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3 (2), 2009, 161 - 164

©BEYKENT UNIVERSITY

A Hierarchical Cluster-Based Fuzzy Rainfall-Runoff

Model

Veysel GÜLDAL

1

and Hakan TONGAL

2

Engineering and Architecture Faculty, Department of Civil Engineering, Süleyman Demirel

University, 32160, Isparta, Turkey, vguldal@mmf.sdu.edu.tr 2Natural and Applied Sciences, Department of Civil Engineering, Süleyman

Demirel University, 32160 Isparta Turkey

Received: 27.01.2009, Accepted: 20.11.2009

Abstract

The estimation of the flows originate from rainfalls in streams is required for designing and sustainable operations of hydraulic structures. The developed models to answer this requirement are named as rainfall-runoff models. The transformation of rainfall to runoff cannot be easily determined with simple models, because many hydrological parameters yields internal uncertainties vary with respect to area and time and these uncertainties cannot be easily represented by simple models. The simple modeling of this transformation, the classical regression analysis which has wide application area. Unfortunately, in the classical regression analysis the internal uncertainties cannot be taken into consideration. In this study, a cluster-based fuzzy system model by using hierarchical clustering algorithm was developed as an alternative to the classical regression approach in the rainfall-runoff relationship. The developed model was applied to the runoff estimation by using the rainfall-runoff records of Köprüçay river basin. It was concluded that the new established cluster-based fuzzy system model yields less forecasting error than the classical regression approach and so, it is recommended that it can be used for the real time runoff estimation in the future.

Keywords: Fuzzy system model, Regression analysis, Hierarchical clustering algorithm, Köprüçay river basin, Real-time runoff forecasting.

Özet

Su yapılarının projelendirilme ve sürdürülebilir işletilmelerinde belli yağışlardan meydana gelen akımların tahmin edilmesine ihtiyaç duyulur. Bu ihtiyaca cevap vermek için geliştirilen modellere yağış-akış modelleri denilir. Yağışın akışa dönüşümünde birçok hidrolojik parametre alansal ve zamana bağlı değişen iç belirsizlikler sergiler ve bu belirsizlikler basit modellerle kolayca temsil edilemez. Bu dönüşümün basit olarak modellenmesinde klasik regresyon analizi yaygın bir kullanım alanına sahiptir. Ancak, klasik regresyon

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analizinde iç belirsizlikler dikkate alınamaz. Bu çalışmada, yağış-akış ilişkisinde klasik regresyon yaklaşımına bir alternatif olarak hiyerarşik kümelendirme algoritmasına dayalı kural tabanlı bir bulanık sistem modeli geliştirilmiştir. Geliştirilen model, Köprüçay Nehri havzasının yağış-akış kayıtları kullanılarak akış tahminine uygulanmıştır. Sonuç olarak, geliştirilen küme-tabanlı bulanık sistem modelinin akış tahmininde klasik regresyon yaklaşımından daha az tahmin hatası verdiği görülmüştür. Geliştirilen modelin gelecekte gerçek zamanlı akım tahmininde kullanılabileceği önerilmektedir. Anahtar kelimeler: Bulanık system modeli, Regresyon analizi, Hiyerarşik kümelendirme analizi, Köprüçay Nehri havzası, Gerçek zamanlı akım tahmini.

1. Introduction

The rainfall runoff models resemble the transformation process of the rainfall which falls on to the river basin, to the runoff, is crucial in the flood hydrology. In literature, as a simple approach the classical regression analysis in determining rainfall-runoff relationships is widely used. The simplicity of this approach lies in its fundamental data requirements, for instance rainfall and runoff records in rainfall-runoff modeling. Besides, the following hydrological assumptions, comments and simplifications should be considered prior to its use.

(i)The uniform distribution of rainfall over the drainage area and its constant intensity assumptions are valid for small basins rather than wide basins. So, more uncertainties have been included in the overall rainfall runoff process. (ii) The runoff generated by any precipitation at any time, not only depends on rainfall intensity and its period but also the antecedent moisture state, affected by interception, depression and detention, and the basin conditions. This situation indicates that the rainfall-runoff process is a dynamic process depends on environmental effects rather than stationary process. The same rainfall will generate different runoffs according to dry and wet soil states. It must be noted that dry and wet terms are linguistically fuzzy in content. (iii) Logically, one won't expect that the runoff depth measured in the same storm is greater than the rainfall depth. The rainfall runoff transformation ratio will change 0 and 1 interval. This rational coefficient is inconsistent through a year according to the basin's condition before the storm. By regression analysis in which all data are considered, it is impossible to examine the variation in this coefficient. Even using multi-regression analysis it is also impossible to consider the variation in the parameter. But using one of the artificial intelligence methods, fuzzy approach and stochastic methods, vagueness like this type can be handled. Again in here "runoff depth measured in same storm is not greater than rainfall depth" and "the rainfall run off transformation ratio will change 0 and 1 interval" statements are linguistically fuzzy in content.

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The regression analysis when applied to rainfall-runoff as random variables doesn't take into account the dynamic behavior of rainfall-runoff process, the variations in the data is accepted deterministic and thus the functional aspect of relationship cannot be determined. It is well known that the proportionality and superposition principles cannot be applied exactly because of existing dynamic and nonlinear phases in the rainfall-runoff process [1]. In such cases the different methods must be searched. For instance, in the statistical approach related to the subject [2], runoff coefficients were grouped with respect to the months from the data, the mean rainfall-runoff values were found and the values plotted in the Cartesian coordinate system. Then the sequence values were connected by lines and irregular polygon was obtained and the authors obtained some conclusions and interpretive features to model rainfall-runoff transformation.

In recent years, the fuzzy logic has gained attention in modeling of the dynamic and nonlinear phases in the hydrological process as an alternative method to the classical regression analysis [3, 4, 5]. The dynamic and non-linear phases and the relationships between these two approaches can be reflected through some linguistic variables in fuzzy rule base in fuzzy model. The aim of this study is to show, these internal relationships especially with nonlinear characteristic can be captured cluster-based approach in establishing fuzzy rule base in the fuzzy model as an alternative to the conventional regression approach which requires a set of restrictive assumptions. The model was applied to the Köprüçay basin which is one of the important basins of Turkey in the point of view energy potential.

2. Fuzzy logic approach strengthened with hierarchical

clustering technique

The first detailed account of fuzzy logic and systems is proposed by Zadeh [6]. The fuzzy modeling is the new way of determining of systems involves the fuzzy structure. The system modeling based on the classical mathematical approaches is not appropriate for the systems consist of linguistically uncertain expresses rather than numerical uncertainty measures. However, the fuzzy approach can model the qualitative aspects of human knowledge and experiments without using sensitive quantitative analysis. Recently, many researchers have applied the fuzzy approach to various engineering problems [3, 7, 8]. In some of these researches contain high dimensional fuzzy modeling, the sample recognizing in medical monitoring, the fuzzy logic based on the event recognizing for the intelligent highway etc.

The internal vagueness and the scarce data in the considering system, increase the fuzziness. While searching these systems, one can obtain meaningful and beneficial results inferences from fuzzy input and output information by using fuzzy logic rules. Many systems can be modeled and even can be copied by the help of fuzzy systems. Fuzzy logic approach gains more and more attention in

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the examining of hydrological events. [2, 3, 7, 9, 10, 11]. In the basis of the fuzzy logic hydrological variables such as rainfall, runoff and sediment load, can be considered as having linguistically uncertain manner, in the forms of subsets which are labeled successive fuzzy word attachment as "low", "medium" and "high", etc. So, the variables are considered not as a global and numerical quantity but in partial groups which provide more possibilities to make out sub-relationships between two or more variable on the basis of fuzzy words. For instance, rainfall and sediment load can be considered as seven partial subsets labeled as "very low", "low", "medium low", "medium", "medium high", "high" and "very high". Small sub-grouping of the variables leads to unrepresentative preventative, however a large number imply unnecessary huge calculations. At above, seven subsets in each variable imply that there are 7x7=49 different partial relationship pairs that may be considered between the rainfall and sediment load variables. In physically most of these fuzzy rules are impossible. For instance if the rainfall is very high on a basin, it is not possible to state that very low sediment load. One can obtain a few fuzzy rules by logically decreasing like this. The each of fuzzy subsets must be represented by membership functions. Generally triangle, Gaussian and trapezoidal membership functions are used for the relative position of fuzzy labels. For each input variables some of membership functions are triggered and fuzzy inference subset is obtained. It is necessary to deduce a single value from this fuzzy inference subset through "defuzzification" phase. There are various defuzzification methods but the most common method is centroid defuzzification [12]. Generally, given a fuzzy set with membership degree m( x) defined on the interval [a, b] of variable x, the centroid defuzzification prediction x is defined as;

in continuous form. One of the most crucial problems is defining the fuzzy rule base. Since, the fuzzy uncertainties and interpretations can be operated via this fuzzy rule set. This means that nonlinear relationships can be represented by this rule set which cannot be captured classical regression approach. This paper offers, clustering method can be used in building up the fuzzy rule set that represents nonlinear relationships. It should be remembered that the results obtained by clustering analysis can be used in establishing fuzzy rule base which determines non-linear relationships between variables. This model is named as cluster-based fuzzy model.

Clustering is the classification of objects into different groups, or more precisely, the partitioning of a data set into subsets (clusters), so that the data

b

b (1)

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in each subset share some common trait - often proximity according to some defined distance measure. By decreasing the clusters some important details can be disregarded but more simple definition can be achieved. In literature there are many clustering algorithms. [13, 14, 15]. The commonly used approach having two good representative examples to clustering are "the agglomerative" and "the divisive" algorithms also named as hierarchical approaches that produce dendrogram. In this study "the hierarchical clustering algorithm" was used. The general outline of hierarchical clustering algorithm can be written as follow. Initially, each data points forms a cluster by itself. Then the algorithm repetitively merges the two closest clusters until the hierarchical structure constructed [16].

In the agglomerative methods, some of the clusters are combined into a new aggregate cluster depending on how similar they are, in each step that leads to cluster reducing. The clustering process in the divisive algorithms split the one cluster into the sub clusters dissimilar to each other.

The similarity or dissimilarity level(s) has/have to be selected carefully with respect to the distinguishing characteristics and the size of the sample. The most used method regarding to specify the difference between two samples is using defined distance level in the marked space from which the samples were selected. The most efficient distance measurement which is known for the samples having continual characteristics is the Euclidian distance [16] which is a special form of the equation given below.

f n m~\ 1/n

di j = dm (X , , X , )

E

x,k-

x

jk (2)

In this formula, m is changeable according to the condition of the distance measure that designates the similarity. The m = 1,2 and 3 conditions of that equation give respectively the City block distance, the Euclidean distance and the Minkowski distance. The Euclidean distance is generally used while calculating the proximity of the objects within two or three dimensional space. There are also some distance measures that regard the points containing a central point or effects of the neighbour points to a given point within the space.

By operating hierarchical clustering algorithm, generally satisfactory clustering results can be found according to various resemblance and different necessities. These results have a hierarchical structure. Hierarchical algorithm shows the groupings in which samplings are intensifying and gives a dendrogram that involves the groupings' changing similarity levels [13, 17]. The dendrogram can be easily broken at different level to obtain cluster or groups of desired importance or radius.

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In the hierarchical clustering algorithm single linkage (nearest neighbour) and complete linkage (farthest neighbour) are the most often used to measure dissimilarity (distance) between clusters. Each of these methods characterizes the similarity between the cluster binaries in different ways. In the single-linkage method, the dissimilarity between two clusters is the minimum distance between the two closest points x. and x . of the clusters C. and C ., '

J

'

J

respectively, [15], i.e.,

dist (C

i

, Cj) = min {dist (x

i

, x

J

)| x

i

e C

i

, x

J

e C

}

-} (3)

while the complete-link method uses the distance of the two farthest points of the two clusters, i.e.,

dist (C

i

, C

J

) = max {dist (x

i

, x

J

) x

i

e C

i

, x

J

e C

J

} (4)

In the first method, the long chain which form loose, straggly clusters is obtained where as the second algorithm tends to produce very tight clusters of similar cases. There are also different linkage methods such as average, centroid and Ward's etc.

At the end of the cluster analysis, the discussion of meaningful of the results gets importance. One of the important problems of the cluster analysis is the determining the validity of the clusters, objectively. There isn't objective and standard method for determining the cluster number. In the hierarchical methods, the cluster analysis is ended in case of the distance between the cluster is too high and the cluster number at this point is accepted or cluster number is determining by the help of some developed tests. Since there isn't statistical dispersion of the cluster analysis, the appropriate cluster number cannot be determined by any statistical test. It is advised that, the determined cluster number should be checked with the validity indices such as root-mean-square standard deviation (RMSSTD), R-root-mean-squared (RS), semi-partial R-root-mean-squared (SPR) [18] and test statistics (pseudo F, pseudo T).. In this study, RMSSTD and RS were used in determining of number of clusters. A more detailed description of these validity indices was given in the following.

RMSSTD measures the homogenity of the formed clusters at each step of the hierarchical algorithm, that is, it is a measure of homogeneity within clusters. Since the aim of the cluster analysis is to form homogenous sets, the RMSSTD of a cluster should be as small as possible. The RMSSTD statistic is the value of {(SS1 +... + SSn)/(df + . . . + dfn)}1/2. That is the pooled standard deviation of all variables. The term SSj is the within group sum of squares and refers to the equation: SSj = ^;= j ( X j — Xj)2 J = 1,...,n. The

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higher RMSSTD values before the previous steps means that the inhomogeneous clusters were obtained.

RS may be considered as a measure of the degree of difference between clusters. It also measures the degree of homogeneity between groups. That means, RS indicates that which clusters are different from each other. RS of the new cluster is SSb / SSt. Where, SSb is the difference between

SSt and SSj; SSt is the total sum of squares of the whole data set. It is

evident that, SSb is a measure of the variation between clusters whereas

SSt is a measure of total variation. The values of RS range between 0 and 1.

In the RS equal to zero case, no difference exists among groups, when it equals to one it indicates the significant difference among groups. More details about these validity indices can be found in [18, 19].

5. Application and Discussion

The applications were carried out for the Köprüçay Basin which is one of the important basins of Turkey in the point of view of energy potential. The Köprüçay Basin is located between 370 10'-370 56' northern latitudes and 300 58'-310 18' eastern longitudes, in the border of Isparta and Antalya provinces and has an approximately 2500 km2 rainfall area. At the west of Köprüçay basin Aksu River Basin, at the east Manavgat River and Beyşehir Lake Basin, at the north Eğirdir Lake Basin are located. The Köprüçay Basin has shown typical Mediterranean climate features [20]. The basin has been observed since 1941 and hence simultaneous measurements of monthly rainfall-runoff averages are available for the application of regression and cluster-based fuzzy model. The statistics of the rainfall and runoff for the catchment were given in Table 1.

In Figure 2, the classical regression line fitting for the determination of the rainfall-runoff relationship in the catchment for a monthly term was given. The determination coefficient and the scatter diagram show that the relationship between rainfall and runoff isn't stable besides the relationship has variance inconsistency. Because, the deviations from regression line are small for the small values but they increase as the rainfall runoff values increase. This means that the classical regression isn't trustful for determining rainfall- runoff relationship. In other words, this non-linearity in the relationship between rainfall and runoff can't be captured with the classical regression approach.

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Table 1. The statistics ( m e a n ,m ; standard deviation, S x ; coefficient of

C

variation vx ) of monthly variables

Stats. Rainfall Runoff

m

C„

m

C„

Oct 13.4 8.0 0.6 6.5 3.4 0.5 Nov 4.3 4.0 0.9 2.1 1.4 0.7 Dec 24.6 40.9 1.7 1.4 1.2 0.9 Jan 63.0 52.7 0.8 4.3 5.0 1.2 Feb 139.6 97.0 0.7 35.4 37.9 1.1 Mar 166.8 57.6 0.3 77.4 24.2 0.3 Apr 42.1 32.0 0.8 25.5 14.9 0.6 May 90.0 51.6 0.6 50.7 29.2 0.6 Jun 89.6 56.8 0.6 79.2 42.2 0.5 July 81.1 43.2 0.5 73.0 43.1 0.6 Aug 87.0 25.4 0.3 49.2 26.8 0.5 Sep 24.4 18.8 0.8 19.5 10.8 0.6 160 ^ 120 S 80 40 100 200 300 R a i n f a l l ( m m / m o n t h )

Figure 2. The regression analysis of monthly rainfall-runoff variables

To obtain qualitative conclusions related to the rainfall-runoff process, the cluster analysis was applied to identify the groups of similar monthly rainfall-runoff process.

In the clustering, the hierarchical algorithm with the complete linkage method was used. To determine the number of clusters, the RMSSTD and the RS validity indices were utilized. The hierarchical algorithm results with the statistics and the validity graph were given in Table 2 and Figure 3, respectively.

0 0

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Table 2. The hierarchical algorithm results (complete linkage method) and the indices values.

Agglomeration Schedule Validity Indices Stage Combined Cluster Coefficients First Appears Stage Cluster Stage Next RMSSTD RS

Clus- 1 Clus- 2 Clus- 1 Clus- 2 Next Stage 1 5 8 3.354 0 0 7 1.6772 0.9998 2 10 11 10.018 0 0 5 5.0093 0.9987 3 6 7 10.929 0 0 7 5.4644 0.9984 4 9 12 18.128 0 0 5 11.8643 0.9957 5 9 10 26.510 4 2 9 11.2732 0.9883 6 1 4 29.840 0 0 9 13.7675 0.9866 7 5 6 30.150 1 3 10 14.9204 0.9801 8 2 3 49.930 0 0 10 21.4363 0.9672 9 1 9 59.041 6 5 11 24.9354 0.9275 10 2 5 86.112 8 7 11 36.4223 0.7906 11 1 2 179.552 9 10 0 56.2881 0.0000

Based on these indices values and the validity graph, the optimum number of clusters was determined. From RMSSTD and RS indices, eight clusters can be selected as an optimum cluster number. As it can be seen from the Figure 3, when the cluster number is eight, the significant change is observed in the RMSSTD plot, the homogeneity between groups is obtained. Also, RS plot propose the eight cluster as the best partitioning at which the values converges to the unity. These groups of similar monthly rainfall-runoff process were considered as an input in descriptions of the cluster-based fuzzy rainfall-runoff model.

RMSSTD RS

C l u s t e r s

Figure 3. Validity graph (bu grafik matlap olacak)

In developing of the fuzzy system model, the triangle membership functions having same intervals for each one of the rainfall (20 mm) and runoff (10 mm) variables were constructed, for simplicity. The results obtained through clustering analysis were reflected by second input beside rainfall input, via

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eight membership functions that represents the groups which denote the homogeneity in rainfall runoff process. The interval of each groups adjusted based on their minimum and maximum values. Depending on these optimal clusters and the membership functions and their intervals the relationship domain was constructed as it was given in Figure 4.

To cover each variable by neglecting the groups, 30X30=900 rules must be written as "If rainfall is PI then runoff is RI; If rainfall is P1 then runoff is R2". But by considering the groups after clustering analysis it is possible to write the fuzzy rule-base includes 33 rules for the description of fuzzy rainfall runoff modeling. These rules are in the following form

Rule 1: If the group is C1 and the rainfall is P1 then the runoff is R1. Rule 2: If the group is C1 and the rainfall is P2 then the runoff is R2. Rule 3: If the group is C2 and the rainfall is P2 then the runoff is R3.

Figure 4. Relationship of cluster domain with fuzzy sets.

where, C, P and R denote the cluster which identify the homogeneity in rainfall runoff process, rainfall and runoff, respectively. With this model, the fuzzy regions (Figure 5) of the rainfall runoff relationship can be taken into account, whereas by the classical regression model, assumes direct proportionality between the rainfall-runoff, these regions cannot be represented.

The cluster-based fuzzy system model, explained in this study, was compared with the classical regression estimations and the real values and model estimations were given in Figure 6 for the visual comments/presentation. As it can be seen from Figure 6, the observed and predicted values follow each other very closely which means that the predicted runoff series have almost the same statistical parameter with the observed runoff series. The forecasting

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results were evaluated by the commonly used RMSE and relative error (RE) criteria. It is obvious that the cluster-based fuzzy model gave practically acceptable better results and smaller prediction errors (RMSE: 1.27; RE: %18) than the classical regression analysis (RMSE: 2.30; RE: %46).

Rainfall

Figure 5. The cluster-based fuzzy rule base space

Each error criteria value for the fuzzy model is less than the regression analysis. It can be said that the cluster-based fuzzy model is better as twice as the regression model. Therefore, the cluster-based fuzzy model can be preferable. The observed and the predicted runoff values using the cluster-based fuzzy model and the classical regression were given in Figure 7. It is clear that almost all the predicted points were closely scattered around the 450 line than the regression solution. The predictions were successful at low and high runoff values and have high correlation with the observations (R2=0.87).

Months

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(a) (b)

Figure 7. Model verification for (a) the regression model (b) the cluster-based

fuzzy model

6. Conclusions

The regression application is widely used practically in the rainfall-based runoff estimations. These applications can still be used for the various aims in small basins. But, there are a hidden and dangerous points in the regression approach. These are; the basic assumptions of the regression approach and the vagueness in the rainfall-runoff relationship. These assumptions cannot be changed also, in the multiple regression analysis. To get rid of these assumptions, the cluster-based fuzzy model was developed based on the qualitative conclusions from rainfall-runoff process and the groups which denote the homogeneity in rainfall runoff process, obtained through clustering analysis and compared with the classical regression application. The models were applied to the rainfall-runoff observations of Koprugay, one of the important rivers of the Middle Mediterranean region in Turkey. The cluster-based fuzzy model estimations are much more successful with acceptable errors than the parametric regression model. The fuzzy system model, proposed in this paper, can be successfully applied to all size basins to produce runoff estimations from the certain rainfalls. Especially, in the medium and large basins, the application of this model should be considered as an alternative to the regression approach.

Acknowledgement

We are grateful to EiE and DMi employers and authorities who provided the data used in this study.

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REFERENCES

[1] Kundzewicz, Z.W. and Napiorkowski, J.J.; "Nonlinear Models of Dynamic Hydrology". Hydrol. Sci. J., 37 (1986), 163-185.

[2] Şen, Z., and Altunkaynak, A.; "Fuzzy Awakening in Rainfall-Runoff Modeling", Nordic. Hyrology, Vol. 35 (1) (2003), 31-43.

[3] Ba'rdossy, A., and Disse, M.; "Fuzzy Rule Based Models for Infiltration" Water Res. Res.,29(2) (1993), 373-382.

[4] Chang, F. J. and Chen, Y. C.; "A Counter Propagation Fuzzy-Neural Network Modeling Approach to Real Time Streamflow Prediction", Journal of Hydrology, 245 (2001), 153-164.

[5] Hong, Y-S. T. and White, P.A.; "Hydrological modeling using a dynamic neuro-fuzzy system with on-line and local learning algorithm", Advances in Water Res., Vol. 32 (1) (2009), 110-119.

[6] Zadeh, L.A.; "Fuzzy Sets", Information and Control, 8 (1965), 338-353. [7] Tatlı, H., and Şen, Z.; "Prediction of Daily Maximum Temperatures via Fuzzy Sets", Turk J Engin Environ Sci., 25 (2001), 1-9.

[8] Vernieuwe, H., Verhoest, N.E.C., De Baets, B., Hoeben, R., De Troch, F.P.; "Cluster-Based Fuzzy Models for Groundwater Flow in the Unsaturated Zone" Advances in Water Resources, 30 (2007), 701-714.

[9] Russel, S. O. and Campbell, P. F.; "Reservoir Operating Rules with Fuzzy Programming", Journal of Water Resources Planning and Management, ASCE, 122, 3 (1996), 165-170.

[10] Tayfur, G., Özdemir, S., and Sing, V.P.; "Fuzzy Logic Algorithm for Runoff-Induced Sediment Transport from Bare Soil Surfaces" Advanced Water Res., 26 (2003),1249-1256.

[11] Nayak P.C., Sudheer K.P. and Ramasastri K.S.; "Fuzzy Computing Based Rainfall-Runoff Model for Real Time Flood Forecasting" Hydrol. Process. 19 (2005), 955-968

[12] Ross, J.T.; Fuzzy Logic with Engineering Applications. McGraw-Hill, Inc, New York, (1995), 593 pp.

[13] Jain, A.K. and Dubes, R.C.; Algorithms for Clustering Data. Prentice-Hall advanced reference series. Prentice-Hall, Inc.,Upper Saddle River,NJ, (1988). [14] Buhmann, J.M.; Data Clustering and Learning, Handbook of Brain Theory and Neural Networks, M. Arbib (ed.), 2nd edition, MIT Press (2002).

[15] Lin, C-Ru. and Chen, M.; "Combining Partitional and Hierarchical Algorithms for Robust and Efficient Data Clustering with Cohesion Self-Merging", IEEE Transactions On Knowledge And Data Engineering, 17 (2) (2005), 145-159.

[16] Jain, A.K., Murty, M.N. and Flynn, P.J.; Data Clustering: A Review, ACM Computing Surveys, 31 (3) (1999), 264-323.

[17] Güldal, V and Tongal, H.; "Clustering analysis in search of wind impacts on evaporation". Applied Ecology and Environmental Research, 6(4): 69-76, (2008).

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[18] Halkidi, M., Batistakis, Y., Vazigiannis, M.; "On Clustering Validation Techniques", J. of Intelligent Inf. Sys. 17:2/3 (2001), 107-145.

[19] Ritz, C. and Skovgaard, I.; "Module 2: Cluster Analysis" Master of Applied Statistics, http://www2.imm.dtu.dk/~pbb/MAS/ST116/module02/module.pdf (2005), 1-20.

[20] EIE.; "Master Plan Report of Upper Part of Köprüfay Catchment". EIE Gen. Admin. Pub. No:15 (2001).

Şekil

Table 1. The statistics  ( m e a n , m ; standard deviation,  S x ; coefficient of  C
Table 2. The hierarchical algorithm results (complete linkage method) and the indices  values
Figure 4. Relationship of cluster domain with fuzzy sets.
Figure 6. Observed and predicted values for the cluster-based fuzzy model
+2

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