Fuzzy exemplar-based inference system for flood forecasting

Li-Chiu Chang

Department of Water Resources and Environmental Engineering, Tamkang University, Taipei, Taiwan

Fi-John Chang and Ya-Hsin Tsai

Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei, Taiwan

Received 18 January 2004; revised 9 November 2004; accepted 23 November 2004; published 2 February 2005.

[1] Fuzzy inference systems have been successfully applied in numerous fields since theycan effectively model human knowledge and adaptively make decision processes. In thispaper we present an innovative fuzzy exemplar-based inference system (FEIS) for floodforecasting. The FEIS is based on a fuzzy inference system, with its clustering abilityenhanced through the Exemplar-Aided Constructor of Hyper-rectangles algorithm, whichcan effectively simulate human intelligence by learning from experience. The FEISexhibits three important properties: knowledge extraction from numerical data, knowledge(rule) modeling, and fuzzy reasoning processes. The proposed model is employed topredict streamflow 1 hour ahead during flood events in the Lan-Yang River, Taiwan. Forthe purpose of comparison the back propagation neural network (BPNN) is alsoperformed. The results show that the FEIS model performs better than the BPNN. TheFEIS provides a great learning ability, robustness, and high predictive accuracy for floodforecasting.

Citation: Chang, L.-C., F.-J. Chang, and Y.-H. Tsai (2005), Fuzzy exemplar-based inference system for flood forecasting, Water

Resour. Res., 41, W02005, doi:10.1029/2004WR003037.

1. Introduction

[2] Flood forecasting is one of the most important andchallenging works for the hydrologist. It is importantbecause an efficient flood forecasting system may signifi-cantly enhance public safety, promote reservoir manage-ment, and alleviate damages caused by inundations. It is achallenge due to high nonlinearity and complexity of therainfall-runoff processes in a watershed. Conceptual rain-fall-runoff models traditionally are basic tools for floodforecasting in river basin management. However, thesemodels generally have a large number of parameters, whichcannot be directly obtained from measurable quantities ofwatershed characteristics [Cheng et al., 2002]. Anotherproblem that arises in traditional methods is the updatingwhich is unavoidable in field application where the model’sstructure needs to be reconstructed and parameters must berecalibrated even if only one item of data is changed oradded to the database; consequently, it cannot be efficientlyimplemented in a real-time operation.[3] Over the last decade, exemplar-based artificial neural

networks (ANN) have become more popular and beenincreasingly used in water resource studies, especially inhydrological forecasting [e.g., Maier and Dandy, 2000;ASCE, 2000a, 2000b]. The preliminary results obtained fromANNs have shown the ability of ANNs to establish suitablerelationships between rainfall and runoff [Karunanithiet al., 1994; Minns and Hall, 1996; Shamseldin, 1997;Furundzic, 1998; Chang et al., 2003]. To implement anANN for flood forecasting, its structure must be identifiedfirst and then its mass connecting weights must be trained

based on a great number of input-output pairs. Even thoughthe concept of the ANN seems easy to capture, the learningand optimizing algorithms used for constructing an ANNmight still be too sophisticated for most of engineers.[4] To overcome these shortcomings, the aim of this

paper is to suggest a novel model called fuzzy exemplar-based inference system (FEIS) for flood forecasting. Withits simple algorithm and human-like learning process with-out prior expertise, it would be more easily accepted in fieldapplications. The entire process is mainly based on twosimple, yet powerful algorithms: Exemplar-Aided Construc-tor of Hyper-rectangles (EACH) and the fuzzy inferencesystem. EACH is a categorization algorithm, which isanalogous to the idea of ‘‘learning by examples’’ (the sameas ANNs); it identifies objects by their similarity to one ormore of the stored examples. This method is one of thefundamental modules of human learning theory: that humanintelligence is built based on learning from experience andadjusting in time [Medin and Schaffer, 1978; Hintzman,1986; Schank and Leake, 1989; Minton et al., 1989].EACH is a very powerful method for category formation,especially in complex, real world situations where completecausal information is not available, and it has many suc-cessful applications [Aha et al., 1991; Cost and Salzberg,1993; Chang and Chen, 1995; Chang and Lin, 1997].[5] A fuzzy inference system (FIS) or a fuzzy rule-based

system is a nonlinear mapping of a given input vector to ascalar output vector by using fuzzy logic. Fuzzy rules haveproven to be effective for specifying how a given systemshould operate. Inference, then, is defined as a procedure fordeducing new facts out of existing ones on the basis offormal deduction rules. Classical mathematical tools (e.g.,two-valued predicate logic, differential equations) are not

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well suited for dealing with ill-defined and uncertain sys-tems. By contrast, a fuzzy inference system employingfuzzy IF-THEN rules allows for a higher degree of flexi-bility and expressivity to cope with problems that are toocomplex for exact solution but do not require a high degreeof precision [Zadeh, 1973; Jang, 1993]. FISs have beensuccessfully applied in many fields, such as control systems[Chang and Chang, 2001; Pal et al., 2002], time seriesforecasting [Mendel, 2000], intelligent robots [Wu, 1996],decision analysis [Chaneau et al., 1987], expert systems,and computer vision.[6] In our proposed FEIS system, EACH serves as a

collection of rules, which are converted into a set of fuzzyIF-THEN rules with appropriate membership functions, andthen FIS provides an inference engine to formulate therelationship between inputs and outputs. Fusing these algo-rithms (EACH and FIS), the system is designed to takeadvantage of simple algorithms and powerful performancein classification and fuzzy reasoning and to promote itsaccuracy over time as more data is processed. The system isapplied to predict streamflow 1 hour ahead during floodevents in the Lan-Yang River, Taiwan.

2. Proposed Fuzzy Exemplar-Based InferenceSystem (FEIS) Methodology

[7] The FEIS is based on fuzzy inference system with itsclustering ability enhanced through the EACH algorithm,which can effectively simulate human intelligence by learn-ing from experience. The FEIS exhibits three importantproperties: knowledge extraction from numerical data,knowledge (rule) modeling, and fuzzy reasoning processes.The following sections would descript these algorithms indetail.

2.1. Exemplar-Aided Constructor of Hyper-rectangles(EACH) Algorithm

[8] Exemplar-based classifiers are pattern recognizersthat encode their collected evidence with the use ofexemplars. The representation of these exemplars couldbe a geometric shape (e.g., hyper-rectangle in our case).The strategy of the exemplar-based learning (EBL) algo-rithm is based on storing points (or examples) in Euclid-ean n space, En, where n is the number of variables orfactors in an example, then comparing the new exampleto those stored examples and finding the most similarexample in memory. Proposed by Salzberg [1988, 1989,1991], EACH is a module of EBL where the points inthe EBL are generalized into hyper-rectangles. The basiclearning strategy of EACH is (1) seeding the examples,(2) comparing with existing examplars (past experience),(3) forecasting possible results by choosing the mostsimilar experience, (4) seeing if the forecast matchesthe real outcome, and (5) the adjustment the relativeparameters and/or structure.[9] EACH uses some existing events as a foundation to

predict the outcomes of other events by building thestructure of hyper-rectangles. This process includes adjust-ing the weights of the model’s parameters in time andmaking the system learn [Chang and Chen, 1995]. Themain procedure is described below and the flowchart isshown in Figure 1.

2.1.1. Initial Work[10] EACH must have a history of examples as knowl-

edge base for future predictions. Consequently, the first stepis to seed some examples which are randomly chosen fromthe training set. This process simply stores each selectedexample in memory without attempting to make any pre-dictions. An example is a vector of factors (including relatedinput factors x and target y).2.1.2. Matching[11] After initialization, every new example is matched to

memory by a matching process. This process uses thedistance metric to measure the distance (or similarity)between a new data point and each exemplar. The bestmatch is the one which has the smallest distance and is thenused to make a prediction. The prediction can be done bysetting the new example that belongs to the same categoryas the closest matching exemplar.[12] Let the new example be E and the existing hyper-

rectangle or the point beH. Thematch score between E andHis calculated by measuring the Euclidean distance betweenthe two objects. The distance is determined as follows:

DEH ¼ WH

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXmi¼1

Wi

Efi � Hfi

maxi �mini

� �2

vuut ; ð1Þ

where WH is the weight of the exemplar H; Wi is the weightof the factor i; Efi is the value of the ith factor in example E;Hfi is the value of the ith factor in exemplar H; mini andmaxi are the minimum and maximum values of the ithfactor, respectively; m is the number of factors. Let Hlower

be the lower end of the range, and Hupper be the upper end,then the distance metric becomes

Efi � Hfi ¼

Hfi � Hupper when Efi > Hupper

Hlower � Efi when Efi < Hlower

0 others

8>>>><>>>>:

: ð2Þ

The distance measured by this formula is equivalent to thelength of a line dropped perpendicularly from the point Eto the nearest surface, edge, or corner of H as shown inFigure 2. Note that points internal to a hyper-rectangle havedistance 0 to that rectangle. The formula divides distancesalong each dimension by (mini � maxi) in order tostandardize them to the interval [0, 1] so that each factorin the exemplars has the same basis.[13] There are two meaningful weights in the distance

metric, WH and Wi. WH measures how frequently theexemplar, H, has been used to make a correct prediction.If a hyper-rectangle H is used many times, but nearly alwaysmakes the wrong prediction, the weight WH will grow verylarge, and H will tend not to be chosen. If H is a noisy point,then it will eventually be ignored asWH increases. The otherweight Wi is the weight of the ith input factor. It is adjustedto reflect the fact that all factors do not normally have equalimportance in a category decision. The calculating detailsare discussed in the following section.2.1.3. Feedback[14] EACH adjusts weights Wi and WH after each match-

ing to make the model fit closely to the learning data set.

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Figure 1. Flowchart of the Exemplar-Aided Constructor of Hyper-rectangles (EACH) algorithm forestablishing the rule-based relationship between inputs and outputs.

Figure 2. An illustration of distance measurement.

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WH is an indicator to tell if the exemplar H is reliable. If Hmakes a correct prediction, the algorithm ‘‘praises’’ its goodperformance; if it does not, the algorithm ‘‘penalizes’’ itswrong prediction. The method to praise or penalize is

WH ¼ number of times H has been used

number of times H has resulted in a correct prediction:

ð3Þ

Thus if a hyper-rectangle H is used many times, but nearlyalways makes wrong predictions, the weight WH will growvery large, and H will tend not to be chosen as the closestmatch in the future. If H is a noisy point, then it willeventually be ignored as WH increases. The minimum valueof WH is unity for perfect prediction.[15] The other weight Wi is the indictor of each factor.

The adjustment is

Wi ¼ Wi 1� Dfð Þ correct prediction

Wi ¼ Wi 1þ Dfð Þ wrong prediction

;

8<: ð4Þ

where Df can be any value and relates to the fluctuation ofWi; larger Df results in bigger fluctuation. However, Wi isnot strongly affected by each single example [Chang andLin, 1997]. A smaller value 0.02 is set for Df, which isgained by trial and error process.[16] The criterion to decide if an exemplar makes a correct

prediction depends on the error tolerance. For classification,in which the target values are limited in certain types, theerror tolerance is zero only if the data with the same targetvalue can be assigned in the similar group. For continuousvalue forecasting, the targets could be any value and rangemuch wider than classification; the error tolerance cansignificantly affect the model performance. If it is setsmaller, most of the training data would be treated asrepresenting variant types; thus the whole structure wouldbecome storing each single point without generating tohyper-rectangles. On the other hand, if error tolerance isset larger, the learning data would be divided into smallamount of groups and the forecasting results would belimited in these groups. The values of error tolerance canbe assigned according to the number of training data and the

wanted accuracy. To our best knowledge and experience, atrial and error process through simulating several differentvalues of error tolerance would usually come out a satisfac-tory result. The error tolerance can be a constant value or aratio. The error tolerance ratio is executed as follows. If anexemplar has a target value y = 10.0 and the error toleranceratio is 0.2, all the values within 8 and 12 are treated ascorrect predictions, that is, an exemplar which has a largertarget value would have a wider range of tolerance. The errortolerance ratio is used in this study because in our applicationcase most of the data (water level) belong to low flow andonly a few events the water level are greater than average. Aswe have just mentioned above, if a large value of constanterror tolerance is assigned, then most of low flows would beclustered into one hyper-rectangle (cluster). On the otherhand, if a small value of error tolerance is assigned, then itwould have lots of clusters and some of them might onlyhave one or two data.

2.2. Fuzzy Inference System

[17] An FIS is based on fuzzy set theory, fuzzy IF-THENrules and fuzzy inference theory, the architecture of whichincludes five major components: fuzzifier, rules, member-ship functions (MFs), inference engine and defuzzifier (asshown in Figure 3). Rules are the core of an FIS and can beextracted from historical data or generated from humanknowledge. To extract rules from the numerical trainingdata through learning or clustering algorithms is commonlyused for industrial applications and academic researches. Inthis paper, the rules are expressed as a collection of fuzzyIF-THEN rules and extracted from the training data. IF isthe antecedent of an FIS, and THEN is the consequent. Thestatements of the antecedent and the consequent parts maybe expressions of linguistic statements/knowledge or fuzzysets, such as[18] If water level is high, then the amount of water

release is large, or

if x1 is F1; then y1 is H1;

where water level and the amount of water release arelinguistic variables, high and large are linguistic labels thatare characterized by appropriate membership functions; x1and y1 are mathematic variables, F1 and H1 are linguisticvalues defined by fuzzy sets or equations.

Figure 3. Architecture of fuzzy inference system.

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[19] On the basis of a set of fuzzy IF-THEN rules, theinference engine maps fuzzy input sets into fuzzy output setsthrough the fuzzy operations, such as T-norm and T-conorm.In this paper, we only focus on algebraic product, one kindof T-norm operations, for fuzzy intersection. The T-normoperators can be thought of as the extension for fuzzyAND operations and have many kinds of operations, e.g.,algebraic product, logical product and bounded product[Tsoukalas and Uhrig, 1997]. The mathematical form ofthese operations are shown as follows:

algebraic product ¼ mA xð Þ � mB xð Þ;

logical product ¼ mA xð Þ ^ mB xð Þ;

bounded product ¼ max 0; mA xð Þ þ mB xð Þ � 1ð Þ;

ð5Þ

where mA(x), mB(x) are fuzzy sets. Membership functions areused in the terms that appear in the antecedents andconsequents of fuzzy IF-THEN rules. The common shapesof membership functions are triangular, trapezoidal, piece-wise-linear, Gaussian and bell-shaped functions. Owing tothe statements of IF-THEN rules, the fuzzifier maps a crispinput set into a fuzzy input set that a crisp variable ischaracterized by an appropriate membership function.[20] The final component, the defuzzifier, converts a

fuzzy output of inference engine to a crisp output. Throughthe aforementioned procedure, FIS formulates the mappingfrom an input vector into a scalar output, whose relationshipcan be expressed as y = f(x). The two most common FISsare Mamdani [Mamdani and Assilian, 1975] and Takagi-Sugeno-Kang (TSK) [Takagi and Sugeno, 1985; Sugenoand Kang, 1988]. They both have the same antecedentstructures, but have the different consequent structures. Theconsequent parts of Mamdani’s fuzzy rules are fuzzy sets,while those of TSK’s fuzzy rules are functions defined byequation (6) [Mendel, 2000]:

if x is F; then y is f xð Þ; ð6Þ

where F is fuzzy sets in the antecedent, while f(x) is a crispfunction in the consequent. Usually f(x) is a polynomial. Forcomputational simplicity, we choose the first-order TSKmodel as the consequent part.

2.3. Proposed FEIS Algorithm

[21] In engineering applications, we often face somespecial cases that cannot obey the law but can be grouped

as an exceptional rule. In FISs, all rules are general rules orthe inference engine handles the way in which fuzzy IF-THEN rules are activated and combined without consider-ing exceptional conditions. For example, there are fourfuzzy rules built as shown in Figure 4; rules C and D areexceptional rules of rule B. If a new input point (denoted bystart) falls into rule C, all rules are activated and combined.In this case, we may expect that rule B is not activated orhas a very small value of MF; however, the effect of rule Bis even much larger than that of rule C because the point iscloser to the center of rule B than that of rule C. Lack ofspecial handling for exceptional conditions causes thiscontradiction. In this paper, we present how to systemati-cally extract the general rules with exceptional rules fromnumerical data and how to infer fuzzy output sets from thefuzzy rules base with exceptional rules included.[22] The procedure of the FEIS model is divided into three

parts: building fuzzy exemplar-based rules (included excep-tional rules), embedding the way of exceptional handling inthe inference engine, and using the first-order TSK model asthe output of the consequent part of the IF-THEN rule. First,we choose EACH to systematically build the exemplar-basedrules and construct their membership functions. The EACHlearning algorithm may build nested hyper-rectangles, whichcould be inside another one (hyper-rectangle) to arbitrarydepth and serve as exceptions to surrounding rectangles.Membership functions of the antecedent part are the gener-alized bell functions defined in equation (7):

mj xið Þ ¼ 1

1þ xi�cjiaji

��� ���2bji: ð7Þ

All parameters of bell functions, a, b and c, have their ownphysical meaning. The center of the corresponding member-ship function is c; a is the distance between the center pointand the point of inflection (where the value of MF is 0.5);and �(b/2a) is the slope at the point of inflection. In theFEIS model, c is set to the center of the corresponding hyperrectangle; a is set to the half width of the correspondingside; and let the slope �(b/2a) be a reasonable value. Theillustration is shown in Figure 5. Second, in order to avoidthe aforementioned contradiction in Figure 4, we proposetwo strategies for handling which rules are activated. One isto ignore a rule (e.g., rule B), when the input example fallsinto exceptional rules (a nested hyper-rectangle, e.g., rule Cor D), and then all rules but this rule are fired and combined;the other is to let all rules be fired and combined when the

Figure 4. Four rules with exceptional rules in a two-dimensional space.

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point falls anywhere that excludes exceptional rules. Finally,for simplicity, the first-order TSK fuzzy model is chosen asthe output of IF-THEN rules, the consequent part. Forinstance, suppose that the model contains two-dimensionalinput data x, one output f, and three built fuzzy IF-THENrules (hyper-rectangles):

Rule 1

if x is A1; then f1 ¼ a1x1 þ b1x2 þ c1;

Rule 2

if x is A2; then f2 ¼ a2x1 þ b2x2 þ c2;

Rule 3

if x is A3; then f3 ¼ a3x1 þ b3x2 þ c3;

where Ai are the built hyper-rectangles and converted intoappropriate membership functions. Compare the input datawith the membership functions on the antecedent part to getthe weight of each rule (wi). Then, calculate a linearcombination of all rules’ outputs and weights to get the crispoutput y (the output of FEIS). Given an input data (x1, x2)and parameters determined (ai, bi, ci), the crisp output canbe obtained from equation (8):

y ¼ f1w1 þ f2w2 þ f3w3

w1 þ w2 þ w3

: ð8Þ

To obtain an optimal solution, the least squares estimator(LSE) can be used to identify the linear parameters (ai, bi,ci) of the first-order TSK model.

[23] Under this hybrid system, EACH could be treatedas a clustering tool to collect rules; while for constructinga suitable membership function for each cluster (hyper-rectangle) the fuzzy inference system provide an inferenceengine to formulate the relationship between inputs andoutputs. The proposed FEIS process can be summarized asfollows, (1) giving the input-output patterns (the trainingdata sets); (2) constructing IF-THEN fuzzy clustered rules(hyper-rectangles) through EACH algorithms; (3) calculat-ing the corresponding parameters of the membershipfunctions of the above hyper-rectangles; and (4) producingthe crisp output by using the first-order TSK fuzzy model,where the linear parameters of the TSK model are obtainedby the least squares estimator.

3. Results and Discussion

[24] The Lan-Yang River is located in northeasternTaiwan with a drainage area of approximately 978 km2.The Lan-Yang River is about 73 km in length and theaverage channel slope is 1/55. Locations of the study areaand gauge stations used are shown in Figure 6, where therain gauge and water level gauge stations are denoted bycircles and by triangles, respectively. Hourly water levelmeasurements were available from the gauge station locatedat Lan-Yang Bridge. The hourly data were collected fromthese gauge stations during the period from 1991 to 2001,consisting of 15 storm events with 1213 observations.These events are arbitrarily divided into three differentsubsets, namely training (933 data sets), validation (162 datasets), and testing subsets (118 data sets). Since the predic-tion focuses on optimum prediction by the minimum meansquared error, the criteria of root mean square error(RMSE), mean absolute error (MAE) and the goodness-

Figure 5. Bell antecedent membership functions for a two-dimensional hyper-rectangle.

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of-fit with respect to the benchmark (Gbench) [Nash andSutcliffe, 1970; Seibert, 2001; Chiang et al., 2004] are usedfor the purpose of comparison. There are defined as shownbelow:

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNt¼1

Hi � Hi

�2N

vuuuut; ð9Þ

MAE ¼

XNi¼1

Hi � Hi

�� ��N

; ð10Þ

Gbench ¼ 1�

XNi¼1

Hi � Hi

�2XNi¼1

Hi � ~Hi

�2 ; ð11Þ

where N is the number of data points, Hi is the observedvalue in i step, Hi is the forecasted value in i step, and ~Hi isthe previous observed value in i step, e.g., ~Hi = Hi�1 in ourcase.[25] Water level forecasting 1 hour ahead is performed by

the proposed FEIS. The input variables include current andprevious (t and t � 1) water level and rainfall at two raingauge stations, and the output is next step (t + 1) waterlevel. By using 933 training data sets with the toleranceratio = 0.5, the FEIS would only produce five hyper-

rectangles (clustered rules). The computation time is short,and the structure of the model is quite simple. For thepurpose of comparison, we use the back propagation neuralnetwork (BPNN). The BPNN was first developed byWerbos [1974] and nicely presented by Rumelhart andMcClelland [1986]. It becomes the most widely used neuralnetwork in the areas of science and engineering today. TheBPNN is performed by using MATLAB toolbox. The sameinput-output patterns are used to train and validate thenetwork. A great number of trial and error processes wereperformed to select the best fitted model. It appears whenthe number of hidden node is 13; the network can have bestperformance.[26] The results of both methods are summarized in

Table 1 and Figures 7–10. The results in all three sets(training, validation, and testing) indicate that the FEIS hasmuch smaller MAE and RMSE and much higher Gbench

values than those of BPNN. The most incredible ability ofthe FEIS identified in this study is its efficient ability forlearning and high predictive accuracy. We can easy to tellthat the Gbench value of FEIS is close to 0.9 in the trainingset, and the Gbench value of FEIS in the testing set is still ashigh as 0.6972, while the Gbench values of BPNN are only0.6136 and 0.1886, respectively.

Figure 6. Location of study area.

Table 1. Result of the Lan-Yang River Flood Forecasting

Data Set

FEIS BPNN

RMSE MAE Gbench RMSE MAE Gbench

Training 0.0363 0.0212 0.8938 0.0693 0.0375 0.6136Validation 0.0605 0.0349 0.7199 0.0857 0.0480 0.4380Testing 0.0314 0.0188 0.6972 0.0514 0.0307 0.1886

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[27] Figures 7 and 8 show the FEIS and BPNNperformances of the training set, respectively. Figures 9and 10 present the difference between the FEIS andBPNN performances of the validation set and testingset, respectively. Apparently, the FEIS nicely fits thetarget values in all the cases, while the BPNN also fitsto the target values in the training set, but it is not so

accurate in the validation set and has shift problems inthe testing set.

4. Conclusion

[28] Exemplar-based machine learning paradigms, such asEACH andANN, have hadmuch success on awide variety of

Figure 7. Flood forecasting: training result by fuzzy exemplar-based inference system (FEIS) model.

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domains, especially on signal processes, pattern recognition,and time series forecasting. The EACH learning model hasbeen effectively applied mostly in classification. In theapplication process, it is found that once the error toleranceis large, the predictive accuracy is reduced, but if the errortolerance is reduced, most of the learning examples will bestored independently. A fuzzy inference system employingfuzzy IF-THEN rules allows for a higher degree of flexibility

to cope with problems which are too complex for exactsolution but do not require a high degree of precision. In thispaper, we propose a hybrid version of EACH and fuzzyinference system, called FEIS, for flood forecasting. Owingto its simple structure and human-like learning process, itwould be more easily accepted in field applications.[29] In this study, the FEIS is applied to predict stream-

flow 1 hour ahead during flood events in the Lan-Yang

Figure 8. Flood forecasting: training result by the back propagation neural network (BPNN) model.

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River, Taiwan. For the purpose of comparison, the BPNN isalso performed. The hourly data are collected, which consistof 15 storm events with 1213 observations. These events arearbitrarily divided into three different subsets, namelytraining, validation, and testing subsets. The results in allthree sets indicate that the FEIS has much smaller MAE,

RMSE and much higher Gbench values than those of BPNN.The results show its ability to learn and save memory spacealong with its high predictive accuracy (much better androbust than BPNN in this study cases). This researchsuggests that the FEIS may be a practical and robust methodof flood forecasting.

Figure 9. Flood forecasting: validation result by the (a) FEIS and (b) BPNN models.

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[30] Acknowledgments. This paper is based on partial work sup-ported by National Science Council, R.O.C. (grant NSC 92-2313-B-002-032). In addition, the authors are indebted to the reviewers for their valuablecomments and suggestions.

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����������������������������F.-J. Chang and Y.-H. Tsai, Department of Bioenvironmental Systems

Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road,Taipei, 106, Taiwan. (changfj@ntu.edu.tw; r91622004@ntu.edu.tw)

L.-C. Chang, Department of Water Resources and EnvironmentalEngineering, Tamkang University, No. 151 Ying-chuan Road, Tamsui,Taipei County, 251, Taiwan. (changlc@mail.tku.edu.tw)

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W02005 CHANG ET AL.: FEIS FOR FLOOD FORECASTING W02005