Soft computing techniques toward modeling the water supplies of Cyprus

Soft computing techniques toward modeling the water supplies of Cyprus

Neural Networks 24 (2011) 836–841 Contents lists available at SciVerse ScienceDirect Neural Networks journal homepage: www.elsevier.com/locate/neune...

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Neural Networks 24 (2011) 836–841

Contents lists available at SciVerse ScienceDirect

Neural Networks journal homepage: www.elsevier.com/locate/neunet

2011 Special Issue

Soft computing techniques toward modeling the water supplies of Cyprus L. Iliadis a,∗,1 , F. Maris a , S. Tachos b a

Democritus University of Thrace, Department of Forestry & Management of the Environment & Natural Resources, 193 Pandazidou street, 68200 N Orestiada, Greece

b

Electrical and Computer Engineering, Aristotle University of Thessaloniki, Greece

article

info

Keywords: Support vector machines Water supply Kernel algorithms 5-fold cross validation Artificial neural networks

abstract This research effort aims in the application of soft computing techniques toward water resources management. More specifically, the target is the development of reliable soft computing models capable of estimating the water supply for the case of ‘‘Germasogeia’’ mountainous watersheds in Cyprus. Initially, ε -Regression Support Vector Machines (ε -RSVM) and fuzzy weighted ε -RSVMR models have been developed that accept five input parameters. At the same time, reliable artificial neural networks have been developed to perform the same job. The 5-fold cross validation approach has been employed in order to eliminate bad local behaviors and to produce a more representative training data set. Thus, the fuzzy weighted Support Vector Regression (SVR) combined with the fuzzy partition has been employed in an effort to enhance the quality of the results. Several rational and reliable models have been produced that can enhance the efficiency of water policy designers. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Learning from experimental data and transferring obtained knowledge into analytical models is one of the main concerns of soft computing. It is a fact that carrying over such tasks is succeeded by the use of artificial neural networks (ANN), fuzzy logic and Support Vector Machines (SVM). These approaches constitute the structures that stand behind the idea of learning (Kecman, 2001). This paper presents the design and the application of a soft computing modeling approach that aims in the estimation of the annual water supply in one of the most important mountainous watersheds of Cyprus. More specifically, this research employs not only ANN but also Support Vector Machines Regression (SVMR) and fuzzy weighted SVM in an effort to obtain various reliable models. 1.1. Area of study Cyprus faces a serious problem of lack of water. The situation is getting worse due to the climate change effect. The Germasogeia watershed is located northeast of Lemesos and it has an area of 157 km2 . The length of its main water stream is 25 Km and its average altitude is 575,18 m whereas its maximum altitude is



Corresponding author. Tel.: +30 2552041135. E-mail addresses: [email protected] (L. Iliadis), [email protected] (F. Maris), [email protected] (S. Tachos). 1 Cooperating Educational stuff of the School of Positive Sciences and Technology, Hellenic Open University. 0893-6080/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.neunet.2011.05.016

1400 m. The annual rain height is 638 mm, whereas the average annual load of sediments is 22.5 millions m3 . The maximum water supply is 0.42 m3 /s whereas its forest cover is 57.7% (33,7% brush lands). The cultivations, the urban areas and the areas without significant forest cover are almost 8.6%. A large dam with a capacity of 13 millionm3 has been constructed in the watershed of Germasogeia in Cyprus. Proper management of the dam requires the daily estimation of the water supply. It is a fact that water demand is growing and water shortage is increasing in the island of Cyprus and in the Lemesos area. Whereas groundwater initially was the primary source of water, surface water reservoirs have now taken over this role (Integrated Project D2.2-3 FP6, 2007). The overabstraction has decreased groundwater levels and caused salinization problems in all coastal areas. In the Lemesos region, the only aquifer not subject to salt water intrusion ‘‘Germasogeia’’ receives almost 80% of its water from artificial recharge (Integrated Project D2.2-3 FP6). 1.2. Literature review The UBC model constitutes a typical approach that creates a computational representation of the hydrological behavior of a watershed (Kotoulas, 2001). UBC divides the watershed into altitude zones, whereas the daily rain height and daily temperature values (min and max) are used as input. It is a semi-distributed model. A simplified algorithm which is based on the preservation of energy is used for the estimation of snow concentration and melting. The flow due to the waterfall–snowfall is distinguished in four distinct parts with a mechanism that controls the soil moisture. The traditional approaches for the estimation of water supply use too many data features; they are complicated and they

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data is the output parameter. According to the above, the values of each feature were scaled based on the following Eq. (1): Zj =

Fig. 1. Graphical display of the data for 1997.

often have to make assumptions, and they are time consuming. Soft computing offers tools that can overcome the above limitations. This motivated our research team to work in this direction. This research effort uses the five main input features of the UBC approach, whereas it produces much simpler, cheaper and faster models (Loukas, 2009). Fuzzy logic approaches (Spartalis, Iliadis, & Maris, 2004) such as SVMR (Behzad, Asghar, Eazi, & Palhang, 2009; Han, Chan, & Zhu, 2007; Iliadis, Spartalis, & Tachos, 2010; Lin, Cheng, & Chau, 2006), Artificial Neural Networks (Campolo, Andreussi, & Soldati, 1999; Maris, Iliadis, & Vasileiou, 2005), real time machine learning models (Lobbrecht & Solomatine, 1999), and Adaptive approaches have been used recently by other research scientists in hydrological applications and for climate change estimation (Tripathi, Srinivas, & Nanjundiah, 2006). Also Iliadis and Maris (Iliadis & Maris, 2007) have worked recently on a general hydrological modeling of Cyprus. However, the modeling approach described here is more specific due to the fact that it concentrates in one of the major watersheds of the island. Since all of the hydrological research efforts that use SVM have been carried out during the past two or three years, this research can be considered a step ahead toward soft computing application in water resources management. 2. Materials and methods 2.1. Data The input data features used in this modeling approach were meteorological, such as daily rain height in the three meteorological stations of the basin, maximum and minimum daily temperatures. The data were measured by the three meteorological stations located in 75,100 and 995 m respectively. For the control of the output, actual measurements of the water supply of the main water stream were used. The data concern the period 1986–1997 (see Fig. 1) . Totally 4020 data records were used in this modeling effort. The output of this research verified its ability to analyze and model the specific knowledge domain and to successfully forecast the levels of water supply in the examined watershed. The obtained modeling approach could have a potential successful application to other similar cases as well. 2.2. Data scaling The actual amount of data comprised of 4018 records, whereas the actual values of the parameters were located in the interval [0, 33]. Due to the very wide range of the features’ values, they were normalized by a scaling process before their potential use as input vectors. More specifically, this was done so that the features with a wide range of values should not prevail over the ones with a narrow range. In this research, scaling was carried out by following the well-known autoscaling method. Standardization (or autoscaling) is the scaling procedure which results in a zero mean and unit variance of any descriptor variable according to Dogra and Shaillay (Dogra & Shaillay, 2010). The first column in the tabular

X j − µj

(1)

σj

where Xj is the jth parameter, Zj is the scaled variable following a normal distribution and σj , µj are the standard deviation and the mean value of the jth parameter (Dogra & Shaillay, 2010). After the calculation of the estimated value Yˆ of the scaled data, it was restored in the actual range of the output parameter based on the following Eq. (2): yˆ = σ1 · Yˆ + µ1 .

(2)

It should be clarified that the average value in function 2 is always related to the first field data. 3. ε-Support Vector Regression Support Vector Machines were initially developed aiming in pattern recognition (Boser, Guyon, & Vapnik, 1992). The support vectors are a small subset of the training set. Vapnik (Kecman, 2001; Vapnik, Golowhich, & Smola, 1997) introduced the following loss function (Eq. (3)) that ignores errors less than a predefined value ε > 0

|y − f (⃗x)|ε = max{0, |y − f (⃗x)| − ε}.

(3)

In this way, the ε -Support Vector Regression (ε -SVR) algorithm was developed which offers the optimal function of the form (Eq. (4)): f (⃗ x) = k(w, ⃗ ⃗x) + bw, ⃗

⃗x ∈ RN , b ∈ R.

(4)

This is achieved based on the training set (x⃗1 , y1 ), . . . , (x⃗p , yp ) ∈ RN × R. The target of this process is the search of a function f with a small testing error which is described by the above function 3. However, it is not possible to minimize function 3 because the probability distribution P is unknown. Thus, the solution is the minimization of the following normalized risk function (Eqs. (5) and (6)). 1 2

‖w‖ ⃗ 2 + CSVR · Rεemp [f ]

(5)

where Rεemp [f ] is the following function of empirical risk Remp [f ] =

p 1−

p i =1

L(f , x⃗i , yi )

(6)

and the loss function is given by Eq. (7) that ignores errors less than

  ε L(f , x⃗i , yi ) = |yi − f x⃗i |ε .

(7)

It should be clarified that ‖w‖ ⃗ 2 is related to the complexity of the model, whereas CSVR is a constant value determining the point that relates Rεemp [f ] to ‖w‖ ⃗ 2 (Davy, 2005). The minimization of function 5 is based on the following optimization problem with constraints: Constraint: Minimize

τ (w, ⃗ ξ⃗,ξ⃗∗ ) =

1 2

‖w‖ ⃗ 2 + CSVR ·

p 1−

p i=1

(ξi + ξi∗ )

(8)

subject to the following Eqs. (9) and (10):

(k(w, ⃗ x⃗i ) + b) − yi ≤ ε + ξi

(9)

yi − (k(w · xi ) + b) ≤ ε + ξi



where ξi, ξi ≥ 0. ∗

(10)

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The vectors x⃗i of the training set that correspond to nonzero values of (ai − a∗i ) are called SVM. If in Eq. (14), the Radial Basis function (RBF) kernel is employed, then the equation estimated by the ε -SVR will be the following:

  p − −‖⃗x − x⃗i ‖2 ∗ f (⃗ x , a, a ) = (ai − ai ) exp + b. 2 2 · σRBF i =1 ∗

(16)

The following parameters {σRBF , γ , ε} play a significant role in the success of the ε -SVR where σRBF is the RBF kernel’s standard deviation, γ is a constant defining the point where the empiric error is related to complexity and ε is the width of the ε -zone. 4. Fuzzy weighted SVR with fuzzy partition

Fig. 2. Regression with ε -SVR.

As can be seen in Fig. 2, ξi∗ , ξi are the distances of the training data set points from the zone where the errors less than ε are ignored. If the i-training point is located above the zone then its distance from the zone is represented by ξi , whereas if it is below its distance is denoted as ξi∗ . Using Lagrange multipliers, the problem can be faced as a double optimization one as follows: Maximize the following value in Eq. (11) W (a, a∗ ) = −ε

p −

(a∗i + ai ) +

i=1

p −

(a∗i − ai )yi

i=1

p 1− ∗ − (ai − ai )(a∗j − aj )k(xi , xj ) 2 i,j=1

subject to (12) :

p −

(11)

(ai − a∗i ) = 0 and also

i=1

(∗)

ai

[

∈ 0,

CSVR p

]

.

(12)

According to Vapnik and to Kecman (Kecman, 2001; Vapnik et al., 1997) the solution of the above problem is found in the form of the following linear extension of the kernel functions (Eqs. (13) and (14)).

w=

p − (a∗i − ai )x⃗i

(13)

i =1

f (⃗ x, a, a∗ ) =

p − (a∗i − ai )k(⃗x, x⃗i ) + b

(14)

i=1

where b is estimated by the following Eq. (15). b = averagei

 ×

 − ∗ ε · sign (ai − ai ) + yk − (aj − aj )k(x⃗j , x⃗i ) . ∗

j

(15)

The Fuzzy Weighted Support Vector Regression with a Fuzzy Partition (FWSVRFP) has been introduced by Chuang, (Chuang, 2007). This approach manages to take advantage of the local behavior of a model (meaning the response differentiation between two or more inputs which differ to each other slightly), due to the use of fuzzy C -means clustering (Cox, 2005). The initial problem is divided into many smaller ones, which results in the more enhanced study of the training set and also the independent study of each partition under a separate ε -SVR. On the other hand, it might result in the appearance of boundary effects. Boundary effects exist when sudden and significant differentiations in the response between neighboring points occur. In order to extinguish such problems, the suggested method integrates the partial responses of the ε -SVR by using the Takagi–Sugeno–Kang fuzzy model (Schnitman, Felippe de Souza, & Yoneyama, 1998). In Fig. 3, in case (a) the model’s local behavior output is not good. The blue line corresponds to the training set whereas the black line is the model’s output. In the case b of the same figure, the blue line is related to the training set, the black line is the output of group 1, and the red the output of group 2. The first step in this process is the application of fuzzy c-means clustering. Each cluster has its own center β⃗j and its own width for each input dimension δjs (Cox, 2005).

For example, if we have a two-dimensional input, then δ31 is the width of the third cluster in the first dimension. In this way, local Regression Models (LRMs) are created by using ε -SVR. The local results of the LRMs are used for the integration of the overall output through a fuzzy weighted approach that employs Triangular membership functions to weight the output of each LRM, by using the center and the width of its corresponding cluster. The integration of the partial output of the LRM kSVR , requires knowledge of the degree of membership of each training vector to each of the clusters. Based on the cluster centers β⃗k and on the corresponding widths of the clusters, the k (k = 1, . . . , C ) triangular membership functions are created. From the triangular membership functions the weights wks (xsi ) are obtained for i = 1, 2, . . . , p, k = 1, 2, . . . , C , s = 1, . . . , q   s s   xi (βk − η · δks ) (βks + η · δks ) − xsi , , 0 . (17) wks (xsi ) = max min βks − (βks − η · δks ) (βks + η · δks ) − βks

Fig. 3. (a) System with bad performance of the model’s local behavior. (b) System with boundary effects phenomena.

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Table 1 Performance using the Global SVR and FWSVR approaches with {σ , γ , ε} {9, 6, 0.02}, respectively and η = 2. Method

Fig. 4. Diagram of the FWSVM approach.

Fig. 5. Triangular membership functions for the estimation of the weight νwks of point x⃗i . A case with a two-dimensional input and C = 3 LRMs.

The degree of belonging of x⃗i to the k-cluster is obtained by Eq. (18). q

q

Wk (x⃗i ) = wk1 (x1i ) · wk2 (x2i ) · · · wk (xi ).

(18)

Fig. 4 shows the structure of the FWSVM model whereas Fig. 5 shows the use of fuzzy logic in this case. The overall output of the proposed fuzzy weighted SVR with fuzzy partition is obtained by the following De-fuzzifier. C ∑

yˆ (x⃗i ) =

Wk (x⃗i )LRM kSVR x⃗i

 

k=1 C ∑

.

(19)

  Wk x⃗i

k=1

The Root Mean Square Error (RMSE) is used as an evaluation instrument, where N is the number of the training vectors, yj is the actual output of the j-training or testing data and yˆ j is the corresponding estimated output (Eq. (20)).

RMSE =

 ∑  N  (yj − yˆ j )2  j =1 N

.

(20)

5. Results 5.1. Application of SVM and FWSVM The 5-Fold Cross Validation modeling method was applied (McLachlan, Do, & Ambroise, 2004; Ron, 1995; Seymour, 1993). The data was split into 5 groups (folds) of equal size and the ε regression algorithm was executed 5 times. Each time a different fold was chosen to serve as the training set. This approach does not allow the existence of extreme values which are not representative for the training set. The division of the data set was done by the use of MATLAB’s crossvalind, which is included in the Bioinformatics Toolbox.

Group

Global SVR



Fuzzy weighted SVR

2 3 4 5 6 7 8 9 10

=

RMSE Training

Testing

1.163 0.72 0.722 0.95 0.906 0.834 0.887 0.952 0.98 0.948

1.167 0.699 0.699 0.934 0.786 0.835 0.864 0.934 0.861 0.9

The following chapter presents the application results of the Global SVR and FWSVR approaches in the total data set. According to Global SVR, a global regression model was obtained for all of the data set, whereas with the FWSVR several local regression models LRMs were constructed. By the use of the 5-fold cross validation, the regression algorithm was executed five times for 3214 training records and 804 testing ones. The regression was performed by the use of the LIBSVM v2.9 (http://www.csie.ntu.edu.tw/~cjlin/libsvm/) (Chang & Lin, 2009) which is encoded in C + + and offers a Matlab Interface. In this specific application the RBF Kernel was applied. In both cases, the SVR-parameter values {σRBF , γ , ε} = {9, 6, 0.02} were used respectively as the optimal ones. They were chosen because after several trial and error experiments they were proven to offer better performance in the model. For the FWSVR, the value of η is equal to 2. Table 1 contains all of the errors obtained in training and in testing for specific values of σ , γ , ε parameters and for n = 2. As can be seen in Table 1, the best performance was achieved for the value C = 2 where RMSE ≈ 0.7 Considering that the range of the output values was [0, 33], one can conclude that the error was more or less equal to the 2.12% of the values’ range. 5.1.1. Partitioning the data based on the output Based on a closer look at the data, it can be seen that from the 4018 data records, the 3660 (the 91,09%) had an output in the range [0, 1]. Consequently, for this data group an error of the 0.7 magnitude was significant. In order to phase this problem, the data set was divided into two groups. The first one contained the data vectors with an output in the interval [0,1] and the second in the interval (1, 33]. The SVM regression algorithms were applied for each group of data separately. 5.1.2. Group in the interval [0, 1] Using the 5-fold cross validation approach, in each of the five execution times of the regression algorithm 2928 training data records and 732 testing ones were obtained and used. In both cases, the SVR-parameters applied were {σRBF , γ , ε} = {9, 6, 0.02} respectively and η = 2. As can be seen in Table 2, the maximum performance was achieved for the FWSVR and for C = 2, where RMSE ≈ 0.2. This error was actually very small. 5.1.3. Group with output in the interval (1, 33] This group contained only 385 data records. By using the 5-fold cross validation, for each of the five execution times there were 308 training data and 77 testing data. The SVR-parameters were {σRBF , γ , ε} = {12, 5, 0.02} respectively and η = 2. It is clearly shown in Table 3 that the best performance was achieved by using the FWSVR and C = 3, where RMSE ≈ 2.45. This error was very low compared to the output values.

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Table 2 Performance using the Global SVR and the FWSVR approaches with {σ , γ , ε} = {9, 6, 0.02} respectively and η = 2. Application for data records having an output in the interval [0, 1]. Method

Group

Global SVR



Fuzzy weighted SVR

2 3 4 5 6 7 8 9 10

RMSE Training

Testing

0.211 0.201 0.202 0.209 0.208 0.208 0.207 0.209 0.211 0.21

0.212 0.202 0.203 0.209 0.209 0.209 0.21 0.211 0.213 0.213

Table 3 Performance using Global SVR and FWSVR with {σ , γ , ε} respectively and η = 2. Application for the region (1, 33]. Method

Global SVR

Fuzzy weighted SVR

Groups

– 2 3 4 5 6 7 8 9 10

=

{12, 5, 0.02}

Fig. 7. Performance of the ANN versus one of the UBC for 1990 and 1988.

RMSE Training

Testing

3.342 2.516 2.547 2.582 3.022 3.016 2.800 2.962 2.987 2.976

3.341 2.442 2.407 2.648 2.617 2.839 2.800 2.466 2.503 2.500

Fig. 6. Performance of the ANN versus one of the UBC for 1993 and 1995.

5.2. Application of artificial neural networks 5.2.1. Multilayer back propagation ANN Several ANN architectures and optimization algorithms were tried in order to determine the optimal ANN model. Initially, multilayer back propagation (BP) artificial neural networks were employed using Sigmoid or Tangent Hyperbolic (TanH) transfer function and the Extended Delta Bar Delta (ExtDBD) learning rule. The existing 4020 data vectors were divided into two distinct subsets, namely the training and the testing one. The training subset contained 70% of the data vectors and the testing 30% of them. The determination of the two sets was done automatically by running the Generate Training Data C + + program in MATLAB. It is a routine contained in the LIBSVM v2.9

Fig. 8. Modular ANN output versus actual output.

(http://www.csie.ntu.edu.tw/~cjlin/libsvm/) (Chang & Lin, 2009) which is encoded in C++ and offers a Matlab Interface. The target in this case was the construction of a unique model that would work reliably for all of the data vectors (no matter if they use large and small output values). The optimal BP ANN had 5 input processing elements, and 6 neurons in the hidden sublayer and one neuron in the output layer. It used a Sigmoid transfer function and a Tangent Hyperbolic learning rule. Figs. 6 and 7 clearly present the performance of the best BP ANN versus the performance of the UBC model. It is clearly shown that the ANN is more reliable. In order to compare the two approaches, the R2 metric was used. In the case of the UBC R2 = 0.8532 in training and R2 = 0.7221 in testing, whereas for the optimal BP ANN the respective values were 0.892 and 0.783. The overall RMS Error for the ANN was equal to 0.53 in testing which is quite low. 5.2.2. Modular ANN In addition, with the BP ANN modular networks were developed. As is well known, Modular ANN (ModANN) uses a Gating network which decides which of the competing ANN is the optimal one. The best ModANN had five neurons in the input layer, 10 in the hidden and one in the output. Its Gating network had four hidden neurons and three output ones. After 6000 iterations, the best ModANN had R2 = 0.9796 and RMS = 0.0131 in training, whereas R2 = 0.6520 and RMS = 0.0495 in testing. Fig. 8 presents the actual output versus the output produced by the ANN. 6. Conclusions From the SVM point of view, this paper uses the state-of-theart modeling techniques that offer flexible and effective regression approaches, capable of manipulating data sets with high levels of differentiations in the expected output. According to the methodology that was employed, the general conclusions for the case of the SVM were the following: every time that the prediction of the performance is required for a new

L. Iliadis et al. / Neural Networks 24 (2011) 836–841

set of input vectors, it is well known that the value of the actual output is located in the closed interval [0, 1] with a probability of 4018 = 91.09%. Respectively, the value of the actual output might 3660

be in the closed interval (1, 33] with a probability of 4018 = 8.91%. 358 So by using a heuristic approach, the user can use his experience in the input data vectors in order to expect the output in the one of the two intervals of values. In the first case, when the output values are expected to be low, the first SVM Regression model should be used with an expected error RMSE ≈ 0.2. In the opposite case, the second SVM regression model should be used with an error more or less RMSE ≈ 2.45. Both error values are low compared to the range of the values used. For the case of the BP ANN, the R2 value in testing is quite good and shows a good level of convergence. However, though the overall RMSE is quite low, it is still worse than the error of the first SVMR model which is specifically developed for low values. Of course, the data could have been partitioned also in the case of the ANN. However, for the ANN the target was to build an overall reliable model and to measure its ability to generalize for all of the values regardless their magnitude. The best modular ANN has the worse overall performance in testing; however, it still has a good testing and thus generalization capacity. From Fig. 8, it is obvious that its moderate overall performance is due to some few extreme values that it fails to catch. However, for the rest of the cases it works satisfactory. If two ANN had been developed instead (one for low range values and one for wide range ones), the results would have been better. However, the aim in ANN development was a model behaving quite well for a wide range of cases. In general, one can claim that both SVMR and ANN could offer reliable approaches for the case of the water supply of Cyprus, and it is obvious from the metrics used and also from Figs. 7 and 8 that the soft computing approaches overcome the UBC classical model for the specific watershed. Of course, a general conclusion cannot be drawn about which methods lends itself better to the specific problem. However, this research has shown that soft computing models can be trusted and they can be used alternatively by authorities and by environmentalists. Similar models can be constructed for other countries as long as there are available data. Future research would involve the performance of the same job for all of the watersheds of Cyprus. However, this would require more and more reliable data vectors. References Behzad, M., Asghar, K., Eazi, M., & Palhang, M. (2009). Generalization performancee of support vector machines and neural networks in runoff modeling. Expert Systems with Applications: An International Journal, 36(4). Boser, B., Guyon, I., & Vapnik, V. (1992). A training algorithm for optimal margin classifiers. In Fifth Annual Workshop on Computational Learning Theory (pp. 144–152). Pittsburgh: ACM.

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