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Modelling of evaporation from the reservoir of Yuvacik dam using adaptive neuro-fuzzy inference systems
ARTICLE IN PRESS Engineering Applications of Artificial Intelligence 23 (2010) 961–967
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Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai
Modelling of evaporation from the reservoir of Yuvacik dam using adaptive neuro-fuzzy inference systems Emrah Dogan a,n, Mahnaz Gumrukcuoglu b, Mehmet Sandalci a, Mucahit Opan c a
Sakarya University, Civil Engineering Department, Esentepe Campus, 54187 Sakarya, Turkey Sakarya University, Environmental Engineering Department, Esentepe Campus, 54187 Sakarya, Turkey c Kocaeli University, Civil Engineering Department, Umuttepe Campus, 41380 Sakarya, Turkey b
a r t i c l e in fo
abstract
Article history: Received 4 May 2008 Received in revised form 7 February 2010 Accepted 25 March 2010 Available online 15 April 2010
Adaptive neuro-fuzzy inference system (ANFIS) models are proposed as an alternative approach of evaporation estimation for Yuvacik Dam. This study has three objectives: (1) to develop ANFIS models to estimate daily pan evaporation from measured meteorological data; (2) to compare the ANFIS model to the multiple linear regression (MLR) model; and (3) to evaluate the potential of ANFIS model. Various combinations of daily meteorological data, namely air temperature, relative humidity, solar radiation and wind speed, are used as inputs to the ANFIS so as to evaluate the degree of effect of each of these variables on daily pan evaporation. The results of the ANFIS model are compared with MLR model. Mean square error, average absolute relative error and coefficient of determination statistics are used as comparison criteria for the evaluation of the model performances. The ANFIS technique whose inputs are solar radiation, air temperature, relative humidity and wind speed, gives mean square errors of 0.181 mm, average absolute relative errors of 9.590% mm, and determination coefficient of 0.958 for Yuvacik Dam station, respectively. Based on the comparisons, it was found that the ANFIS technique could be employed successfully in modelling evaporation process from the available climatic data. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Daily pan evaporation Adaptive neuro-fuzzy inference systems Multiple linear regression model Yuvacik Dam station Model performances
1. Introduction Evaporation from dams’ surface is crucial in terms of water resources management and planning. Thus, dam water surface evaporation should be taken into account for the planning of water budget of dams. Evaporation can be either measured with an evaporation pan (pan evaporimeter) or estimated from meteorological data. The Class A pan evaporimeter is one of the most widely used instruments for the measurement of evaporation. It is impractical to place evaporation pans at every point where there is a planned or existing reservoir and irrigation project. It is also highly unlikely to have in inaccessible areas where accurate instruments cannot be established or maintained. A practical means of estimating the amount of pan evaporation where no pans are available is of considerable significance to the hydrologists, agriculturists, and meteorologists. A number of researchers have attempted to estimate the evaporation values from climatic variables (Stephens and Stewart, 1963; Linarce, 1967; Reis and Dias, 1998; Coulomb et al., 2001; Gavin and Agnew, 2004), and most of these methods require data that are not easily available. Simple methods that are reported (e.g., Stephens and Stewart, 1963) try to fit a linear relationship
n
Corresponding author. Tel.: + 90 264 295 5749; fax: + 90 264 346 0359. E-mail address: [email protected] (E. Dogan).
0952-1976/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engappai.2010.03.007
between the variables. However, the process of evaporation is highly, nonlinear in nature, as it is evidenced by many of the estimation procedures. Many researchers have emphasized the need for accurate estimates of evaporation in hydrologic modelling studies (Sudheer et al., 2002). This requirement could be addressed through better models that will address the inherent non-linearities in the process. In the past few decades, artificial neural networks (ANNs) have been extensively used in a wide range of engineering applications. Recently, ANNs have recently gained increasing applications in modelling daily soil evaporation (Han and Felker, 1997), daily evapotranspiration (Kumar et al., 2002; Trajkovic et al., 2003; Sudheer et al., 2003; Trajkovic, 2005, Kisi, 2006a,b), daily pan evaporation (Sudheer et al., 2002; Keskin and Terzi, 2006) and daily actual crop evapotranspiration (Sudheer et al., 2003).Despite a number of advantages, traditional neural network models have several drawbacks including possibility of getting trapped in local minima, and subjectivity in the choice of model architecture (Suykens, 2001). To overcome these drawbacks of traditional neural networks and to increase their reliability, many new training algorithms have been proposed (Bianchini and Gori, 1996; Neocleous and Schizas, 2002). In this paradigm, one of the significant developments is a adaptive neuro-fuzzy inference system (ANFIS). The research presented in this study is motivated by a desire to explore the potential of ANFIS estimation of pan evaporation. The ANFIS model applied for estimating pan evaporation owing to its
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or
ability to provide good generalization performance in capturing nonlinear regression relationships between predictors and the predictand. The performance of the ANFIS model is compared with MLR. Comparison of the results shows that the ANFIS model is superior to MLR model.
O1,i ¼ mBi2 ðyÞ,
This section of this paper provides description of the ANFIS and MLR techniques in detail for evaporation modelling.
mAi ðxÞ ¼
2.1. Adaptive neuro-fuzzy inference system (ANFIS) Jang, (1993) introduced for the first time an architecture and a learning procedure for the fuzzy inference systems (FIS) that uses a neural network learning algorithm for constructing a set of fuzzy ifthen rules with appropriate membership functions (MFs) from the specified input–output pairs. This procedure of developing a FIS using the framework of adaptive neural networks is called an adaptive neuro-fuzzy inference system (ANFIS). There are two methods that ANFIS learning employs for updating membership function parameters: (1) backpropagation for all parameters (a steepest descent method); and (2) a hybrid method consisting of backpropagation for the parameters associated with the input membership and least squares estimation for the parameters associated with the output MFs. As a result, the training error decreases, at least locally, throughout the learning process. Therefore, the more the initial MFs resemble the optimal ones, the easier it will be for the model parameter training to converge. Human expertise about the target system to be modeled may aid in setting up these initial membership function parameters in the FIS structure. ANFIS architecture is shown in Fig. 1. For instance, assume that the FIS has two inputs x and y and one output f. For the first order Sugeno fuzzy model, a typical rule set with two fuzzy if-then rules can be expressed as Rule 1. If x is A1 and y is B1, then ð1Þ
1 2bi xci 1 þ ai
ð5Þ
where {ai, bi, ci} is the parameter set which changes the shape of the MF degree with maximum value equal to 1 and minimum equal to 0.
Layer 2. Every node in this layer is a fixed node labeled &, whose output is the product of all incoming signals O2,i ¼ wi ¼ mAi ðxÞmBi ðyÞ,
i ¼ 1,2
O3,i ¼ wi ¼
wi , w1 þ w2
i ¼ 1,2
ð7Þ
Layer 4. Every node i in this layer is an adaptive node with a node function, O4,i ¼ wi fi ¼ wi ðpi x þqi y þri Þ,
ð8Þ
where wi is the output of layer 3 and {pi, qi, ri}is the parameter set of this node.
i
ð2Þ
where A1, A2 and B1, B2 are the MFs for inputs x and y, respectively, p1, q1, r1 and p2, q2, r2 are the parameters of the output function. The functioning of the ANFIS is described as
Layer 1
Layer 4
Layer 2
Layer 3
x y Layer 5
x A2 B1 y
ð9Þ
If it is assumed that the dependent variable Y is affected by m independent variables X1, X2, y, Xm and a linear equation is selected for the relation among them, the regression equation of Y
ð3Þ
A1
Si wi fi Si wi
2.2. Multiple linear regression (MLR)
Layer 1. Every node in this layer produces membership grades of an input parameter. The node output O1, i is explained by, i ¼ 1,2,
ð6Þ
Layer 3. The ith node of this layer, labeled N, calculates the normalized firing strength as
f ¼ Overall outputt ¼ O5,i ¼ Swi fi ¼
f2 ¼ p2 x þ q2 y þ r2 ,
for
ð4Þ
Layer 5. The single node in this layer is a fixed node labeled S, which computes the overall output as the summation of all incoming signals
Rule 2. If x is A2 and y is B2, then
O1,i ¼ mAi ðxÞ,
i ¼ 3,4,
where x (or y) is the input to the node i; Ai (or Bi 2) is a linguistic fuzzy set associated with this node. O1,i is the MFs grade of a fuzzy set and it specifies the degree to which the given input x (or y) satisfies the quantifier. MFs can be Gaussian, generalized bell shaped, triangular and trapezoidal shaped functions. A generalized bell shaped function is described as
2. Methods
f1 ¼ p1 x þq1 yþ r1 ,
for
Π
w1
Ν
w1
w1 f1 Σ
Π
w2
Ν
w2
w2 f2
B2 x y Fig. 1. ANFIS architecture.
f
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can be written as y ¼ a þb1 x1 þ b2 x2 þ ::: þ bm xm
ð10Þ
y in this equation shows the expected value of the variable Y when the independent variables take the values X1 ¼x1, X2 ¼x2, y, Xm ¼xm. The regression coefficients a, b1, b2, y, bm are evaluated, similar to simple regression, by minimizing the sum of the eyi distances of observation points from the plane expressed by the regression equation (Bayazıt and Oguz, 1998): N X i¼1
e2yi ¼
N X
ðyi ab1 x1i b2 x2i bm xmi Þ2
ð11Þ
i¼1
In this study, the coefficients of the regressions were determined using least square method.
963
ANFIS and MLR models are daily measurements obtained from the Directorate of State Hydraulic Works, Turkey. The daily statistical parameters of each data are given in Table 1. In this table, xmean, Sx, Cv, Csx, xmin, and xmax denote the mean, standard deviation, variation, skewness coefficient, minimum and maximum of the data, respectively. It is clearly seen from Table 1 the mostly varied (Cv ¼0.89 mm) data is E. Its minimum value is 0.0023 while its mean and maximum values are 2.45 and 8.475, respectively. All the data sets have high correlation between E except the wind speed. The highest correlation coefficient with the E (0.904) belongs to the Rs. There is an inverse proportion between RH and E with a negative correlation of 0.706. 3.1. Application of ANFIS model In this study, before the training of the model both input and output variables were normalized within the range 0.1–0.9 as follows:
3. Compilation of data The study area is the Yuvacik Dam Basin which is in the eastern part of Marmara Region of Turkey at about 401320 – 401410 North and 291290 –301080 East and located 12 km south of Kocaeli city. Yuvacik dam was built in 1999. The drainage area of Yuvacik Dam basin is about 257.8 km2. The location of the station is depicted in Fig. 2. Yuvacik Dam Reservoir is used as a drinking and irrigation water source. The water level of Yuvacik Dam is decreased below the level of active volume due to little precipitation in 2006. Meteorological data from Yuvacik Dam station, consisting of 352 daily records (the drought year of 2006) are used to develop the models of daily pan evaporation. The data set of January 01, 2006 to December 31, 2006 (352 data set) were used for modeling daily evaporation. To develop an ANFIS and MLR models for estimating E, the available data set was partitioned into a training set and a test set according to station. About 232 daily data set of the available record was selected for training while the remaining 120 daily data set was used for testing. The daily solar radiation (Rs), air temperature (T), relative humidity (RH), wind speed (U2) data obtained from the Automated GroWeather Meteorological Yuvacik Dam Station which are logged every 2 h are used in the study. Two hourly data are integrated subsequently to obtain daily data, because the pan evaporation (E) values used as output in the
xi ¼ 0:8
ðxxmin Þ þ0:1 ðxmax xmin Þ
ð12Þ
where xi is the normalized value of a certain parameter, x is the measured value for this parameter, xmin and xmax are the minimum and maximum values in the database for this parameter, respectively. 3.2. Selection of input parameters for the ANFIS The selection of the input parameters is a very important aspect for the neural network modeling. In order to use ANFIS structures effectively, input variables in the phenomenon must be selected with a great care. This highly depends on the better understanding of the problem. In a firm ANFIS architecture, in order not to confuse training process key variables must be introduced and unnecessary variables must be avoided. For this purpose, a sensitivity analysis can be used to find out the key parameters. Also sensitivity analysis can be useful to determine the relative importance of the parameters when sufficient data are available. The sensitivity analysis is used to determine the effect of changes and to determine relative importance or effectiveness of a variable on the output. The input variables that
Fig. 2. The location of the Yuvacik Dam Station in Kocaeli, Turkey.
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Table 1 The daily statistical parameters of each data set. Data set
xmean
Sx
Cv(Sx/xmean)
Csx
xmin
xmax
Correlation with E
Rs(W/m2) T(1C) RH(%) U2(m/s) E(mm/day)
57.40 14.50 72.77 1.75 2.45
44.41 7.40 14.08 0.74 2.17
0.77 0.51 0.19 0.42 0.89
0.49 0.10 0.13 0.25 0.70
0.01 0.11 34.4 0.11 0.0023
155.0 29.5 100.0 4.30 8.475
0.904 0.854 0.706 0.518 1.000
model (mm)
T
8 6
E (mm)
y = 0.8136x + 0.3564 R2 = 0.7952
8
4 2
6 4 2 0
0 0
60 day observed
0
120
E (mm)
model (mm)
Rs
6 4 2 0 0
60 day observed
y = 0.8301x + 0.3474 R2 = 0.8467
4 2 0
8
8
6 4 2
y = 0.3207x + 1.5639 R2 = 0.2798
6 4 2 0
0 0
60 day observed
0
120
2 4 6 observed (mm)
8
estimated 8 model (mm)
RH
8 E (mm)
2 4 6 observed (mm)
estimated
model (mm)
E (mm)
6
0
120
U2
8
8
estimated
8 8
2 4 6 observed (mm)
6 4 2
y = 0.5038x + 1.3709 R2 = 0.5028
6 4 2 0
0 0
60 day observed
120
0
2 4 6 observed (mm)
8
estimated
Fig. 3. Comparison of ANFIS and observed E values depending on each input parameter.
do not have a significant effect on the performance of an ANFIS can be excluded from the input variables, resulting in a more compact network. Then, it becomes necessary to work on
methods like sensitivity analysis to make ANFIS work effectively. Evaporation depends on the some independent parameters and those can be given in this form: E¼f(Rs, T, RH, U2).
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3.3. Sensitivity analysis
These criteria can be computed as
It appears that while assessing the performance of any model for its applicability in estimating E, it is not only important to evaluate the average prediction error but also the distribution of prediction errors. The statistical performance evaluation criteria employed so far in this study are global statistics (R2 and MSE) and do not provide any information on the distribution of errors. Therefore, in order to test the robustness of the model developed, it is important to test the model using some other performance evaluation criteria such as average absolute relative error (AARE). The AARE not only gives the performance index in terms of predicting E but also shows the distribution of the prediction errors.
AARE ¼
RE ¼
Rs
AARE (%) MSE R2
Rs +T
13.60 0.677 0.847
Rs + T +RH
10.77 0.402 0.910
Rs + T + RH +U2
ANFIS Structure
(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi)
Model Performance N
trimf trapmf gbellmf gaussmf trimf trapmf gbellmf gaussmf trapmf gbellmf gaussmf
2 2 2 2 3 3 3 3 2 2 2
2 2 2 2 3 3 3 3 1 1 1
2 2 2 2 3 3 3 3 2 2 2
2 2 2 2 3 3 3 3 1 1 1
R2
MSE
0.902 0.885 0.845 0.930 0.0127 0.0707 0.008 0.029 0.956 0.958 0.958
0.470 0.527 0.860 0.340 181 96.27 194 160 0.192 0.185 0.181
Table 4 The regression parameters of the ANFIS. p
q
r
s
t
Rule Rule Rule Rule
0.06162 0.2652 0.3274 0.00426
0.01403 0.113 0.0875 0.07485
0.00506 0.1137 0.009496 0.04329
0.01506 0.05117 0.09007 0.4878
1.402 18.67 2.273 5.014
Eo Es Es
ð15Þ
n X
ðt p tmean Þ2
ð16Þ
n X
ðt p op Þ2
ð17Þ
where, tmean is the mean E, The mean Square error (MSE) is defined as MSE ¼
N 1X ðt p op Þ2 Ni¼1
ð18Þ
Solar radiation Rs is used as the common parameter for the rest of the sensitivity analysis. Performance evaluation of all possible combination of variables such that each and every combination includes E, was also investigated.
4. Results and discussion The four ANFIS models were established using each independent parameter separately. Sensitivity analysis applied for finding the most effective input parameters. Sensitivity analysis determination coefficient (R2) of the parameters involved in the phenomenon is given in Fig. 3 respectively. It is clearly seen from Fig. 3 that the most effective parameter is determined as Rs, solar radiation.
Performance
Rs
Rs + T
Rs + T + RH
Rs + T + RH+ U2
AARE (%) MSE R2
13.610 0.710 0.839
11.770 0.492 0.889
13.97 0.433 0.900
13.97 0.433 0.901
8 8 6 4 2 0
model (mm)
E
ð14Þ
Table 5 The performances of the MLR in the test period.
Rules 1 2 3 4
tp op 100 tp
i¼1
Table 3 Determination of the efficient ANFIS model (MF: membership function type; N: number of membership functions).
MF
ð13Þ
p¼1
Es ¼
Note: Best results are indicated by ‘ ’.
Model
Eo ¼
9.590 0.181n 0.958n
n
in which,
where
n
9.890 0.236 0.946
n 1 X 9RE9 Np¼1
where RE is the relative error in forecast expressed as percentage, tp is the observed E for the pth pattern; and op is the computed E for the pth pattern which is produced by ANFIS; and N is the total number of the testing patterns. Clearly the smaller the value of AARE is, the better the performance. The performance control of the ANFIS output was evaluated by estimating the determination coefficient (R2) which is defined as R2 ¼
Table 2 Performance evaluation of the effective parameters for sensitivity analysis. Performance
965
0
60
120
y = 0.9499x + 0.1212 R2 = 0.958
6 4 2 0 0
day observed
estimated
Fig. 4. Comparison of ANFIS E vs. observed E.
2 4 6 observed (mm)
8
ARTICLE IN PRESS E. Dogan et al. / Engineering Applications of Artificial Intelligence 23 (2010) 961–967
8
8
6
6
model (mm)
E (mm)
966
4 2 0 0
60 day observed
120
y = 0.9172x + 0.1859 R2 = 0.9008
4 2 0 -2 0
2 4 6 observed (mm)
8
estimated
Fig. 5. Comparison of MLR E vs. observed E.
The findings obtained from the sensitivity analysis are listed in Table 2. Based on the findings, as depicted in Table 2 the ANFIS model has four inputs (Rs,T, RH, U2) gives the best estimation. The final architectures of the ANFIS models that were found after many trials is given in Table 3. In this table, the trimf, trapmf, gbellmf and gaussmf denote the triangle, trapezoid, gumbell and gaussian MFs, respectively. The MSE and R2 values of each ANFIS model in test period are also given in Table 3. It is clearly seen from Table 3 the ANFIS model (xi) has 2, 1, 2 and 1 gaussian MFs gives the best result. The error tolerance in training is used as a stopping criterion. The training stops after the training error exceeds this tolerance. In this study, membership function parameters were iteratively learned using hybrid method. The maximum number of training epochs was set to 2. The rules of the ANFIS model (xi) which showed the highest accuracy are provided in Table 4. After training the ANFIS model, test performance was checked. The performance of ANFIS for prediction of E is demonstrated in Fig. 4. Fig. 4 also shows an analysis between the network outputs (estimations) and the corresponding targets (observed data) for the test data set. It is obvious that the predicted values from the trained ANFIS outputs catch the targets well except a few very small E values. These are 6 very small values. It can be seen clearly From Fig. 4 and their range change from 0.0023 to 0.0127 mm (from day 16 to day 21). These 6 daily sequence data were collected during winter (from 02.21.2006 to 02.25.2006). There is no very low E observation seen either during winter or other seasons of year 2006 except these 6 days. Furthermore, the other meteorological data are not drastically changed during these 6 days. The reason of these low observations could be inaccurate measurements. It is known that the accuracy of measurements from automated pan evaporation devices is highly dependent upon the amount and quality of maintenance they receive. Pressure transducers for measuring water height in evaporation pans are prone to malfunction due to electrical problems. Automated pan evaporation measurement devices which rely on floats can become stuck, significantly decreasing the measured amount of evaporation (Bruton et al., 1998). The performance criteria for the test results of the MLR model is given in Table 5. As can be seen from Table 5, the model has the highest MSE and AARE and the lowest R2 values when Rs is only used as input. However, it is clearly seen that the from Table 5 adding the relative humidity and wind velocity to the input combinations (Rs + T) significantly increases the models’ performances. The MLR estimates for combination Rs +T+ RH+ U2 are demonstrated in Fig. 5. The MLR has better estimates. However, it gives some negative values for the lowest value of E. It is a drawback for the MLR (Fig. 5).
5. Conclusion The present study demonstrates the capabilities of adaptive neuro-fuzzy inference system technique (ANFIS) for evaporation
modeling, however the choice of ANFIS architecture and input parameters are crucial for obtaining good estimate accuracy. Thus, sensitivity analysis had been conducted to determine the degree of effectiveness of the variables by using various performance statistics. From the results obtained, an ANFIS model appears to be a useful tool for prediction of the E. The results showed that the solar radiation (Rs) was found to be more effective on E estimation than the other three parameters. Remaining parameters were used one by one in estimating E. After the application of sensitivity analysis, other effective parameters were determined as air temperature T, relative humidity RH and wind speed U2, respectively. The models whose inputs are the wind speed, solar radiation, relative humidity and air temperature have the best performance criteria among the input combinations tried in the study. This indicates that all these variables are needed for better evaporation modeling. The MLR has also better estimates. However, it is giving some negative values for the lowest value of E estimation. It is a drawback for the MLR. Based on the comparison results, the ANFIS technique was found to be superior to the MLR technique.
Acknowledgment The authors wish to thank Thames Water for making meteorological data available.
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Contents lists available at ScienceDirect
Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai
Modelling of evaporation from the reservoir of Yuvacik dam using adaptive neuro-fuzzy inference systems Emrah Dogan a,n, Mahnaz Gumrukcuoglu b, Mehmet Sandalci a, Mucahit Opan c a
Sakarya University, Civil Engineering Department, Esentepe Campus, 54187 Sakarya, Turkey Sakarya University, Environmental Engineering Department, Esentepe Campus, 54187 Sakarya, Turkey c Kocaeli University, Civil Engineering Department, Umuttepe Campus, 41380 Sakarya, Turkey b
a r t i c l e in fo
abstract
Article history: Received 4 May 2008 Received in revised form 7 February 2010 Accepted 25 March 2010 Available online 15 April 2010
Adaptive neuro-fuzzy inference system (ANFIS) models are proposed as an alternative approach of evaporation estimation for Yuvacik Dam. This study has three objectives: (1) to develop ANFIS models to estimate daily pan evaporation from measured meteorological data; (2) to compare the ANFIS model to the multiple linear regression (MLR) model; and (3) to evaluate the potential of ANFIS model. Various combinations of daily meteorological data, namely air temperature, relative humidity, solar radiation and wind speed, are used as inputs to the ANFIS so as to evaluate the degree of effect of each of these variables on daily pan evaporation. The results of the ANFIS model are compared with MLR model. Mean square error, average absolute relative error and coefficient of determination statistics are used as comparison criteria for the evaluation of the model performances. The ANFIS technique whose inputs are solar radiation, air temperature, relative humidity and wind speed, gives mean square errors of 0.181 mm, average absolute relative errors of 9.590% mm, and determination coefficient of 0.958 for Yuvacik Dam station, respectively. Based on the comparisons, it was found that the ANFIS technique could be employed successfully in modelling evaporation process from the available climatic data. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Daily pan evaporation Adaptive neuro-fuzzy inference systems Multiple linear regression model Yuvacik Dam station Model performances
1. Introduction Evaporation from dams’ surface is crucial in terms of water resources management and planning. Thus, dam water surface evaporation should be taken into account for the planning of water budget of dams. Evaporation can be either measured with an evaporation pan (pan evaporimeter) or estimated from meteorological data. The Class A pan evaporimeter is one of the most widely used instruments for the measurement of evaporation. It is impractical to place evaporation pans at every point where there is a planned or existing reservoir and irrigation project. It is also highly unlikely to have in inaccessible areas where accurate instruments cannot be established or maintained. A practical means of estimating the amount of pan evaporation where no pans are available is of considerable significance to the hydrologists, agriculturists, and meteorologists. A number of researchers have attempted to estimate the evaporation values from climatic variables (Stephens and Stewart, 1963; Linarce, 1967; Reis and Dias, 1998; Coulomb et al., 2001; Gavin and Agnew, 2004), and most of these methods require data that are not easily available. Simple methods that are reported (e.g., Stephens and Stewart, 1963) try to fit a linear relationship
n
Corresponding author. Tel.: + 90 264 295 5749; fax: + 90 264 346 0359. E-mail address: [email protected] (E. Dogan).
0952-1976/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engappai.2010.03.007
between the variables. However, the process of evaporation is highly, nonlinear in nature, as it is evidenced by many of the estimation procedures. Many researchers have emphasized the need for accurate estimates of evaporation in hydrologic modelling studies (Sudheer et al., 2002). This requirement could be addressed through better models that will address the inherent non-linearities in the process. In the past few decades, artificial neural networks (ANNs) have been extensively used in a wide range of engineering applications. Recently, ANNs have recently gained increasing applications in modelling daily soil evaporation (Han and Felker, 1997), daily evapotranspiration (Kumar et al., 2002; Trajkovic et al., 2003; Sudheer et al., 2003; Trajkovic, 2005, Kisi, 2006a,b), daily pan evaporation (Sudheer et al., 2002; Keskin and Terzi, 2006) and daily actual crop evapotranspiration (Sudheer et al., 2003).Despite a number of advantages, traditional neural network models have several drawbacks including possibility of getting trapped in local minima, and subjectivity in the choice of model architecture (Suykens, 2001). To overcome these drawbacks of traditional neural networks and to increase their reliability, many new training algorithms have been proposed (Bianchini and Gori, 1996; Neocleous and Schizas, 2002). In this paradigm, one of the significant developments is a adaptive neuro-fuzzy inference system (ANFIS). The research presented in this study is motivated by a desire to explore the potential of ANFIS estimation of pan evaporation. The ANFIS model applied for estimating pan evaporation owing to its
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or
ability to provide good generalization performance in capturing nonlinear regression relationships between predictors and the predictand. The performance of the ANFIS model is compared with MLR. Comparison of the results shows that the ANFIS model is superior to MLR model.
O1,i ¼ mBi2 ðyÞ,
This section of this paper provides description of the ANFIS and MLR techniques in detail for evaporation modelling.
mAi ðxÞ ¼
2.1. Adaptive neuro-fuzzy inference system (ANFIS) Jang, (1993) introduced for the first time an architecture and a learning procedure for the fuzzy inference systems (FIS) that uses a neural network learning algorithm for constructing a set of fuzzy ifthen rules with appropriate membership functions (MFs) from the specified input–output pairs. This procedure of developing a FIS using the framework of adaptive neural networks is called an adaptive neuro-fuzzy inference system (ANFIS). There are two methods that ANFIS learning employs for updating membership function parameters: (1) backpropagation for all parameters (a steepest descent method); and (2) a hybrid method consisting of backpropagation for the parameters associated with the input membership and least squares estimation for the parameters associated with the output MFs. As a result, the training error decreases, at least locally, throughout the learning process. Therefore, the more the initial MFs resemble the optimal ones, the easier it will be for the model parameter training to converge. Human expertise about the target system to be modeled may aid in setting up these initial membership function parameters in the FIS structure. ANFIS architecture is shown in Fig. 1. For instance, assume that the FIS has two inputs x and y and one output f. For the first order Sugeno fuzzy model, a typical rule set with two fuzzy if-then rules can be expressed as Rule 1. If x is A1 and y is B1, then ð1Þ
1 2bi xci 1 þ ai
ð5Þ
where {ai, bi, ci} is the parameter set which changes the shape of the MF degree with maximum value equal to 1 and minimum equal to 0.
Layer 2. Every node in this layer is a fixed node labeled &, whose output is the product of all incoming signals O2,i ¼ wi ¼ mAi ðxÞmBi ðyÞ,
i ¼ 1,2
O3,i ¼ wi ¼
wi , w1 þ w2
i ¼ 1,2
ð7Þ
Layer 4. Every node i in this layer is an adaptive node with a node function, O4,i ¼ wi fi ¼ wi ðpi x þqi y þri Þ,
ð8Þ
where wi is the output of layer 3 and {pi, qi, ri}is the parameter set of this node.
i
ð2Þ
where A1, A2 and B1, B2 are the MFs for inputs x and y, respectively, p1, q1, r1 and p2, q2, r2 are the parameters of the output function. The functioning of the ANFIS is described as
Layer 1
Layer 4
Layer 2
Layer 3
x y Layer 5
x A2 B1 y
ð9Þ
If it is assumed that the dependent variable Y is affected by m independent variables X1, X2, y, Xm and a linear equation is selected for the relation among them, the regression equation of Y
ð3Þ
A1
Si wi fi Si wi
2.2. Multiple linear regression (MLR)
Layer 1. Every node in this layer produces membership grades of an input parameter. The node output O1, i is explained by, i ¼ 1,2,
ð6Þ
Layer 3. The ith node of this layer, labeled N, calculates the normalized firing strength as
f ¼ Overall outputt ¼ O5,i ¼ Swi fi ¼
f2 ¼ p2 x þ q2 y þ r2 ,
for
ð4Þ
Layer 5. The single node in this layer is a fixed node labeled S, which computes the overall output as the summation of all incoming signals
Rule 2. If x is A2 and y is B2, then
O1,i ¼ mAi ðxÞ,
i ¼ 3,4,
where x (or y) is the input to the node i; Ai (or Bi 2) is a linguistic fuzzy set associated with this node. O1,i is the MFs grade of a fuzzy set and it specifies the degree to which the given input x (or y) satisfies the quantifier. MFs can be Gaussian, generalized bell shaped, triangular and trapezoidal shaped functions. A generalized bell shaped function is described as
2. Methods
f1 ¼ p1 x þq1 yþ r1 ,
for
Π
w1
Ν
w1
w1 f1 Σ
Π
w2
Ν
w2
w2 f2
B2 x y Fig. 1. ANFIS architecture.
f
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can be written as y ¼ a þb1 x1 þ b2 x2 þ ::: þ bm xm
ð10Þ
y in this equation shows the expected value of the variable Y when the independent variables take the values X1 ¼x1, X2 ¼x2, y, Xm ¼xm. The regression coefficients a, b1, b2, y, bm are evaluated, similar to simple regression, by minimizing the sum of the eyi distances of observation points from the plane expressed by the regression equation (Bayazıt and Oguz, 1998): N X i¼1
e2yi ¼
N X
ðyi ab1 x1i b2 x2i bm xmi Þ2
ð11Þ
i¼1
In this study, the coefficients of the regressions were determined using least square method.
963
ANFIS and MLR models are daily measurements obtained from the Directorate of State Hydraulic Works, Turkey. The daily statistical parameters of each data are given in Table 1. In this table, xmean, Sx, Cv, Csx, xmin, and xmax denote the mean, standard deviation, variation, skewness coefficient, minimum and maximum of the data, respectively. It is clearly seen from Table 1 the mostly varied (Cv ¼0.89 mm) data is E. Its minimum value is 0.0023 while its mean and maximum values are 2.45 and 8.475, respectively. All the data sets have high correlation between E except the wind speed. The highest correlation coefficient with the E (0.904) belongs to the Rs. There is an inverse proportion between RH and E with a negative correlation of 0.706. 3.1. Application of ANFIS model In this study, before the training of the model both input and output variables were normalized within the range 0.1–0.9 as follows:
3. Compilation of data The study area is the Yuvacik Dam Basin which is in the eastern part of Marmara Region of Turkey at about 401320 – 401410 North and 291290 –301080 East and located 12 km south of Kocaeli city. Yuvacik dam was built in 1999. The drainage area of Yuvacik Dam basin is about 257.8 km2. The location of the station is depicted in Fig. 2. Yuvacik Dam Reservoir is used as a drinking and irrigation water source. The water level of Yuvacik Dam is decreased below the level of active volume due to little precipitation in 2006. Meteorological data from Yuvacik Dam station, consisting of 352 daily records (the drought year of 2006) are used to develop the models of daily pan evaporation. The data set of January 01, 2006 to December 31, 2006 (352 data set) were used for modeling daily evaporation. To develop an ANFIS and MLR models for estimating E, the available data set was partitioned into a training set and a test set according to station. About 232 daily data set of the available record was selected for training while the remaining 120 daily data set was used for testing. The daily solar radiation (Rs), air temperature (T), relative humidity (RH), wind speed (U2) data obtained from the Automated GroWeather Meteorological Yuvacik Dam Station which are logged every 2 h are used in the study. Two hourly data are integrated subsequently to obtain daily data, because the pan evaporation (E) values used as output in the
xi ¼ 0:8
ðxxmin Þ þ0:1 ðxmax xmin Þ
ð12Þ
where xi is the normalized value of a certain parameter, x is the measured value for this parameter, xmin and xmax are the minimum and maximum values in the database for this parameter, respectively. 3.2. Selection of input parameters for the ANFIS The selection of the input parameters is a very important aspect for the neural network modeling. In order to use ANFIS structures effectively, input variables in the phenomenon must be selected with a great care. This highly depends on the better understanding of the problem. In a firm ANFIS architecture, in order not to confuse training process key variables must be introduced and unnecessary variables must be avoided. For this purpose, a sensitivity analysis can be used to find out the key parameters. Also sensitivity analysis can be useful to determine the relative importance of the parameters when sufficient data are available. The sensitivity analysis is used to determine the effect of changes and to determine relative importance or effectiveness of a variable on the output. The input variables that
Fig. 2. The location of the Yuvacik Dam Station in Kocaeli, Turkey.
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Table 1 The daily statistical parameters of each data set. Data set
xmean
Sx
Cv(Sx/xmean)
Csx
xmin
xmax
Correlation with E
Rs(W/m2) T(1C) RH(%) U2(m/s) E(mm/day)
57.40 14.50 72.77 1.75 2.45
44.41 7.40 14.08 0.74 2.17
0.77 0.51 0.19 0.42 0.89
0.49 0.10 0.13 0.25 0.70
0.01 0.11 34.4 0.11 0.0023
155.0 29.5 100.0 4.30 8.475
0.904 0.854 0.706 0.518 1.000
model (mm)
T
8 6
E (mm)
y = 0.8136x + 0.3564 R2 = 0.7952
8
4 2
6 4 2 0
0 0
60 day observed
0
120
E (mm)
model (mm)
Rs
6 4 2 0 0
60 day observed
y = 0.8301x + 0.3474 R2 = 0.8467
4 2 0
8
8
6 4 2
y = 0.3207x + 1.5639 R2 = 0.2798
6 4 2 0
0 0
60 day observed
0
120
2 4 6 observed (mm)
8
estimated 8 model (mm)
RH
8 E (mm)
2 4 6 observed (mm)
estimated
model (mm)
E (mm)
6
0
120
U2
8
8
estimated
8 8
2 4 6 observed (mm)
6 4 2
y = 0.5038x + 1.3709 R2 = 0.5028
6 4 2 0
0 0
60 day observed
120
0
2 4 6 observed (mm)
8
estimated
Fig. 3. Comparison of ANFIS and observed E values depending on each input parameter.
do not have a significant effect on the performance of an ANFIS can be excluded from the input variables, resulting in a more compact network. Then, it becomes necessary to work on
methods like sensitivity analysis to make ANFIS work effectively. Evaporation depends on the some independent parameters and those can be given in this form: E¼f(Rs, T, RH, U2).
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3.3. Sensitivity analysis
These criteria can be computed as
It appears that while assessing the performance of any model for its applicability in estimating E, it is not only important to evaluate the average prediction error but also the distribution of prediction errors. The statistical performance evaluation criteria employed so far in this study are global statistics (R2 and MSE) and do not provide any information on the distribution of errors. Therefore, in order to test the robustness of the model developed, it is important to test the model using some other performance evaluation criteria such as average absolute relative error (AARE). The AARE not only gives the performance index in terms of predicting E but also shows the distribution of the prediction errors.
AARE ¼
RE ¼
Rs
AARE (%) MSE R2
Rs +T
13.60 0.677 0.847
Rs + T +RH
10.77 0.402 0.910
Rs + T + RH +U2
ANFIS Structure
(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi)
Model Performance N
trimf trapmf gbellmf gaussmf trimf trapmf gbellmf gaussmf trapmf gbellmf gaussmf
2 2 2 2 3 3 3 3 2 2 2
2 2 2 2 3 3 3 3 1 1 1
2 2 2 2 3 3 3 3 2 2 2
2 2 2 2 3 3 3 3 1 1 1
R2
MSE
0.902 0.885 0.845 0.930 0.0127 0.0707 0.008 0.029 0.956 0.958 0.958
0.470 0.527 0.860 0.340 181 96.27 194 160 0.192 0.185 0.181
Table 4 The regression parameters of the ANFIS. p
q
r
s
t
Rule Rule Rule Rule
0.06162 0.2652 0.3274 0.00426
0.01403 0.113 0.0875 0.07485
0.00506 0.1137 0.009496 0.04329
0.01506 0.05117 0.09007 0.4878
1.402 18.67 2.273 5.014
Eo Es Es
ð15Þ
n X
ðt p tmean Þ2
ð16Þ
n X
ðt p op Þ2
ð17Þ
where, tmean is the mean E, The mean Square error (MSE) is defined as MSE ¼
N 1X ðt p op Þ2 Ni¼1
ð18Þ
Solar radiation Rs is used as the common parameter for the rest of the sensitivity analysis. Performance evaluation of all possible combination of variables such that each and every combination includes E, was also investigated.
4. Results and discussion The four ANFIS models were established using each independent parameter separately. Sensitivity analysis applied for finding the most effective input parameters. Sensitivity analysis determination coefficient (R2) of the parameters involved in the phenomenon is given in Fig. 3 respectively. It is clearly seen from Fig. 3 that the most effective parameter is determined as Rs, solar radiation.
Performance
Rs
Rs + T
Rs + T + RH
Rs + T + RH+ U2
AARE (%) MSE R2
13.610 0.710 0.839
11.770 0.492 0.889
13.97 0.433 0.900
13.97 0.433 0.901
8 8 6 4 2 0
model (mm)
E
ð14Þ
Table 5 The performances of the MLR in the test period.
Rules 1 2 3 4
tp op 100 tp
i¼1
Table 3 Determination of the efficient ANFIS model (MF: membership function type; N: number of membership functions).
MF
ð13Þ
p¼1
Es ¼
Note: Best results are indicated by ‘ ’.
Model
Eo ¼
9.590 0.181n 0.958n
n
in which,
where
n
9.890 0.236 0.946
n 1 X 9RE9 Np¼1
where RE is the relative error in forecast expressed as percentage, tp is the observed E for the pth pattern; and op is the computed E for the pth pattern which is produced by ANFIS; and N is the total number of the testing patterns. Clearly the smaller the value of AARE is, the better the performance. The performance control of the ANFIS output was evaluated by estimating the determination coefficient (R2) which is defined as R2 ¼
Table 2 Performance evaluation of the effective parameters for sensitivity analysis. Performance
965
0
60
120
y = 0.9499x + 0.1212 R2 = 0.958
6 4 2 0 0
day observed
estimated
Fig. 4. Comparison of ANFIS E vs. observed E.
2 4 6 observed (mm)
8
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8
8
6
6
model (mm)
E (mm)
966
4 2 0 0
60 day observed
120
y = 0.9172x + 0.1859 R2 = 0.9008
4 2 0 -2 0
2 4 6 observed (mm)
8
estimated
Fig. 5. Comparison of MLR E vs. observed E.
The findings obtained from the sensitivity analysis are listed in Table 2. Based on the findings, as depicted in Table 2 the ANFIS model has four inputs (Rs,T, RH, U2) gives the best estimation. The final architectures of the ANFIS models that were found after many trials is given in Table 3. In this table, the trimf, trapmf, gbellmf and gaussmf denote the triangle, trapezoid, gumbell and gaussian MFs, respectively. The MSE and R2 values of each ANFIS model in test period are also given in Table 3. It is clearly seen from Table 3 the ANFIS model (xi) has 2, 1, 2 and 1 gaussian MFs gives the best result. The error tolerance in training is used as a stopping criterion. The training stops after the training error exceeds this tolerance. In this study, membership function parameters were iteratively learned using hybrid method. The maximum number of training epochs was set to 2. The rules of the ANFIS model (xi) which showed the highest accuracy are provided in Table 4. After training the ANFIS model, test performance was checked. The performance of ANFIS for prediction of E is demonstrated in Fig. 4. Fig. 4 also shows an analysis between the network outputs (estimations) and the corresponding targets (observed data) for the test data set. It is obvious that the predicted values from the trained ANFIS outputs catch the targets well except a few very small E values. These are 6 very small values. It can be seen clearly From Fig. 4 and their range change from 0.0023 to 0.0127 mm (from day 16 to day 21). These 6 daily sequence data were collected during winter (from 02.21.2006 to 02.25.2006). There is no very low E observation seen either during winter or other seasons of year 2006 except these 6 days. Furthermore, the other meteorological data are not drastically changed during these 6 days. The reason of these low observations could be inaccurate measurements. It is known that the accuracy of measurements from automated pan evaporation devices is highly dependent upon the amount and quality of maintenance they receive. Pressure transducers for measuring water height in evaporation pans are prone to malfunction due to electrical problems. Automated pan evaporation measurement devices which rely on floats can become stuck, significantly decreasing the measured amount of evaporation (Bruton et al., 1998). The performance criteria for the test results of the MLR model is given in Table 5. As can be seen from Table 5, the model has the highest MSE and AARE and the lowest R2 values when Rs is only used as input. However, it is clearly seen that the from Table 5 adding the relative humidity and wind velocity to the input combinations (Rs + T) significantly increases the models’ performances. The MLR estimates for combination Rs +T+ RH+ U2 are demonstrated in Fig. 5. The MLR has better estimates. However, it gives some negative values for the lowest value of E. It is a drawback for the MLR (Fig. 5).
5. Conclusion The present study demonstrates the capabilities of adaptive neuro-fuzzy inference system technique (ANFIS) for evaporation
modeling, however the choice of ANFIS architecture and input parameters are crucial for obtaining good estimate accuracy. Thus, sensitivity analysis had been conducted to determine the degree of effectiveness of the variables by using various performance statistics. From the results obtained, an ANFIS model appears to be a useful tool for prediction of the E. The results showed that the solar radiation (Rs) was found to be more effective on E estimation than the other three parameters. Remaining parameters were used one by one in estimating E. After the application of sensitivity analysis, other effective parameters were determined as air temperature T, relative humidity RH and wind speed U2, respectively. The models whose inputs are the wind speed, solar radiation, relative humidity and air temperature have the best performance criteria among the input combinations tried in the study. This indicates that all these variables are needed for better evaporation modeling. The MLR has also better estimates. However, it is giving some negative values for the lowest value of E estimation. It is a drawback for the MLR. Based on the comparison results, the ANFIS technique was found to be superior to the MLR technique.
Acknowledgment The authors wish to thank Thames Water for making meteorological data available.
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