench is negative, the forecasting performance is
poorer than the benchmark; if Gbench is equal to zero,
Figure 3. Location of study area and gauge stations
the performance is as good as the benchmark; If Gbench
is equal to one, it means perfect fit.
In the present study, the input dimensions are mined by the input variables and the lag time. To deter-mine an appropriate feed-forward neural network struc-ture for forecasting the streamflow at time t C 1 and t C 2 in this selected basin, we develop two different models for the BPNN and RBFN, namely,
Model 1: Qt C 1 D fQt, Qt 1, Qt 2, R1t, R1t 1, R1t 2, R2t, R2t 1, R2t 2, R3t, R3t 1, R3t 2, R4t, R4t 1, R4t 2; Model 2: Qt C 2 D fQt, Qt 1, R1t, R1t 1, R2t, R2t 1, R3t, R3t 1, R4t, R4t 1,
where Qt i represents the value of the Lan-Yang Bridge streamflow gauge station at time t i and R1t
i, R2t i, R3t i and R4t i represents the
pre-cipitation of the four rainfall gauge stations at time t i. Because the drainage area of the study watershed is less than 1000 km2and the basin slope is quite steep, the time
of concentration is short. Consequently, the input pattern is focused on the previous two-hours information only.
BPNN
The BP network is trained by using supervised learn-ing to obtain the optimal values of the connected weights such that the energy function is a minimum. The con-jugate gradient method is a well-known numerical tech-nique used for solving various optimization problems. The conjugate gradient algorithm searches a system of conjugate directions on the error surface and updates the weights along these directions. In practice, the process makes good uniform progress towards the solution at every time step and has been found to be effective in finding better optimization than the standard BP algo-rithm (Ham and Kostanic, 2001). We use this algoalgo-rithm to update the connecting weights during the training process.
Finding the optimal size of the networks
Determining the appropriate size of neural networks is very important for validity and efficiency. Since there is no systematic or standard method for finding the optimal number of hidden nodes, the optimal network size is based on trial and error in the circumstance. To determine the optimal BPNN for the given input-output patterns, we investigated several network architectures. Because the input vector is set the same as RBFN, i.e. 15 for one-step ahead and 10 for two-one-step ahead, respectively, and the output vector is 1 (one-step ahead or two-step ahead), only the number of nodes in the hidden layer needs to be determined. The networks with various numbers of hidden nodes, from 1 to 12, are performed; furthermore, each network is executed with 50 initial sets of connected weights and the best result is selected. The search results
Figure 4. The root mean square error versus the number of node (centrr) in the constructed back propagation neural network
of one-step ahead are shown in Figure 4. As shown, when the hidden layer has 9 nodes for one-step ahead, the constructed BPNN has suitable performance in the set and best performance in the validation set. For two-step ahead forecast, as the hidden layer has 13 nodes, the BPNN has the best performance. These results are used as a base to evaluate the adequacy of our proposed RBF neural networks. To determine the optimal RBFN for the given input-output patterns, the OLS method, which offers a systematic way for centre selection and can significantly reduce the size of the RBFN, is used. The Gaussian function is implemented as the Kernel function of the RBFN. The optimal number of nodes in the hidden layer obtained by the OLS method is 11 and 12 for one-step ahead and two-step ahead, respectively.
Results and Discussion
As presented above, the structures of BPNN for river flood forecasting have been identified based on a time-consuming trial and error processes, whereas the pro-posed methodology that can systematically configure RBF network and estimate its parameters using a set of input-output patterns. Based on these constructed net-works, the results of BPNN and RBFN are summarized in Table II. The results show that (1) the RBFN is supe-rior to the BPNN in forecasting ability, where the RBFN obtains much lower forecasting errors and higher Gbench
values as compared with the BPNN and (2) the RBFN can adequately produce one-step ahead streamflow fore-casting with small MAE and high Gbenchvalue. To closely
investigate the performance of the RBFN, Figure 5 shows the hydrographs and scatter plots of observed versus one-step ahead forecasting of the RBFN in all the three phases, i.e. training, validation and testing, respectively. One can easily find that the forecast values of RBFN are highly correlative to the observed values of hydrographs and close to the ideal line in the scatter plots in all three phases.
For further investigating the applicability of the RBFN and BPNN networks on two-hour ahead forecasting,
Table II. One-step ahead flood forecast in three different phases by back propagation neural network (BPNN) and radial basis
function networks (RBFN)
Training Validation Testing
BPNN RMSE (cms) 98Ð18 115Ð58 134Ð58 MAE (cms) 59Ð43 68Ð20 74Ð05 Gbench 0Ð23 0Ð26 0Ð34 RBFN RMSE (cms) 74Ð02 74Ð12 97Ð07 MAE (cms) 37Ð87 38Ð82 51Ð74 Gbench 0Ð56 0Ð70 0Ð38
RMSE, root mean square error; MAE, mean absolute error.
Table III. Two-step ahead flood forecast in three different phases by back propagation neural network (BPNN) and radial basis
function networks (RBFN)
Training Validation Testing
BPNN RMSE (cms) 134Ð2 200Ð6 179Ð3 MAE (cms) 77Ð3 99Ð8 101Ð0 Gbench 0Ð54 0Ð35 0Ð31 RBFN RMSE (cms) 110Ð5 110Ð4 150Ð9 MAE (cms) 57Ð6 57Ð9 73Ð2 Gbench 0Ð69 0Ð80 0Ð60
RMSE, root mean square error; MAE, mean absolute error.
the above procedures were re-executed and the results are shown in Table III. Again, we can easily find that both networks can reasonably make two-hour ahead forecast, while the RBFN is superior to the BPNN, in terms of smaller RMSE and MAE values and higher value of Gbench in all three phases. Figure 6 represents
the two-hour ahead forecast results, where the RBFN reasonably fits the ideal line and slightly wider spread from the ideal line than the one-hour ahead forecasts. It appears that the constructed RBFN has better forecasting performances, on both one-step and two-step ahead, than the constructed BPN in this study case. We demonstrate that the auto-configuring RBFN can effectively capture the essential components of the non-linear dynamic system and accurately forecast the short-term flood.
CONCLUSION
RBF networks have been widely used for modelling com-plex non-linear systems and for time series prediction. The configuration of an RBFN was usually divided into two stages: Using the unsupervised cluster method to select the number of hidden neurons for determining the structure of RBFN, and then search the optimal position of the RBF centres and the width of the RBFs. Even though previous works have demonstrated, or claimed, that it is usually faster to construct and train this type of network than that of multilayer perceptron networks, it could only be classified as a local search method and suf-fers from the time-consuming trial and error procedures. To continuously increase their popularity, we propose a systematic and efficient way to automatically construct
Figure 5. The hydrographs and scatter plots of observed versus one-step ahead forecasting of the radial basis function networks in three phases
the RBFN. Our strategy entails fusing the OLS algorithm, the GA and the LSE method to (1) directly identify the structure of RBFN, (2) search the position of the centres and the width of RBFs and (3) identify the linear weights of output layer. The time-consuming search problem is effectively solved and the resulting networks can provide better performance.
The superiority of the proposed algorithm over the con-ventional method in constructing and training the RBFN is first demonstrated through a benchmark problem— the chaotic Mackey-Glass time series. Our experimental
results show the efficiency and effectiveness of the pro-posed approaches in predicting the chaotic time series. The auto-configuring strategy increases the confidence and promises of the RBFN to be widely and easily used. The proposed method is then applied to flood forecasting in the Lan-Yang River, Taiwan. For the purpose of com-parison, the commonly used BPNN is also performed. The results demonstrate that the RBFN is superior to the BPNN in making one-hour ahead and two-hour ahead flood forecasts. The proposed strategy can systematically construct the structure (the number of hidden nodes)
Figure 6. The hydrographs and scatter plots of observed versus two-step ahead forecasting of the radial basis function networks in three phases
of RBFN and properly identify the network’s relative parameters (centres and their widths of kernel functions and linear weights of output layer) to make accurate flood forecasts.
ACKNOWLEDGEMENT
This article is based on partial work supported by the National Science Council, ROC under NSC-95-2313-B-002-031. In addition, the authors are indebted to the reviewers for their valuable comments and suggestions.
REFERENCES
Aminian K, Ameri S. 2005. Application of artificial neural networks for reservoir characterization with limited data. Journal of Petroleum
Science and Engineering 49(3– 4): 212– 222.
Arifovic J, Gencay R. 2001. Using genetic algorithms to select architecture of a feedforward artificial neural network. Physica A 289: 574– 594.
Billings SA, Zheng GL. 1995. Radial basis function network configura-tion using genetic algorithms. Neural Networks 8(6): 877– 890. Boozarjomehry RB, Svrcek WY. 2001. Automatic design of neural
network structures. Computers & Chemical Engineering 25(7– 8): 1075– 1088.
Box GEP, Jenkins GM. 1976. Time Series Analysis Forecasting and
Broomhead DS, Lowe D. 1988. Multivariable function interpolation and adaptive networks. Complex System 2(3): 321– 355.
Chang LC, Chang FJ, Chiang YM. 2004. A two-step-ahead recurrent neural network for stream-flow forecasting. Hydrological Processes 18(1): 81– 92.
Chang YT, Chang LC, Chang FJ. 2005. Intelligent control for modeling of real-time reservoir operation, part II: artificial neural network with operating rule curves. Hydrological Processes 19(7): 1431– 1444. Chang FJ, Chang LC, Wang YS. 2007. Enforced self-organizing map
neural networks for river flood forecasting. Hydrological Processes 21(6): 741– 749.
Chaves P, Kojiri T. 2007. Conceptual fuzzy neural network model for water quality simulation. Hydrological Processes 21(5): 634– 646. Chen S, Cowan CFN, Grant PM. 1991. Orthogonal least squares learning
algorithm for radial basis function networks. IEEE Transactions on
Neural Networks 2(2): 302– 309.
Chiang YM, Chang LC, Chang FJ. 2004. Comparison of static-feedforward and dynamic-feedback neural networks for rainfall-runoff modeling. Journal of Hydrology 290: 297– 311.
Chiang YM, Chang FJ, Jou JD, Lin PF. 2007. Dynamic ANN for precipitation estimation and forecasting from radar observations.
Journal of Hydrology 334: 250– 261.
Fritzke B. 1994. Fast learning with incremental RBF networks. Neural
Processing Letters 1(1): 2– 5.
Gholipour A, Araabi BN, Lucas C. 2006. Predicting chaotic time series using neural and neurofuzzy models: a comparative study. Neural
Processing Letters 24(3): 217– 239.
Ham FM, Kostanic I. 2001. Principles of Neurocomputing for Science &
Engineering. McGraw-Hill: New York.
Holland JH. 1975. Adaptation in Natural and Artificial System. University of Michigan Press: Ann Arbar, MI.
Hsu KL, Gao X, Sorooshian S, Gupta HV. 1997. Precipitation estimation from remotely sensed information using artificial neural networks.
Journal of Applied Meteorology 36: 1176– 1190.
Jain SK, Nayak PC, Sudheer KP. 2008. Models for estimating evapotranspiration using artificial neural networks, and their physical interpretation. Hydrological Processes 22(13): 2225– 2234.
Jang RJS. 1993. ANFIS: Adaptive-network-based fuzzy inference systems. IEEE Transactions on Systems, Man, and Cybernetics 23(3): 665– 685.
Jeong D, Kim Y-O. 2005. Rainfall-runoff models using artificial neural networks for ensemble streamflow prediction. Hydrological Processes 19(19): 3819– 3835.
Kisi O. 2007. Evapotranspiration modelling from climatic data using a neural computing technique. Hydrological Processes 21(14): 1925– 1934.
Kisi O. 2008. River flow forecasting and estimation using different artificial neural network techniques. Hydrology Research 39(1): 27– 40. Krishna B, Satyaji Rao YR, Vijaya T. 2008. Modelling groundwater levels in an urban coastal aquifer using artificial neural networks.
Hydrological Processes 22(8): 1180– 1188.
Kumarasiri AD, Sonnadara UJ. 2008. Performance of an artificial neural network on forecasting the daily occurrence and annual depth of rainfall at a tropical site. Hydrological Processes 22(17): 3535– 3542. Kuncheva LI. 1997. Fitness functions in editing k-NN reference set by
genetic algorithms. Pattern Recognition 30: 1041– 1049.
Kuo JT, Wang YY, Lung WS. 2006. A hybrid neural– genetic algorithm for reservoir water quality management. Water Research 40(7): 1367– 1376.
Lin GF, Chen GR. 2008. A systematic approach to the input determination for neural network rainfall-runoff models. Hydrological
Processes 22(14): 2524– 2530.
Maier HR, Morgan N, Chow CWK. 2004. Use of artificial neural networks for predicting optimal alum doses and treated water quality parameters. Environmental Modelling & Software 19(5): 485– 494. Moody J, Darken D. 1989. Fast learning in networks of locally-tuned
processing units. Neural Computation 1: 281– 294.
Musavi MT, Ahmed W, Chan KH, Faris KB, Hummels DM. 1992. On the training of radial basis function classifiers. Neural Networks 5: 595– 603.
Poggio R, Girosi F. 1990. Networks for approximation and learning.
Proceedings of the IEEE 78(9): 1481– 1497.
Sahoo GB, Ray C, De Carlo EH. 2006. Use of neural network to predict flash flood and attendant water qualities of a mountainous stream on Oahu, Hawaii. Journal of Hydrology 327: 525– 538.
Tayfur G, Moramarco T, Singh VP. 2007. Predicting and forecasting flow discharge at sites receiving significant lateral inflow. Hydrological
Processes 21(14): 1848– 1859.
Whitely D, Starkweather T, Bogartm C. 1990. Genetic algorithms and neural networks — optimizing connections and connectivity. Parallel
