A counterpropagation fuzzy-neural network modeling approach to real time streamflow prediction

A counterpropagation fuzzy-neural network modeling approach

to real time stream¯ow prediction

Fi-John Chang

_{*, Yen-Chang Chen}

a_{Department of Agricultural Engineering, National Taiwan University, Roosevelt Road, Taipei 10770, Taiwan, ROC} b_{Department of Agricultural Engineering, National Taiwan University, Taipei, Taiwan, ROC}

Received 23 February 2000; revised 7 November 2000; accepted 6 February 2001

Abstract

A counterpropagation fuzzy-neural network (CFNN) is the fusion of a neural network and fuzzy arithmetic. It can auto-matically generate the rules used for clustering the input data. No parameter input is needed, because the parameters are systematically estimated by the approach of converging to an optimal solution. The advantages of the CFNN include the ability to cluster, learn, and construct, and the model presented herein is used to develop a hydrological model. The CFNN can automatically construct a rainfall-runoff model to forecast stream¯ow. The available stream¯ow and precipitation data of the upstream of the Da-cha River, in central Taiwan, is used to evaluate the CFNN rainfall-runoff model. A comparison of the results obtained by the CFNN model and ARMAX indicate the superiority and reliability of the CFNN rainfall-runoff model. q 2001 Elsevier Science B.V. All rights reserved.

Keywords: Arti®cial neural network; Counterpropagation neural network; Fuzzy system; Counterpropagation fuzzy-neural network; Rainfall-runoff; Stream¯ow

1. Introduction

Real time stream¯ow estimation is always a bench-mark problem of hydrologists and water resource engineers, and has received a prominent focus for many decades. One of the dif®cult tasks of reservoir operation is in¯ow forecasting for preventing dam failure. Thus, accurate stream¯ow forecasting is extremely important for on-line reservoir operation. Some sophisticated hydrological models, such as deterministic catchment model (Kraijenhoff and Moll, 1986) and geomorphologic instantaneous unit hydrograph (GIUH) model (Rodriguez-Iturbe and

Valdes, 1979), for describing the rainfall-runoff process are usually very complicated. And a great deal of work, such as ®eld surveying and parameter estimation, should be done before such models can be applied. Obviously, such models show a lack of prac-ticality and are very dif®cult to use for real time stream¯ow forecasting in Taiwan, whose characteris-tics of watersheds are erodible soils, high mountains, steep slopes, subtropical climate, and heavy rainfall during typhoons.

Neural networks and fuzzy systems are created to simulate the nervous system and brain activity. The notion of fuzzy sets was ®rst introduced by Zadeh (Zadeh, 1965) to represent vagueness in linguistics by a mathematical way. Recently, a rapid growth in the use of fuzzy sets in hydrological modeling, such as rainfall forecasting (Yu and Chen, 2000), groundwater

0022-1694/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S0022-1694(01)00350-X

www.elsevier.com/locate/jhydrol

* Corresponding author. Tel.: 223639461; fax: 1886-223635854.

simulation (Dou et al., 1999; Schulz and Huwe, 1997), and drought analysis (Pongracz et al., 1999; Pesti et al., 1996). The prototype of a neural network was ®rst proposed by McCulloch and Pitts (McCulloch and Pitts, 1943). Hebb designed the ®rst learning law for neural networks (Hebb, 1949). After that, a number of neural networks, such as the backpropagation neural network (Rumelhart et al., 1986) and fuzzy-neural network (Nie and Linkens, 1994), were developed to solve a wide variety of problems. Mathematically, neural networks are information processes systems to model the brain as a parallel computation. Neural network is composed of a large number of intercon-nected processing units (nodes), arranged in an input layer, an output layer, and one or more hidden layers. Each layer consists of several nodes. The input layer contacts with the outside environment, the hidden layer transforms the input or hidden layers to a hidden layer by a nonlinear function, and the output layer is the response of the network. Fig. 1 shows a 3-layer feedforward neural network with n input, p hidden,

and m output nodes. The principal advantage of neural networks is their adaptive nature, which learns from the historical data to automatically adjust parameters, in dealing with nonlinear problems. A physical model is not needed when neural networks and fuzzy systems are applied. They estimate functions from training data and then construct numerical estimators that can be applied to deal with stream¯ow prediction (Chang and Suen, 1997; Shamseldin, 1997; Chang and Hwang, 1999; Sajikumar and Thandaveswara, 1999, etc.), rainfall estimation (Kuligowski and Barros, 1998; Zhang et al., 1997), and groundwater modeling (Yang et al., 1997). In this study, a counter-propagation fuzzy-neural networks (CFNN), which couples neural network and fuzzy systems is used to develop a rainfall-runoff model for stream¯ow fore-casting during typhoon. The CFNN is, furthermore, compared with a traditional stochastic forecasting model, autoregressive moving average with exogen-ous variable (ARMAX) model, for showing its impression of performances.

1

i

n

1

j

p

1

k

m

π

π

π

π

π

π

π

π

π

w

w

w

w

w

w

w

w

w

Input layer

Kohonen layer

Grossberg layer

2. Forward-only counterpropagation network structure

The counterpropagation network (CPN), which was ®rst introduced by Hecht-Nielsen (Hecht-Nielsen, 1987a,b, 1988), is of two types, full and forward-only. The CPN was developed to provide an ef®cient method for approximating a function y ˆ f …x†: The full CPN works best only when inverse function f21

exists. To avoid this effort, the forward-only CPN was designed to approximate y ˆ f …x† when f21 is not necessarily needed. The forward-only CPN is adopted to develop the CFNN. The architecture of the forward-only CPN, shown in Fig. 1, shows the infor-mation ¯ows in the feedforward direction only. The forward-only CPN consists of three layers: input, hidden (Kohonen), and Grossberg layers. The input layer with n nodes stores the input values, the Koho-nen layer with p nodes clusters the input values with a similarity measurement, and the Grossberg layer with m nodes calculates the output by summing the weighted Kohonen layer outputs.

The forward-only CPN has a hybrid learning scheme. The learning of CPN can be split into two stages, unsupervised and supervised. During the learning processes, the weights (or parameters) of the neural network will be adjusted automatically. In supervised learning, each input pattern is associated with a speci®cally correct target pattern. On the other hand, unsupervised learning involves no critic to over-see the learning process. Unsupervised learning is used during the ®rst stage for clustering the input vectors to separate distinct sets of input data. During the second stage of learning, the weight vectors between the Kohonen and Grossberg layers are adjusted by supervised learning to reduce the errors between the CPN outputs and the corresponding desired targets.

During the ®rst stage, the distances between the input vector x ˆ …x1; ¼; xi; ¼; xn†T composed of n

input nodes and all of the p Kohonen nodes with n dimensions are determined to compete for the winner. The winning node zj has the weight vector wjˆ

…w1j; ¼; wij; ¼; wnj†T closest to the input vector. The

winner-take-all operation that permits only the hidden node being the most similar to the input vector to be active at a time is implemented here to train the weight vectors from input layer to Kohonen layer.

The winner's weight vector is updated according to (Rumelhart and Zipser, 1985)

Dwijˆ

a…x_i2 w_ij† winning node; 0 otherwise; (

…1† whereais the learning rate subject toa. 0, xiis the

ith node of input vector, and wijis the weight of the ith

input node to the winning node j. The competitive signal, which is a binary variable assuming value 1 for the winner node presented and value 0 for the winner node absented, sent from the winning node to the Grossberg layer is 1, and the competitive signals sent by the other Kohonen nodes are 0s'.

After the weight vectors from the input layer to the Kohonen layer have been determined, the weight vectors between the Kohonen layer and the Grossberg layer are trained by

Dpjkˆ Zjb…yk2pjk† …2†

in which Zjis the competitive signal, b the learning

rate, ykthe target, andpjkis the weight from Kohonen

node j to Grossberg node k. The output node k is given by

ypkˆ

XP jˆ1pjk

Zj …3†

where ypk is the kth computed output.

The CPN can compress the n input data to p sets where in general p p n: It classi®es the similar input vectors as a single output cluster to build a look-up table. Compared to other neural networks, the learn-ing speed of CPN is extremely fast due to the simple network topology and the ef®cient learning algorithm. After the CPN is trained, all of the weights will be kept ®xing. Only input data is needed for the model to operate when the CPN is used for predicting. 3. Fuzzy arithmetic

Fuzzy arithmetic provides a good approach to deal-ing with ambiguity and uncertainty. It is a structured numerical estimator and has successfully attracted a growing interest in applications. Fuzzy arithmetic combines rule base and fuzzy control to describe complex nonlinear processes. The rule base is the collection of rules. A rule contains two statements,

>the premise and the conclusion. It is a logical impli-cation: IF premise THEN conclusion. For example, Fig. 2 shows the rule base of a reservoir operation. H, Q, and O are gage height, in¯ow and out¯ow, respectively. The entry at the center of the rule base de®nes the reservoir operation rule: `IF H ˆ 165 and Q ˆ 100 THEN O ˆ 90.' The rule base of reservoir operation is composed of nine rules (3 different H times 3 different Q).

Classical logic theory can only represent one color in back and white (likes binary number 0 and 1). This logic does not accord well in gray (likes binary system containing no other number except 0 and 1). Fuzzy logic, which can be used to represent vague concepts lets elements be represented by degrees of membership. The degree of membership is a positive real number in

the interval [0,1]. A membership function assigns a degree of membership to an element and can be any shape. Using the centroid defuzzi®cation method to map the crisp output from a space of fuzzy control actions, the fuzzy control output can be determined as follows: IF x is Mi_{THEN y}I _4† Y ˆ Xn iˆ1 Mi_x†yi Xn iˆ1 Mi_x† …5†

in which Y is the fuzzy control output, Mi_{(x) the degree}

of membership of the ith rule, x the input, yi_{the output of}

the ith rule, and n is the number of rule. The fuzzy control is suggested to reduce error when a rule is chosen (Nie, 1989). Fig. 3 shows how the membership function works. It illustrates an example of the reservoir opera-tion according to gage high only. When H ˆ 167, the degrees ofmembership, M1_{(H), M}2_{(H), and M}3_{(H) are 0,}

1/3, and 2/3, respectively. The fuzzy control output of reservoir operation (out¯ow) becomes …0 £ 80 1 …1=3†100 1 …2=3†120†=…0 1 …1=3† 1 …2=3†† ˆ …340=3†: 4. The counterpropagation fuzzy-neural network model

The CFNN is the fusion of CPN and fuzzy arith-metic. A similar type of network was ®rst proposed by Nie (1993), who further investigated its adapting,

O Q H 162 165 168 80 100 125 50 70 85 60 90 100 75 110 120

Fig. 2. A rule base of reservoir operation.

H 159 162 165 167 168 171 M i (H ) 0.00 0.33 0.67 1.00 Membership function 1 Membership function 2 Membership function 3

learning, and reasoning capabilities and applied to a problem of multivariable control of blood pressure (Nie and Linkens, 1994). Two modi®cations have been made to aim at constructing a rainfall-runoff model. First, the type of membership function is chan-ged from triangular to Gaussian function, so that the matching process could be more ¯exible and the weighted averaging for the fuzzy output would be more reasonable. The Gaussian function is a localized function with the property of z…x† ˆ exp…2ix 2 wi2=2D2_{† ! 0 as ux 2 wu ! 1: Its membership degree}

symmetrically and sharply decreases as the difference between the x and the node centroid w increases, but the membership degree is always larger than 0. Second,D the tolerant interval, using in the original model is a constant that would be gradually ampli®ed as the input data cannot be clustered into any rule during the model's application stage. This improves the model performance, especially for those unusual cases.

The CFNN can be split into two stages, training and

application. During the ®rst stage, the CPN is used to build the rule base. The fuzzy arithmetic is introduced in the second stage to improve the performances. Fig. 1 also shows the architecture of the CFNN. Each Kohonen node represents a rule. The connections of input and Kohonen layers, w, represent the `IF' state-ment of a rule. The connection between Kohonen and Grossberg layers is p that is, the `THEN' part of a

H O Q Rule 1 Rule 3 Rule 2 162 168 165 80 125 100 50 90 120

Fig. 4. A schematic diagram of 2-3-1 CPN.

w

∆

Fig. 5. Gaussian membership function. xt, yt

t=1

Determine minimum distance

0 ≤ ∆

Adjust weights, w andπ

End of training data t=t+1 Stability of rule number t=1 Model construction finished Generate a new rule Setup the first

rule Yes Yes Yes No No No No Yes

rule. Thus the statement of each rule is de®ned as: IF x is w THEN y is p (Weigend et al., 1991). A 2-3-1 CPN is shown in Fig. 4. The simple network includes three rules. The inputs of the network are gage height and in¯ow; the only output of the network is out¯ow. Consider Rule 3, that is: `IF H ˆ 168 and Q ˆ 125 THEN O ˆ 120.'

A Gaussian function shown in Fig. 5 is used to represent the degree of membership (Donald et al., 1996). The function is de®ned as

Mj_{x† ˆ exp} 2Xn iˆ1 …xi2 wij†2 D2 0 B B B B @ 1 C C C C A …6†

where wijˆ mean (center of rule j); D2ˆ variance

(width square of rule j); and n ˆ number of input variables (xi).

Fig. 6 summarizes the training processes of the CFNN. TheD is selected before the training of the model. The center of the ®rst rule is the ®rst input. After the ®rst rule is built, the distances between input data and the center of each rule have to be determined as Djˆ Xn iˆ1 …wij2 xi†2 " #_1=2† …7† where Djˆ the distance between input data and rule.

The minimum distance between input data and rules is given by

Dmin ˆ min_jˆ1;p Dj …8†

where Dminˆ the minimum distance between input

data and zj. If Dmin is smaller than D, the center of

rule, wijandpjkhas to be updated as

wnew

ij ˆ woldij 1a‰xi2 woldij Š …9†

pnewjk ˆpoldjk 1b‰yk2poldjk Š …10†

wherea andb are learning rates within the interval [0,1]. If Dmin.D a new rule will be created and

wnew

ij ˆ xi;pnewij ˆ yk: The learning rates usually will

be reduced after the entire training data is presented. The training processes will be iterated until the number of rule is stable.

The CFNN can be applied after training. The appli-cation procedure for the CFNN is shown in Fig. 7 and can be summarized as follows:Step 1: Initialize weights that are trained during the training stage.Step 2: Present input vector x.Step 3: Compute the distances between input x and rules. Eq. (7) is used to represent the distances.Step 4: Determine relative distance, dj, which is given by

d_jˆ Dj

D

 ₂

…11† Step 5: Determine matching degree of rule j that is de®ned as follows:

Sjˆ exp…2dj† …12†

Step 6: Find the rule that is closest to the input vector x. The input x can not be clustered, if the sum of matching degree is less than a selected value, i.e. 1025 _{in our case. In such a condition, D will be}

x

D

)

(

∆

=

D

d

_e

S

=

10

≤

∑

S

)

(

ˆ t

Y

Increase

∆

No

Yes

automatically expanded. Repeat Steps 4 to 6 until the input x can be clustered.Step 7: Compute activation. The fuzzy output is deduced by the weighted aver-aging, de®ned as Y ˆ Xp jˆ1 Sjpj Xp jˆ1 Sj …13† 5. ARMAX model

The autoregressive moving average with exogen-ous variable (ARMAX) model (Yang et al., 1996) for one-hour ahead ¯ood forecasting is used as a basis for comparison with CFNN. A general input± output system of ARMAX can be written as follows: A…q†y…t† ˆ B…q†u…t† 1 C…q†e…t† …14† where y(t) is ¯ood at time t; u(t) is exogenous rainfall input at time t; e(t) is white noise at time t; A(q), B(q)

and, C(q) are parameters of autoregressive, exogen-ous, and moving average parts as the following form: A…q† ˆ 1 1 a1q211 ¼ 1 anq2n …15†

B…q† ˆ b11 b2q211 ¼ 1 bmq2m11 …16†

C…q† ˆ 1 1 c₁q21_{1 ¼ 1 c}

rq2r …17†

where q21_{is back-shift operator; n, m, and r are the}

order of autoregressive; exogenous; and moving aver-age.

6. Application ofthe CFNN

The above methodology is applied to the upstream of the Da-Cha River for predicting real time stream-¯ow. The Da-Cha River is located in central Taiwan with a total catchment size of 1236 km2_{. The length of}

the Da-Cha River is about 140 km and the average channel slope is 1/39. It is the steepest channel in Taiwan. A series of hydraulic structures were constructed to generate power. Locations of the

Pacific Ocean Taiwan

Strait

Taiwan

∆ Streamflow gaging station ♦ Rain gaging station Fig. 8. Locations of study watershed and gage stations.

studied basin (area ˆ 514 km2_{) and used gage stations}

are shown in Fig. 8. A station for the stream¯ow data is denoted by a triangle and the precipitation stations by circles. Son-Mou gage station was established to measure the in¯ow of the De-Chi Reservoir, the upmost and pivotal reservoir in the Da-Cha River. Accurate stream¯ow forecasting is extremely impor-tant for the operation of the De-Chi Reservoir. The stream¯ow (m3_{/s) and precipitation (mm/h) data used}

here are gathered from Taiwan Power Company. A CFNN model is established to model real time stream¯ow prediction and is shown in Fig. 9. Qs stands for the stream¯ow of Son-Mou, and Pc, Pp, Pm, and Ps for precipitation of Chi-Chia-Yan, Pen-Yuan-Shan, Men-Shan, and Son-Fon, respectively. The subscripts t-2, t-1, t, and t 1 1 represent two-hour-before, two-hour-before, present time, and

one-hour-ahead, respectively. The input layer has three nodes representing the stream¯ow at Son-Mou, and 12 nodes for the precipitation of the other four rain gage stations. Only one node in the Grossberg layer represents the one-hour-ahead forecasting stream¯ow of the Da-Cha River at Son-Mou station. The initial number of rules is zero and will be automatically generated to describe the complex hydrological processes. The performances of CFNN and ARMAX are evalu-ated and compared by the normalized root-mean-square error (NRMSE), mean absolute error (MAE), and relative mean absolute error (RMAE) as NRMSE ˆ 1 s 1 T XT iˆ1 …Qi2 Qi†2 " #₁₌₂ …18† Qs_t-2 Pc_t-2 Pp_t-2 Pm_t-2 Qs_t-1 Pc_t-1 Pp_t-1 Pm_t-1 Qs_t Pc_t Pm_t Pp_t Streamflow of Son-Mou Precipitation of Chi-Chia-Yan Precipitation of Pen-Yuan-San Precipitation of Men-Shan Precipitation of Son-Fon Ps_t-2 Pst-1 Pst Real time Streamflow of Son-Mou Qs_t+1

MAE ˆ _T1 XT

iˆ1

uQi2 Qiu …19†

RMAE ˆ MAE_Q …20†

in which sˆ standard deviation of measured stream¯ow; Qiˆ measured stream¯ow; Qiˆ

forecasted stream¯ow; Q ˆ mean of measured stream¯ow; and T ˆ total number of measured stream¯ow. NRMSE indicates the closeness of the forecasted stream¯ow to observed stream¯ow. MAE and RMAE (Maidment, 1992) to represent the prediction error. This implies perfect matching when MAE ˆ 0 or R MAE ˆ 0.

The stream¯ow is forecasted under the conditions that any two of four hours and any three of four rain gage stations have precipitation records. The forecast-ing process is terminated when all of the rain gage stations continue having no precipitation record for 12 h. The initial learning rate a is 1/2. The a is

replaced with a harmonic series, 1/(t 1 1), until the number of rules stabilizes. In this study, b is not adjusted and is always 1/2. The available data sets from 1986 to 1989 are used for establishing the candi-date models. Each set consists of 15 data. Nineteen values of D, which is updated from the initial value 100 with increments of 50 until 1000 are used to construct the model. The ¯ood data between June and September 1990 (heavy rainfall season) is used to determine the bestDfor model construction Fig. 10 shown that Dˆ 250 has the minimum MAE. Thus

Dˆ 250 is selected for constructing the rainfall-runoff models and it will be ®xed for the forecasting. After D is determined, the stream¯ow data from 1990 to 1996 is used for verifying the stream¯ow prediction of model. Owing to the new collected data, the model is updated every year. Table 1 presents the number of training data and the number of rules or hidden nodes generated for constructing models. The training data of every year includes the training data of this year and previous years. The new

∆ 100 200 300 400 500 600 700 800 900 1000 MAE 11.0 11.5 12.0 12.5 13.0

Fig. 10. Effect ofDon MAE for the rainfall-runoff model. Table 1

Number of training data set and number of rule generated for constructing models Year Cumulative number of training data sets Cumulative number of rules 1983±1989 3183 1058 1990 4315 1497 1991 4894 1577 1992 5768 1899 1993 6624 2069 1994 7560 2366 1995 8590 2534 Table 2

Summary of the result of the CFNN and ARMAX

Year Q (m3_{/s) Q (m}3_{/s) NRMSE MAE (m}3_{/s) RMAE}

CFNN ARMAX CFNN ARMAX CFNN ARMAX CFNN ARMAX

1990 155.2 159.0 153.1 0.09 0.17 7.09 15.78 0.05 0.12 1991 23.0 24.2 24.6 0.11 0.28 1.38 3.27 0.06 0.17 1992 115.1 119.1 114.7 0.07 0.20 4.05 11.37 0.03 0.12 1993 48.1 49.9 49.0 0.09 0.20 1.46 5.27 0.03 0.12 1994 110.5 114.5 111.1 0.13 0.19 6.80 12.41 0.06 0.14 1995 34.5 35.6 35.8 0.14 0.31 1.40 4.34 0.04 0.14 1996 74.4 77.4 75.2 0.11 0.14 3.81 7.41 0.05 0.12

incoming data sometimes cannot be clustered into any rule; then a new rule is generated. Thus, the number of rules increases with the number of training data. A rule base with enough rules could accurately cluster the input data. The CFNN and ARMAX are applied for the prediction of stream¯ow. Table 2 gives the comparative results of two models. Apparently, the CFNN has better performance than the ARMAX in terms of smaller NRMSE, MAE, and RMAE. The results of the stream¯ow underestimated and overes-timated are shown in Tables 3 and 4, respectively. The de®nitions of underestimation and overestimation are that the differences between the forecasted and observed stream¯ows are more than 10 and 5%, respectively. These results show the CFNN can predict stream¯ow more accurate than the ARMAX and reveal that the CFNN can be used to forecast one-hour-ahead stream¯ow. To provide a impression of

the accuracy of stream¯ow prediction using the CFNN, four typhoon events including the largest event have been extracted to show the performance of the CFNN and ARMAX. Fig. 11 shows the

Table 3

Results of underestimated stream¯ow

Year Events Events underestimated Percentage of events underestimated CFNN ARMAX CFNN ARMAX 1990 842 33 137 3.92 16.27 1991 370 21 40 5.68 10.82 1992 646 14 118 2.17 18.27 1993 590 14 70 2.37 11.86 1994 693 37 106 5.34 15.30 1995 725 16 98 2.21 13.52 1996 692 24 104 3.47 15.03 Table 4

Results of overestimated stream¯ow

Year Events Events overestimated Percentage of events overestimated CFNN ARMAX CFNN ARMAX 1990 842 34 92 4.04 10.93 1991 370 18 52 4.86 14.05 1992 646 24 61 3.72 9.44 1993 590 15 85 2.54 14.41 1994 693 29 96 4.18 13.85 1995 725 21 87 2.90 12.00 1996 692 30 92 4.34 13.36 Time (hr) 0 10 20 30 40 50 Strea m flo w (m 3 /s) 0 200 400 600 800 1000 1200 1400 1600 1800 Observed ARMAX CFNN Time (hr) 0 10 20 30 40 50 Strea m flo w (m 3 /s) 0 200 400 600 800 1000 1200 1400 1600 1800 Observed ARMAX CFNN Time (hr) 0 5 10 15 20 25 30 35 40 45 50 55 Strea mflo w (m 3 /s) 0 200 400 600 800 1000 1200 1400 1600 1800 Observed ARMAX CFNN Time (hr) 0 10 20 30 40 50 Strea mflo w (m 3 /s) 0 200 400 600 800 1000 1200 1400 1600 1800 Observed ARMAX CFNN Typhoon Ofelia, June 23, 1990

Typhoon Dot, Sept. 7, 1990

Typhoon Tim, July 11, 1994

Typhoon Herb, Aug. 1, 1996

Fig. 11. Comparison of CFNN and ARMAX using four typhoon events.

comparison of CFNN and ARMAX during four typhoon events. Fig. 12 shows the accuracy of fore-casted stream¯ow by the CFNN model. All the points in the ®gure nicely fall around the line of agreement. These results indicate the accuracy and reliability of the model presented herein.

7. Conclusions

Neural networks have a complex connection struc-ture and simple computing elements that can solve

problems with natural mechanisms. The fuzzy arith-metic combines a rule base with fuzzy logic to form a structure of fuzzy if-then rules. This study proposes the CFNN, which is the integration of a neural network and fuzzy arithmetic. The CFNN presents three advantages: the ability to learn, construct, and cluster. The membership function of original CFNN is triangular, which, in this study, is replaced by Gaus-sian function. The concept of automatically increasing

D for clustering input data in the application stage is introduced. They improve the model performance, especially the unusual events.

Da-chia River at Son-mou Taiwan Power Company Data, 1990-1996

Sequence (a) 0 500 1000 1500 2000 2500 Strea m flo w (m 3 /s) 0 200 400 600 800 1000 1200 1400 1600 Observed CFNN

Da-Chia River at Son-Mou Taiwan Power Company Data, 1990-1996

Sequence (b) 0 500 1000 1500 2000 2500 Strea m flo w (m 3 /s ) 0 200 400 600 800 1000 1200 1400 1600 Observed CFNN 1990 1991 1992 1993 1994 1995 1996

The CFNN rainfall-runoff model is successfully applied to forecast one-hour-ahead stream¯ow of the Da-Cha River at Son-Mou. Compared to sophisticated hydrological models, such as GIUH, the principal advantages of the proposed method is that the CFNN can automatically construct a rainfall-runoff model and estimate the needed parameters by an approach converging to an optimal solution. The comparative results obtained by the CFNN and ARMAX provide evidences that the CFNN can offer a higher degree of reliability and accuracy than ARMAX in stream¯ow forecasting.

Acknowledgements

This paper is based on partial work supported by Taiwan Power Company and National Science Coun-cil, ROC (Grant no. NSC 89-2313-13-002-041). In addition, the authors are indebted to the reviewers for their valuable comments and suggestions. References

Chang, F.-J., Hwang, Y.-Y., 1999. A self-organization algorithm for real-time ¯ood forecast. Hydrologic Processes 13, 123±138. Chang, F.-J., Suen, J.-P., 1997. A study of the arti®cial neural

network for rainfall-runoff process. Journal of Chinese Agricul-tural Engineering 43 (1), 9±25 (in Chinese).

Donald, K., Wedding, I., Cios, K.J., 1996. Time series forecasting by combining RBF networks, certainty factors, and the Box± Jenkins model. Neurocomputing 10, 149±168.

Dou, C., Woldt, W., Gogardi, I., 1999. Fuzzy rule-based approach to describe solute transport in the unsaturated zone. Journal of Hydrology 220, 74±85.

Hebb, D.O., 1949. The Organization of Behavior. Wiley, New York.

Hecht-Nielsen, R., 1987a. Counterpropagation networks. Applied Optics 26, 4979±4984.

Hecht-Nielsen, R., 1987b. Nearest matched ®lter classi®cation of spatiotemporal patterns. Applied Optics 26 (10), 892±1899. Hecht-Nielsen, R., 1988. Applications of counterpropagation

networks. Neural Networks 1, 131±139.

Kraijenhoff, D.A., Moll, J.R., 1986. River Flow Modelling and Forecasting. Dordrecht, Netherlands.

Kuligowski, R.J., Barros, A.P., 1998. Experiments in short-term precipitation forecasting using arti®cial neural networks. Monthly Weather Review 126, 470±482.

Maidment, D.R., 1992. Handbook of Hydrology. McGraw-Hill, New York (pp. 262±263).

McCulloch, W.S., Pitts, W., 1943. A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophy-sics 5, 115±133.

Nie, J., 1989. A-class of new fuzzy control algorithms. Proceedings of IEEEInternationalConferenceonControlandApplications,Israel. Nie, J., 1993. Constructing rule-bases for multivariable fuzzy control by self-learning Ð Part I: system structure and learning algo-rithms. International Journal of System Science 24, 111±127. Nie, J., Linkens, D.A., 1994. Fast self-learning multivariable fuzzy

controllers constructed from a CPN network. International Jour-nal of Control 6, 369±393.

Pesti, G., Shrestha, B., Duckstein, L., Gogardi, I., 1996. A fuzzy rule-based approach to drought assessment. Water Resources Research 32 (6), 1741±1747.

Pongracz, R., Bogardi, I., Duckstein, L., 1999. Application of fuzzy rule-based modeling technique to regional drought. Journal of Hydrology 224, 100±114.

Rodriguez-Iturbe, I., Valdes, J.B., 1979. The geomorphologic struc-ture of hydrologic response. Water Resources Research 15 (6), 1409±1420.

Rumelhart, D.E., Hinton, G.E., Willianms, R.J., 1986. Learning internal representation by error propagation. Parallel Distribu-tion Processing 1, 318±362.

Rumelhart, D.E., Zipser, D., 1985. Feature discovery by competi-tive learning. Cognicompeti-tive Science 9, 75±112.

Sajikumar, N., Thandaveswara, B.S., 1999. A non-linear rainfall-runoff model using an arti®cial neural network. Journal of Hydrology 216, 32±55.

Schulz, K., Huwe, B., 1997. Water ¯ow modeling in the unsaturated zone with imprecise parameters using a fuzzy approach. Journal of Hydrology 201, 211±229.

Shamseldin, A.Y., 1997. Application of a neural network technique to rainfall-runoff modelling. Journal of Hydrology 199, 272± 294.

Weigend, A.S., Rumelhart, D.E., Huberman, B.A., 1991. General-ization by weight-elimination with application to forecasting. Advances in Neural Information Processing System 3, 875±882. Yang, C.C., Prasher, S.O., Lacroix, R., Sreekanth, S., Patni, N.K., Masse, L., 1997. Arti®cial neural network model for subsurface-drained farmland. Journal of Irrigation and Drainage Engineer-ing, ASCE 123, 285±292.

Yang, H.-Z., Huang, C.-M., Huang, C.-L., 1996. Identi®cation of ARMAX model for short term load forecasting: an evolutionary programming approach. IEEE Transactions on Power Systems 11 (1), 403±408.

Yu, P.-S., Chen, C.-J., 2000. Application of gray and fuzzy methods for rainfall forecasting. Journal of Hydrological Engineering, ASCE 5 (4), 339±345.

Zadeh, L.A., 1965. Fuzzy sets. Information Contributions 8, 338± 353.

Zhang, M., Fulcher, J., Sco®eld, R.A., 1997. Rainfall estimation using arti®cial neural network group. Neurocomputing 16, 97±115.