Evaporation estimation is very essential for planning and development of water resources. The study investigates the ability of new method, dynamic evolving neuralfuzzy inference system (DENFIS), in modeling monthly pan evaporation. Monthly maximum and minimum temperatures, solar radiation, wind speed, and relative humidity data obtained from two stations located in Turkey are used as inputs to the models. The results of DENFIS method were compared with the classical adaptive neuralfuzzy inference system (ANFIS) by using root mean square error (RMSE), mean absolute relative error (MARE), and NashSutcliffe Coefficient (NS) statistics. Cross validation was applied for better comparison of the models. The results indicated that DENFIS models increased the accuracy of ANFIS models to some extent. RMSE, MARE, and NS of the ANFIS model were increased by 11.13, 11.45, and 6.83% for the Antalya station and 20.11, 12.94%, and 8.29% for the Antakya station using DENFIS.
One of the major issues that may negatively affect the agriculture of dry regions such as Iran is the evaporation of water resources. It has been estimated that evaporation is responsible for annual loss of more than 40% of water resources. Such loss of water may severely undermine not only the productivity of agricultural sector but environmental projects as well. The pan evaporation tests are prone to multiple errors originating from pan size and material, water depth in the pan, sun and wind exposure, and animal activity in the vicinity. The factors effectively deciding the degree of evaporation in given area are air and soil temperature, relative humidity, sunshine and wind speed, vapor pressure deficit, and atmospheric pressure. Thus, empirical evaporation prediction formulas [
Evaporation pans are valuable in hydrology since they are simple strong instruments that incorporate the important physical factors, in particular radiation, temperature, humidity, and wind speed, into a single measure of evaporative demand [
Research has shown the artificial intelligence methods such as ANFIS and ANN to be very successful in civil engineering and especially in water resources analysis applications [
In the present study, a new method, dynamic evolving neuralfuzzy inference system (DENFIS), was applied for evaporation modeling. This is the first study that uses DENFIS for solving this problem.
In the current study, monthly maximum and minimum temperatures, solar radiation, wind speed, relative humidity, and pan evaporation data measured in Antalya (latitude 36.42°N, longitude 30.44°E, and altitude 64 m) and Antakya (latitude 36.33°N, longitude 36.30°E, and altitude 100 m) stations located in Mediterranean Region of Turkey were used (Figure
The location of the Antalya and Antakya stations.
Data used in the present study include 203 monthly values from 1983 to 2010 for the Antakya and 362 values from 1967 to 2006 for the Antalya station. Whole data were divided into three equal subsets and each subset was used for testing the applied models. Basic statistical properties of the used data are provided for each data set in Table
The monthly statistical parameters of pan evaporation data sets.
Data set 






Antalya  
1967 Jan–1985 Dec  4.95  2.45  0.66  1.3  11.0 
1986 Jan–1995 Dec  5.18  2.51  0.51  1.7  11.4 
1996 Jan–2006 Nov  5.78  2.81  0.40  1.5  12.4 


Antakya  
1983 Sep–1989 Dec  4.21  2.19  0.09  0.9  8.1 
1990 Jan–2002 Nov  4.39  2.30  0.28  1.1  9.8 
2003 Apr–2010 Sep  5.27  1.75  −0.40  1.6  7.9 
The principles of evolving neural networks and in particular evolving neurofuzzy method were first introduced by Kasabov and Song [
Input and output neurons can be fuzzified by a fuzzy quantization approach. Thus, fuzzy neural networks can be seen as connectionist structures with rules expressed with fuzzy logic [
The notable difference of this approach from other evolving fuzzy systems is the method of making a prediction for a new sample. In this respect, DENFIS follows a modelbased lazy learning approach, where network assesses the position of the input vector in the feature space, and forms, accordingly, a fuzzy inference system for predicting the output through a dynamic process based on the nearest fuzzy rules created during the incremental learning. In other words, the classical lazy learning of this network employs a samplebased approach, where local samples taken from the area closest to the query point are used to construct a small local model on demand [
Use of a fuzzy model with TakagiSugeno architecture: this means converting the model to a neurofuzzy format in a way that produces the same learning problem, which means linear consequent, nonlinear antecedent parameters, rules, or neurons need to be evolved on demand.
Use of a clusteringbased process for evolution of rules or neurons (evolving a new cluster means evolving a new rule).
Local learning of consequent parameters.
DENFIS uses TakagiSugeno type fuzzy inference engine [
A zeroorder TakagiSugeno type fuzzy inference system is defined as a system where resultant functions are crisp fixed parameters; that is,
The conclusion of inference of the output,
Here:
ANFIS is a multilayer feedforward neurofuzzy network capable of combining the linguistic flexibility of fuzzy logic with the numeric capabilities of artificial neural networks (ANNs). Desirability of ANFIS lies in its ability to synergize the merits of ANN and fuzzy logic to map an input space to an output space more efficiently than either approach and achieve more effective forecasting models through its enhanced learning and data classification capabilities. Given the outstanding ability of ANFIS to infer fuzzy rules or expert knowledge from numerical data, this technique has found countless applications in classification, rulebased process control, and pattern recognition, for example, to analyze and predict the wind speed, dynamic load of power systems, and faults in engines. In a way similar to ANN’s mechanism of solving function estimation problems, before making any prediction, ANFIS model needs to undergo a training phase specific to the data at hand and the target application [
From the functional perspective, ANFIS architecture is, on the one hand, an equivalent of fuzzy model as defined by TakagiSugenoKang (TSK model) and, on the other hand, a rough equivalent of Radial Basis Function Networks (RBFNs). But equivalence of RBFN and TSK fuzzy model is subject to the following requirements [
For both models, the aggregation method used to extract the overall outputs must be the same (weighted average or weighted sum).
There must be an equal number of activation functions and fuzzy IFTHEN rules.
For the rule bases consisting of several inputs, each activation function must be equal to a composite input membership function. This can be achieved by several methods, the simplest of which is to incorporate Gaussian membership functions with the same variance into the rule base, while using the algebraic product for the “AND” operation. The product of the Gaussian membership functions will yield a multidimensional Gaussian RBFN.
The activation functions on the output of neurons should have the same functions as their corresponding fuzzy rules.
The DENFIS method was employed for estimating pan evaporation based on the climatic data of maximum and minimum temperatures, solar radiation, wind speed, and relative humidity. The results of the proposed model were compared with those of the classical ANFIS model. The employed models were evaluated by using three commonly applied comparison criteria, namely, root mean square error (RMSE), mean absolute relative error (MARE), and NashSutcliffe Coefficient (NS). The expressions of the applied statistics are
First, whole data set was divided into three equal parts. Two parts were used in training of the applied models while the remaining one part was used for testing. Thus, three applications were obtained for each method. Then, the models were compared with each other according to the mean of the used statistics. Table
The training and test data sets used for each model.
Cross validation  Training data set  Test data set 

Antalya  
M1  1967 Jan–1995 Dec  1996 Jan–2006 Nov 
M2  1986 Jan–2006 Nov  1967 Jan–1985 Dec 
M3  1967 Jan–1985 Dec and 1996 Jan–2006 Nov  1986 Jan–1995 Dec 


Antakya  
M1  1983 Sep–2002 Nov  2003 Apr–2010 Sep 
M2  1990 Jan–2010 Sep  1983 Sep–1989 Dec 
M3  1983 Sep–1989 Dec and 2003 Apr–2010 Sep  1990 Jan–2002 Nov 
The optimal Dthr values obtained for the DENFIS models are 0.02, 0.01, and 0.013 for the M1, M2, and M3 data sets, respectively. For the ANFIS models, different number of membership functions (MFs) were tried and 2 Gaussian MFs for each input provided the best accuracy. 100 iterations were used for each model following the suggestions of [
Comparison of the DENFIS and ANFIS models in modeling pan evaporation—Antalya station.
Statistics  Cross validation  Test data set  Method  

DENFIS  ANFIS  
RMSE (mm)  M1  1996–2006  0.976  0.962 
M2  1967–1985  1.036  1.483  
M3  1986–1995  0.862  0.788  
Mean 





MARE (%)  M1  1996–2006  15.96  14.46 
M2  1967–1985  20.50  30.34  
M3  1986–1995  15.27  13.62  
Mean 





NS  M1  1996–2006  0.878  0.882 
M2  1967–1985  0.820  0.631  
M3  1986–1995  0.881  0.901  
Mean 


The observed and estimated pan evaporation using DENFIS and ANFIS methods for the M1, M2, and M3 data sets—Antalya.
Comparison of the DENFIS and ANFIS in estimating pan evaporation of Antakya station is made in Table
Comparison of the DENFIS and ANFIS models in modeling pan evaporation—Antakya station.
Statistics  Cross validation  Test data set  Method  

DENFIS  ANFIS  
RMSE (mm)  M1  2003–2010  0.659  0.740 
M2  1983–1989  0.528  0.914  
M3  1990–2002  1.054  1.152  
Mean 





MARE (%)  M1  2003–2010  10.76  12.60 
M2  1983–1989  12.15  14.93  
M3  1990–2002  20.29  22.08  
Mean 





NS  M1  2003–2010  0.856  0.818 
M2  1983–1989  0.941  0.823  
M3  1990–2002  0.788  0.747  
Mean 


The observed and estimated pan evaporation using DENFIS and ANFIS methods for the M1, M2, and M3 data sets—Antakya.
The results of the DENFIS and ANFIS models are tested by using oneway analysis of variance (ANOVA) and reported in Table
Analysis of variance (ANOVA) for pan evaporation estimation in the test period.
Method  Antalya  Antakya  


Resultant sig. level 

Resultant sig. level  
DENFIS  
M1  1.574  0.211  0.035  3.920 
M2  1.784  0.183  0.009  0.926 
M3  2.345  0.127  0.046  0.830 


ANFIS  
M1  2.008  0.158  0.006  0.938 
M2  3.992  0.047  0.894  0.346 
M3  2.098  0.149  0.152  0.697 
This study investigated the accuracy of DENFIS as a new method in pan evaporation modeling. Monthly climatic data obtained from two stations from Turkey were used for testing the used models. Results were compared with those of the classic ANFIS models by using cross validation and RMSE, MAE, and NS statistics. According to the obtained results DENFIS model improved the ability of ANFIS by 11.13, 11.45, and 6.83% from the viewpoints of RMSE, MARE, and NS, respectively, for Antalya station, while, in Antakya station, the ANFIS accuracy was improved by 20.11% for RMSE, 12.94% for MARE, and 8.29% for NS. ANOVA test results indicated the superior accuracy of the DENFIS to the ANFIS models. Comparison with previous studies suggested the use of this method (DENFIS) in pan evaporation modeling.
The ability of DENFIS was compared with ANFIS method using two stations’ data. The proposed method can also be compared with other datadriven methods in modeling pan evaporation using more data having different climatic conditions.
The authors declare that they have no conflicts of interest.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2017R1A2B2010120).