Estuary water-stage forecasting by using radial basis function
neural network
Fi-John Chang*, Yen-Chang Chen
Department of Bioenvironmental System Engineering, National Taiwan University, Roosevelt Road, Taipei 10770, Taiwan, ROC Received 31 October 2001; revised 1 September 2002; accepted 6 September 2002
Abstract
The Radial basis function neural network (RBFNN) has been successfully applied to many tasks due to its powerful properties in classification and functional approximation. This paper presents a novel RBFNN for water-stage forecasting in an estuary under high flood and tidal effects. The RBFNN adopts a hybrid two-stage learning scheme, unsupervised and supervised learning. In the first scheme, fuzzy min – max clustering is proposed for choosing best patterns for cluster representation in an efficient and automatic way. The second scheme uses supervised learning, which is a multivariate linear regression method to produce a weighted sum of the output from the hidden layer. Since this network has only one layer using a supervised learning algorithm, its training process is much faster than the error back propagation based multilayer perceptrons. Moreover, only one parameter, u, must be determined manually. The other parameters used in this model can be adjusted automatically by model training. The water-stage data of the Tanshui River under tidal effect are used to construct a water-stage forecasting model that can also be used during flood. The results show that the RBFNN can be applied successfully and provide high accuracy and reliability of water-stage forecasting in an estuary.
q2002 Published by Elsevier Science B.V.
Keywords: Estuary; Neural network; Nonlinear; Radial basis function; Tidal effect; Water-stage forecasting
1. Introduction
An estuary is a semi-enclosed coastal body of water which has free connection to the open sea and within which sea water is measurably diluted with fresh water derived from land drainage (Cameron and Pritchard, 1963). The hydrological systems in an estuary are unique and the water-stage is continually changing under the interaction of riverine and marine processes. The most obvious factors having a
profound influence on the water-stage include shape of the estuary, astronomical tide, wind, salinity, temperature, sediment, river discharge, storm surge, and others, are too complex to model directly. Consequently, the hydrodynamic processes of estu-aries are manifestly complex and notoriously nonlinear.
Water-stage forecasting in a river under tidal effects is among the most important outstanding problems of flood management. It is never an easy task, because in order to develop a water-stage forecasting model one must know the behavior of the physical processes. The river flow conditions under tidal effects are rarely steady or uniform. www.elsevier.com/locate/jhydrol
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* Corresponding author. Tel.: 2363-9461; fax: þ886-2-2363-5854.
However, many of the concepts and principles derived from other watercourses can be applied. For example, the momentum equation is used to study surges and floodwaves, stochastic hydraulics is employed to estimate discharge (Chen and Chiu, 2002) and the diffusion equation with kinetics is applied to evaluate the pollutant movement in estuarine systems. Some empirical equations also have been developed to characterize the flow (Wright et al., 1973; Prandle, 1991; Van Dongeren, 1992). Many deterministic and stochastic models have been developed to forecast the water-stage (Lin and Lee, 1996; Perumal and Ranga Raju, 1998). One of the major disadvantages of using these models is that the parameters are usually difficult to determine from the observed data. Owing to the lack of practicality and difficulty in use, the application of such sophisticated models in Taiwan has so far been unsuccessful.
Even with all these difficulties and challenges, artificial neural networks (ANNs), a relatively new computational tool that has found wide acceptance in many disciplines, provide an alternative way to make important contributions to one step ahead under-standing and/or managing these hydrological pro-cesses. The attractiveness of ANNs comes from their information processing characteristics, such as non-linearity, parallelism, noise tolerance, and learning and generalization capabilities (Basheer and Hajmeer, 2000). Recently, ANNs have been successfully used for modeling hydrological processes (Hsu et al., 1995; Rogers et al., 1995; Kuligowski and Barros, 1998; Sajikumar and Thandaveswara, 1999; Govindaraju and Rao, 2000; Chang and Chen, 2001; Chang et al., 2002).
Most of the water-stage forecasting models built by ANNs are based on training the rainfall – runoff data set where the previous stages of rainfall and/or water-stage could dominate the current water-water-stage. In estuary problems, there are many effects that could influence the water-stage. Moreover, the poorly defined or even misunderstood riverine and marine interaction in the estuary makes it impossible to model its hydrological processes in the circumstance. To develop an estuary water-stage forecasting neural network, a great amount of relative information (data) would be involved, and thus massive network structure. In this study, we present a novel Radial
basis function neural network (RBFNN) for solving this poorly defined and complex problem.
2. Radial basis function neural network
RBFNNs rank among the most popular tools for function approximation and have currently been widely applied in many areas, such as nonlinear control, speech processing, and pattern recognition (Gorinevsky and Vukovich, 1997; Pedrycz, 1998). An important property of RBFNNs is that a high-dimensional space nonlinearity problem can be easily broken down through a set of linear combination of radial basis functions. Another important feature of an RBFNN is the ability to be quickly trained. For the purpose of faster training speed, RBFNNs with the hybrid learning scheme
applied herein, which is suggested by Moody and
Darken (1989), have a feedforward structure that involves three layers. The input layer is composed of n input nodes. The only hidden layer consists of J locally tuned units and each unit has a radial basis function acting like a hidden node. The hidden node output zj(x ) calculates the closeness of
the input and projects the distance to an activation function. The activation function of the jth hidden node used in this study is the Gaussian function given by zjðxÞ ¼ exp 2 kx 2mjk2 2s2 j ! ð1Þ where x is the n-dimensional input vector; mj is the
center of the radial basis function for hidden node j; and kx 2mjkdenotes the Euclidean distance between the center of the radial basis function and input; sj
is a parameter for controlling the smoothness properties of the radial basis functions. The third layer of the network is the output layer with L nodes that are fully interconnected to each hidden node. The output of the network is the sum of linear weighted zj(x ) yl¼ XJ j¼0 wljzjðxÞ ð2Þ z0ðxÞ ¼ 1 ð3Þ
where ylis the lth component of the output layer; wljis
the synaptic weight between the jth node of hidden layer and the lth node of output layer. Eq. (3) denotes the constant, wl0, in the regression Eq. (2).
3. Network training
Training an RBFNN for a specific problem involves selecting the type of basis functions with associated center location uand widths, the number of functions K and the weights. Apparently, this is a non-convex optimization problem. A great number of studies have been done to explore the efficient ways. For batch mode training, RBFNNs with localized basis functions offer a very attractive way that, in practice, the estimation of parameters can be decoupled into a two-stage pro-cedure: (1) determine the centers and widths, and (2) based on the results obtained in step (1), determine the weights to the output units.
The two-stage training scheme is shown inFig. 1. In the first stage only the input values are used for determining the centers and the widths of the radial basis functions. Thus, learning is unsupervised. Once the function parameters are fixed, supervised training (i.e. with target information) can be used for determining the second layer weights. More detail of the two-stage scheme is as follows.
3.1. Unsupervised training—fuzzy min – max clustering
The key to determining the locations and widths of the localized basis functions is to view them as representing the input data density. An efficient approach is to cluster the input vectors and then locate the basis functions at the centers. A variety of clustering techniques can be used to cluster the data with simultaneous water-stage conditions into the same hidden node. The K-means clustering algorithm, which minimizes the sum of squares error between the inputs and hidden node centers, is commonly used to locate the centers of the radial basis functions (Oukhellou and Aknin, 1999); however, this algor-ithm must be given or input the number of radial basis functions J, and the widths sj of every radial basis
function usually have to be the same. One has to guess this number J andsjbefore starting training the neural
network. Instead of the K-means clustering method, the fuzzy min – max clustering method (Simpson, 1993) is employed in this study. The advantage of the fuzzy min – max clustering method is that the number, centers and sj of radial basis functions can all be
determined systematically and automatically. The fuzzy min – max clustering algorithm involves three phases: expansion of a hyperbox, overlap test, and contraction of a hyperbox. During the network training process, a large number of n-dimensional hyperboxes that range from 0 to 1 along each dimension will be generated. Each hyperbox is
Fig. 1. General flowchart of algorithm for constructing a rainfall – runoff model using RBFNN.
viewed as a hidden unit. The boundary of a hyperbox is defined by the max and min points of the form
vj¼ ðvj1; vj2; …; vjnÞ ð4Þ
uj¼ ðuj1; uj2; …; ujnÞ ð5Þ
where vjand ujare the min and max points for the jth
hyperbox; vjnand ujnare the min and max value for
the nth dimension. The membership function for measuring the degree of the hth input xhfalling within
the hyperbox j is defined as Hjðxh; vj; ujÞ ¼ 1 n Xn i¼1 ½1 2 f ðxhi2 ujiÞ 2 f ðvji2 xhiÞ ð6Þ f ðjÞ ¼ 1 j. 1 j if 0 #j# 1 0 j, 0 8 > > < > > : ð7Þ
where xhiis the ith node of the hth input; Hj(·) is the
membership value setting to the unit interval [0,1];j
is either xhi2 ujior vji2 xhi. The membership values
are used to determine which hyperbox needs to be expanded.
At the beginning, the max and min points of the first hyperbox are set to be the first input data. In the subsequent, the degree of membership values will be calculated for every new input, and the hyperbox with the highest degree of membership is tested for possible hyperbox expansion if
Xn i¼1
ðmaxðuji; xhiÞ 2 minðvji; xhiÞÞ # nu ð8Þ in whichuis a user-defined value and 0 #u# 1: A
smallumeans more hyperboxes will be created. If the hyperbox is expanded, the min and max points of the hyperbox will be adjusted as
vnewji ¼ minðvold
ji ; xhiÞ ð9Þ
unewji ¼ maxðuoldji ; xhiÞ ð10Þ
If no hyperbox can be expanded Xn
i¼1
ðmaxðuji; xhiÞ 2 minðvji; xhiÞÞ . nu !
;
a new hyperbox will be generated. vnewji ¼ u
new
ji ¼ xhi ð11Þ
After the hyperbox is expanded, the hyperbox overlap test will be used to determine whether hyperboxes overlap or not. If overlapping is found between hyperboxes, the max and/or min points of each dimension of hyperbox j could be contained within another hyperbox k. Thus, the hyperboxes will be contracted with the minimal disturbance principle, and only the one dimension that has the minimum overlap is adjusted.
The entire training data will be presented for clustering again and again until no hyperbox needs to be adjusted. The parametersjis determined by half of
the width of the hyperbox, as given:
sj¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðvj12 uj1Þ2þ ðvj22 uj2Þ2þ … þ ðvjn2 ujnÞ2 q 2 ð12Þ 3.2. Supervised learning—multivariate linear regression
After the clustering is completed, the radial basis functions are fixed. In the second scheme, the weights wlj are determined to let the output of network yhl approximate to the target yphl: The supervised training algorithm aims to minimize the following sum of squares error, SSE ¼ 1 2 XN h¼1 XL l¼1 ðyp hl2 yhlÞ 2 ð13Þ where yhland y p
hlare the lth node of the hth set output and target, respectively. Since the outputs of the network are linear combinations of the outputs of the hidden layer, a multivariate linear regression model, given as
yp¼ zw þe ð14Þ
can be used to determine the weights. In Eq. (14) ypis the target that is an N £ L matrix; z is the output of the hidden layer that is an N £ (J þ 1) matrix; w is the weight between hidden and output layers that is a (J þ 1) £ L matrix; and e is independent noise with zero mean. The method of least squares selecting
weight matrix wpto minimize SSE is given by
wp¼ ðzTzÞ21zTyp ð15Þ
4. Study watersheds and data
To illustrate the foregoing with practical appli-cations, the Tanshui River will be considered. The Tanshui River, composed of three major tributaries, the Ta-han Creek, the Hsin-tien Creek, and the Keelung River, as shown in Fig. 2, is situated near the city of Taipei. The third largest river in Taiwan, it
drains 1183 km2 into the Taiwan Strait and its
downstream section for approximately 25 km from the river mouth is under tidal effects. The water-stage in this section is needed as the boundary condition for flood forecasting after heavy rainfalls brought by tropical storms every year. Recently, a flood control project was undertaken on the Tanshui River, including forecasting models, costly levees and bank protection works that can reduce the frequency of inundation of Taipei, mitigate losses by flood
damages, and safeguard lives and properties. Water-stage forecasting, especially during a flood, is one of the most crucial parts of the project. Unfortunately, because of the complex hydrological processes in these highly unsteady flows, the traditional physics-based models and time series models cannot be used. The RBFNN can be then used to test its applicability. Six gaging stations equipped with automatic water-stage recorders are located in the study watershed. The available water-stage data collected by Taiwan Water Conservancy Agency are the hourly data measured at the gaging stations.
5. Practical applications
The travel times of flow between the Taipei Bridge Station and the other gaging stations are less than 3 h because of the small watershed. For water-stage forecasting, the inputs of the model include the lunar month, lunar day, time and the water-stages of the six gaging stations for up to 3 h before the study. The only output is the 1-h-ahead water-stage of the
Taipei Bridge Station. In order to employ fuzzy min – max clustering, the different units and scales of the input data must be scaled to have values be between 0 and 1. The model using the RBFNN can be represented by Gtðt þ 1Þ ¼ XJ j¼0 wjzj½TmðtÞ;TdðtÞ;ThðtÞ;GtðtÞ;Gtðt 2 1Þ; Gtðt 2 2Þ;GhðtÞ;Ghðt 2 1Þ;Ghðt 2 2Þ;GcðtÞ;Gcðt 2 1Þ; Gcðt 2 2Þ;GsðtÞ;Gsðt 2 1Þ;Gsðt 2 2Þ;GwðtÞ;Gwðt 2 1Þ; Gwðt 2 2Þ;GgðtÞ;Ggðt 2 1Þ;Ggðt 2 2Þ ð16Þ where G(t þ 1) is the water-stage at time t þ 1; the subscripts t, h, c, s, w, and g represent gaging stations of the Taipei Bridge, Ho-kou, the Chung-cheng Bridge, the Hsin-hai Bridge, Wu-du, and the Guan-du Bridge, respectively; Tm(t ), Td(t ), and Th(t ) are
lunar month, lunar day, and time. The water-stages of Ho-kou and the Guan-du Bridge are used to describe storm surge, ebb and flood flows. The water-stages of the Chung-cheng Bridge, the Hsin-hai Bridge and Wu-du represent the inflows of the Hsin-tien Creek, the Ta-han Creek and the Keelung River, respectively. The effect of astronomical tide is related to lunar month, lunar day and time.
The data of eight typhoon events during 1994 were obtained, and the continuous hourly water-stage data were measured from July 1996 to December 2000 by Taiwan Water Conservancy Agency. Totally, 22 864 sets data are used in this study. The data are split into three independent subsets: the training, evaluation, and testing subsets, respectively. The training subset includes 13 269 sets of data, the evaluation subset has 3469 sets, while the test subset has the remaining 6126 sets. The training subset is used for model develop-ment and parameter estimation. In this phase, a great number of models (networks) are created due to different values of the predetermined parameteru. In the second phase, the best model from the above candidates is chosen by simulating the performance of evaluation subset data. Neither networks’ structure nor their parameters can be adjusted at this stage. As the best model is determined in the previous phase, the testing subset data which is never part of training and validation subsets is then devoted to access the performance of the selected model without any modification.
Correlation coefficient and root-mean-square error (RMSE) are used to evaluate the performance of the networks. The correlation coefficient indicates the strength of relationships between observed and estimated water-stages. The RMSE evaluates the residual between observed and forecasted water stages.
Fig. 3shows the effect ofuon RMSE and number of hidden nodes. It appears thatucontrols the number of hidden nodes (or radial basis functions) where the number of hidden nodes decreases as u increases. According to the evaluation subset data, the best model, in terms of minimum RMSE, isu¼ 0.15 and
Fig. 4. Observed and forecasted water-stages of the model at training phase.
450 hidden nodes. That means, on average, each node might include 332 data sets. Compared with the training data sets (13 269), 450 is relative small and can be acceptable. Fig. 4 shows the observed and forecasted water-stages at training phase with
u¼ 0.15. Gobs and Gest denote the observed and
forecasted water-stages, respectively. All the data points nicely fall onto the line of agreement. Three typhoon events at the training phase are shown in
Fig. 5. Fig. 6 is similar to Fig. 4 and shows the accuracy of the water-stages at the evaluation phase
forecasted by the RBFNN.Fig. 7shows the perform-ance of one of the typhoon events at the evaluation phase that the hydrograph is well predicted. Finally, in the testing phase, the best model identified in previous phase is directly implemented without any adjustment to its structure or parameters. The forecasted and observed water-stages at the testing phase are
compared in Fig. 8. Again, it shows that the
performance of the RBFNN is very good. The water-stage forecasting model is also exhibited in
Fig. 9, which includes four typhoon events at the
Fig. 5. Accuracy of water-stage forecasting during typhoons at training phase.
Fig. 6. Accuracy of 1 h ahead water-stage forecasting in the Tanshui River at evaluation phase.
Fig. 7. Accuracy of water-stage forecasting phase during typhoons at evaluation phase.
testing phase. The forecasts are very satisfactory and accurate; in particular, the peaks of the hydrographs are captured by the model. These figures indicate that the water-stages forecasted by RBFNN agree quite well with the observed water-stages. The RMSEs of the training, evaluation and testing subsets are 0.059, 0.07 and 0.059, respectively. Moreover, the corre-lation coefficients of the training, evaluation and testing are 0.998, 0.997 and 0.998. They results show that the model performance is accurate and consistent in these different subsets. All correlation coefficients are very close to unity, and all RMSEs are relatively smaller. It demonstrates that the RBFNN can be successfully applied to establish the model and provide accurate and reliable one-step-ahead water-stage forecasting.
6. Summary and conclusions
The hydrological processes in an estuary are nonlinear and extremely complicated, and it is difficult to quote physically model. The RBFNN, an intelligently adaptive model, has been successfully used for many tasks. In this study, we proposed a novel RBFNN, which employs hybrid unsupervised
Fig. 8. One hour ahead water-stage forecasting in open channel under tidal effect.
Fig. 9. Accuracy and reliability of RBFNN during typhoons at testing phase.
and supervised training schemes, for water-stage forecasting in an estuary. During the first scheme, the commonly used K-means clustering method is replaced by the fuzzy min – max clustering for determining the characteristics of the radial basis functions. The advantage of using fuzzy min – max clustering is that the number, centers and s of the radial basis function can be determined systematically and automatically. It is not necessary to estimate many parameters. Only one parameter,u, needs to be pre-decided. A small u, which will generate more radial basis functions and might overfit the data at the training phase, does not necessary guarantee better performance in forecasting. The weights between the hidden layer and output layer are obtained by multivariate linear regression method. The output of the RBFNN is simply the sum of the weighted output of the hidden layer. Several candidate models are built by using the training subset data. The model, which applies to the validation subset data and has minimum RMSE, is chosen from those candidate models. The chosen model is then verified, without any further change, through the test subset data to evaluate its applicability and suitability. The water-stage data of the Tanshui River under tidal effect is used for developing the water-stage forecasting model. The results demonstrate that RBFNN networks can successfully model nonlinear hydrological systems and accurately forecast 1-h-ahead water-stage in rivers under tidal and typhoon effects.
Acknowledgements
This paper is based on partial work supported by the Water Resources Bureau and National Science Council of Republic of China (Grant no. NSC 89-2313-13-002-041). The first author thanks Professor Sorooshian, S and Dr. Hsu, K.-L. for their kindly support and valuable advise during their visit of University of Arizona (fall, 2001).
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