Differential evolution based radial basis function neural network model for reference evapotranspiration estimation

The present study is an effort to examine the capability of a differential evolution based radial basis function neural network (RBFDE) to model weekly reference evapotranspiration (ET0) as a function of climatic parameters in different agro-climatic zones (ACZs) of a moist sub-humid region in East-Central India. The ET0 computed using the empirical equation of Penman–Monteith suggested by the Food and Agricultural Organization (FAO56-PM) is considered as a target variable for investigation. The performance of the proposed RBFDE model is compared with particle swarm optimization based radial basis function (RBFPSO), radial basis function neural network (RBFNN), multilayer artificial neural network (MLANN) models and conventional empirical equations of Hargreaves, Turc, Open-Pan, and Blaney-Criddle. Weekly ET0 estimates that are obtained using RBFDE, RBFPSO, and RBFNN and MLANN are observed to be more consistent than equivalent empirical methods. For a critical analysis of simulation results, mean absolute percentage error (MAPE), root means square error (RMSE), determination coefficient (R2) and Nash–Sutcliffe efficiency factor (NSE) is computed. Low MAPE and RMSE values along with higher R2 and NSE close to 1, obtained with soft computing models exhibit that, soft computing models produce better estimates of ET0 than empirical methods. Among the soft computing models, RBFDE provides improved results as compared to RBFPSO, RBFNN, and MLANN models. This method can be extended for ET0 estimation in other ACZs.


Introduction
In response to atmospheric demand, soil surface evaporation and transpiration from plant occurs simultaneously in a cropping field and is termed as evapotranspiration (ET) in a combined manner [1]. Approximately two-thirds of the total precipitation is consumed by the atmosphere in the form of ET [2]. Therefore, ET is considered one of the most important water balance components for the determination of crop water requirement, length of the crop growing season, and associated agro-climatic studies. Hence, accurate measurement or estimation of ET is essential for the planning and effective implementation of irrigation and water management practices for practical applications. Accurate measurement of ET by volumetric and gravimetric lysimeter is practically very difficult because various factors affect the ET process, which includes climatic parameters, crop characteristics, soil properties, and management practices. Therefore, consumptive use of water from a uniformly distributed grass reference crop under nonlimiting conditions is estimated for practical purposes and termed as ET 0 [3]. In general, ET 0 is computed employing empirical equations as climatic parameters being the only factor affecting the ET process.
wavelet regression (WR) and Hargreaves (HG) methods for the studied locations in different agro-ecological regions of India. Sanikhani et al. [25] have applied several artificial intelligence models including multi-layer perceptron (MLP), generalized regression neural network (GRNN), integrated ANFIS systems with grid partitioning (ANFIS-GP) and subtractive clustering (AFNIS-SC), radial basis neural network (RBNN) and GEP for modeling ET 0 in a cross-station scenario for different locations in Turkey and demonstrated that AI-based models performed better than the empirical equation of Hargreaves-Samani (HS) and its calibrated version (CHS).
It is also observed from the literature review that researchers have successfully implemented various types of hybrid soft computing models combining conventional neural networks along with evolutionary computing algorithms for estimation of ET 0 . Application of nature-inspired algorithms such as genetic algorithm (GA), particle swarm optimization (PSO), artificial bee colony (ABC), etc., in combination with conventional neural networks like ANN and RBNN are investigated in some research publications for ET 0 estimation [26][27][28][29][30]. A study conducted by Feng et al. [31] for estimating FAO56-PM ET 0 in a humid region of Southwest China reveals that ELM and ANN optimized by genetic algorithm (GANN) has resulted in better ET 0 estimates than WNN and empirical approaches of Hargreaves, Makkink, Priestley-Taylor and Ritchie models. Gocić et al. [32] have analyzed the potential of genetic programming (GP), support-vector machine-firefly algorithm (SVM-FFA), ANN, and SVM-Wavelet soft computing approaches and found SVM-Wavelet resulted in improved FAO56-PM ET 0 estimates in Serbia. Mehdizadeh et al. [33] have evaluated the performance of gene expression programming (GEP) and MARS along with two SVM based hybrid models, SVM-Polynomial and SVM-RBF for estimation of monthly mean ET 0 and reported SVM-RBF and MARS outperformed GEP and SVM-Poly and also performed better than 16 other empirical equations considered for comparison. However, Mattar and Alazba [34] have confirmed that the GEP model performed better than the conventional multilinear regression (MLR) approach in Egypt. Most of the soft computing models discussed above are developed under a given scenario in terms of study location, the combination of available input climatic parameters, time scale and duration of climatic data, model structure, learning parameters, and an optimization algorithm, etc. Therefore, practically it becomes very difficult to employ these models in a new location without proper calibration and validation of the model parameters.
To examine the potential of an evolutionary optimized soft computing technique, RBFNN in combination with the differential evolution algorithm (RBFDE) is introduced here for the estimation of ET 0 under three different ACZs in the Chhattisgarh region of East-Central India. Differential evolution (DE) is considered because it is a simple algorithm in comparison to GA which requires intensive calculations. Due to its simplicity, DE is used in various applications [35][36][37]. Technical analysis of DE parameters, hybridization of DE with other soft computing techniques, and its practical applications have been discussed by Das et al. [38]. Different variants over state-of-the-art DE have also been presented in the literature. Among these, Hui and Suguntham [39] suggested ensemble and arithmetic recombinationbased speciation DE for multimodal optimization of common benchmark problems. Ramdas et al. [40] developed a reconstructed mutation strategy for DE and applied the same with multilevel image thresholding for improved weather radar image segmentation [41]. A DE variant with multi-donor mutation strategy and annealing-base local search has been developed by Ghosh et al. [42] for optimization of Lennard-Jones potential function-based molecular clustering. The effect of DE-based constraint handling techniques has been evaluated by Biswas et al. [43] for the optimization of power flow systems. One of the authors of this investigation has also been engaged in DE based training of adaptive autoregressive moving average (ARMA) model for exchange rate forecasting [44] and development of a hybrid system using functional link artificial neural network (FLANN) and DE for Odia handwritten numeral recognition [45]. The proposed evolutionary optimized hybrid structure of RBFDE is developed and used for the first time to model FAO56-PM ET 0 , and therefore it may be considered as a novel scientific approach for such application. Conventional soft computing techniques like MLANN, RBFNN along with empirical methods of Hargreaves, Turc, Open Pan, and Blaney-Criddle are considered for comparison purposes. Results obtained with RBFDE is also compared with RBFPSO under similar condition. This paper is organized into different sections. Section 1 introduces the problem formulation, literature reviews, and motivation behind the investigation. The detailed description of the data sets, soft computing techniques, and empirical methods are described in the Materials and methods of Sect. 2. Simulation results and comparative performance evaluation of different models are outlined in the results and discussion of Sect. 3. The salient findings of the study are summarized in the conclusion section.

Study area and dataset
This investigation is carried out to model weekly ET 0 using soft computing techniques. Long term weekly meteorological data (2001 to 2019) of maximum temperature (T max ), minimum temperature (T min ), bright sunshine hours (BSS), wind speed (WS), morning relative humidity during (RH 1 ), afternoon relative humidity (RH 2 ) and weekly cumulative pan evaporation (EP) are collected from Raipur, Jagdalpur and Ambikapur stations located in three distinct ACZs of Chhattisgarh region in central India (Fig. 1). The climate of Chhattisgarh is moist sub-humid in general with an average annual rainfall of 1200-1400 mm and annual ET 0 losses between 1400 and 1600 mm in different ACZs. Data sets are collected from the India Meteorological Department (IMD) (https ://mausa m.imd.gov.in/) certified observatories located in these stations. These surface meteorological observatories follow the World Meteorological Organization (WMO) guidelines for data collection [46]. WMO guidelines for the observational procedure and quality control are adopted uniformly in these surface meteorological observatories while data acquisition, tabulation, and computation. The online data entry system, itself has an inbuilt quality control mechanism to test the errors like data format, duplicate records, and incorrect units of measurement, impossible values, extremes, and outliers.
Descriptive statistics of different meteorological parameters in terms of mean, high, low, range, standard deviation (SD), and coefficient of variation (CV) are also computed to understand data patterns and to ensure the quality check of data (Table 1). To measure the strength and direction of a linear relationship between two variables, correlation coefficient (R) between meteorological parameters (T max . T min , BSS, WS, RH 1 , RH 2 , and EP) with FAO56-PM ET 0 are also computed ( Table 1). Weekly totals of ET 0 are computed using the FAO56-PM equation which is considered as the target output for model development [1].
The pattern of different meteorological parameters considered as input variables for model development along with target variable FAO56-PM ET 0 in selected stations is represented as box plot arrangements in Fig. 2. The middle line of the box plot signifies the median value while the upper and lower edges signify 75% and 25% of the data set respectively. The highest and lowest limits of the upper and lower vertical lines indicate the highest and lowest values respectively. The square depicts the simulated mean, and the straight-line shows the observed mean.

Radial basis function neural network (RBFNN) based estimator
RBFNN is a category of feed-forward neural network with a single hidden layer and an output layer formulated by Broomhead and Lowe [47]. Pictorial representation of the RBFNN is given in Fig. 3. The processing units termed as neurons in the hidden layer are associated with centers, c = c 1 , c 2 , c 3 , ., ., c h , and their width = 1 , 2 , 3 , ., ., . h , where h is the number of neurons in the hidden layer. Each neuron in the hidden layer receives the same set of input data X = x 1 , x 2 , x 3 , ., ., ., x n . The centers of every hidden neuron have the same dimension as that of the input data, i.e. c i ∈ R n , X ∈ R n . The output of hidden layer neurons Finally, the response of the RBFNN at the output layer, for a given set of input data is linear in terms of weights and computed using the following expression.
Development of the RBFNN for each instant of input data and its corresponding output {X, y} is obtained recursively by updating the network parameters w i , c i , i to minimize the instantaneous error cost function given as.
The weight update rules to optimize the network parameters w i , c i , i at time t are given by following equations which are derived using gradient descent algorithm [48].
where y d desired output or target value, c ij j th element of the i th center, η 1 , η 2 , η 3 learning rates for network parameters w i , c i , i respectively.

Differential evolution based RBF neural network estimator
Differential evolution (DE) [49,50] is a simple and efficient global optimization technique based on a heuristic method for minimizing a nonlinear function. Using this efficient heuristic approach a hybrid structure, RBFDE is developed in which total d number of network parameters, represented by a parameter vector, ⃗ x i = w i , c i , i , is optimized by the differential evolution algorithm (DE). DE algorithm involves three basic operations viz., mutation, recombination, and selection. The step-wise procedure for the development of RBFDE is described below.
Step 1 Randomly initialize i = 1, 2, 3, ., ., .NP number of target or population vectors, ⃗ x i,G between 0 to 1, where each ith individual of the population vector represents parameters of the RBFDE model. The ith target vector of Gth generation, ⃗ x i,G is given as ⃗ Step 2 Repeat step 3 with each target vector ⃗ x i,G for i = 1, 2, 3, ., ., .NP.
Step 3 a) Give K numbers of input patterns to the RBF network sequentially with each pattern having dimension n. b) For each one of the K input patterns, obtain corresponding network output using ith target vector ⃗ x i,G as the parameters of the network and compare it with the corresponding desired output to get an error using (3). For K patterns, the K number of error values will be obtained. c) Calculate f ⃗ x i,G using (7), where f ⃗ x i,G represents the fitness function i.e. mean square error (MSE).
Step 4 Obtain f min ⃗ x i,G and represent the corresponding ⃗ x i,G as the ⃗ x best,G for Gth generation.
Step 5 Choose a scaling factor F ∈ [o, 1] and a cross over ratio CR, ∈ [o, 1] and repeat step 6 to step 15 until the desired minimum MSE is obtained.
Step 8 Compute the mutant vector v i,G for each target vector ⃗ x i,G for Gth generation as Step 9 Repeat the steps from 10 to 11 for i = 1, 2, 3, ., ., .NP times Step 10 Randomly choose an index r 3 between 1 to d and repeat step 11 for j = 1 to d, where d is the dimension of the target or population vector.
Step 11 Generate a random number rand ∈ [o, 1] and compute the trial vector ⃗ U j,i,G by recombination operation, which replaces the previously successful individuals with mutant vector as Research Article SN Applied Sciences (2021) 3:56 | https://doi.org/10.1007/s42452-020-04069-z Step 12 For each trial vector ⃗ U j,i,G, i = 1, 2, 3, ., ., .NP , evaluate f U i,G , which is a mean square error (MSE). (Similar to step 3) Step 13 Repeat the step14 for i = 1, 2, 3, ., ., .NP Step 14 Finally, the next generation of NP number of target/population vector ⃗ x i,G+1 is selected based on survival of the fittest criteria as Step 15 Obtain f min ⃗ x i,G+1 and represent it as ⃗ x best,G+1 for the next generation.
Step 16 Stop Pictorial representation of the DE algorithm is shown in Fig. 4.

Particle swarm optimization based RBF neural network estimator
In this approach parameters of the RBFNN model i.e. {w i , c i , σ i }, as described in Sect. 2.2.1, are updated using the PSO algorithm. The PSO [51][52][53] is a metaheuristics optimization algorithm inspired by the paradigm of swarm intelligence which mimics the social behavior of animals like fish and birds. It is successfully applied to various applications in engineering and science [54][55][56]. The algorithm uses a fixed number of particles that represent the parameters of RBFNN. Each particle updates its current velocity and position by its own experience called personal best (p-best) and by the social experience of the swarm called global best (g-best). Steps involved in PSO are briefly described below: Step 1 Initialize fix number of particles with random position and velocity uniformly distributed over the search space.
Step 2 Evaluate the fitness of each particle according to the objective function Step 3 Record pbest for each particle and g-best of the swarm.
Step 4 Update velocity of each particle Step 5 Update the position of each particle.
Step 6 Update pbest and gbest Step 7 Repeat the steps from 2 to 6 until the termination condition is satisfied and stop.
Pictorial representation of the PSO algorithm is shown in Fig. 5.

Multi-layer artificial neural network (MLANN)
MLANN, suggested by Haykin [57] is successfully employed in many applications to solve the regression problem. MLANN architecture considered for this proposed investigation consists of an N-5-1 structure. N represents the number of input features. Optimum results are obtained with 5 neurons in the intermittent hidden layer. Desired ET 0 estimates are obtained at output neurons. Hyperbolic tangent (tanh) is used as an activation function in every processing neuron. The training of the network is done by a conventional back-propagation algorithm which is based on the error-correcting learning rule to update the weights and bias of each neuron in different layers.

Empirical models
Weekly ET 0 for the study locations is also computed using empirical methods of FAO56-PM, Blaney-Criddle, Open Pan, Turc, and Hargreaves from available meteorological data. A brief description regarding empirical approaches considered in this investigation and the corresponding input meteorological parameter requirement are listed in Table 2. The description regarding different climate based empirical methods considered in this investigation is not included in this paper. More details regarding these empirical approaches can be obtained from basic references [1, 5-7].

Performance evaluation measures
Comparative analysis of estimated ET 0 obtained with different soft computing models and empirical methods considered for the investigation is carried out by computing performance evaluation measures, namely, mean square percentage error (MAPE), root mean square error (RMSE), determination coefficient (R 2 ) and efficiency factor (EF) proposed by Nash and Sutcliffe (NSE) [58]. The mathematical expression of different evaluation measures is as follows.
R 2 and EF values close to 1 are also indicators of a higher accuracy level of the model.

Results and discussion
The key objective of this investigation is to examine the potential of different evolutionary optimized hybrid (RBFDE, RBFPSO) and conventional (RBFNN, MLANN) soft computing approaches with available climatic features for estimation of ET 0 comparable to FAO56-PM ET 0 . Input features combination of different models is decided based on empirical approaches of Hargreaves, Turc, Open Pan, Blaney-Criddle, and FAO56-PM ET 0 listed in the previous section. These soft computing models are categorized  Simulation studies are carried out with a different input features combination to test the sensitivity of the soft computing approach to control parameters until a satisfactory accuracy level is achieved for estimation of FAO56-ET 0 for different study locations. Detailed information regarding modeling strategies and respective control parameters that produce optimum results during the simulation process are shown in Table 4 for different soft computing models.
Calibration of RBFDE, RBFPSO, RBFNN, and MLANN models is done using the above-listed network parameters with training datasets of all the three study locations, Raipur, Jagdalpur, and Ambikapur. During the training process, input patterns are given to the model sequentially and the corresponding estimated output is obtained at the output layer after completion of the forward pass (Fig. 3). The estimated output is compared with the corresponding target FAO56-ET 0 output to compute the instantaneous error cost function. Real-time update of the model parameters is done in each instance to minimize the squared error using respective evolutionary (DE and PSO) and conventional back-propagation learning algorithms (RBFNN and MLANN). The process continues until all the available training input patterns for model calibration gets  exhausted. This completes one cycle called an epoch. At the end of each epoch, the mean square error is computed and stored for each epoch to examine the learning characteristic of soft computing models. The iterative process is repeated several times until MSE is minimized to a desired low value nearly close to zero. This completes the supervised learning process and model parameters are then fixed to constitute soft computing models. A similar calibration process is adopted for all soft computing approaches.
To test the performance of different soft computing models, test data patterns are then presented sequentially at the input layer of the model and through forward pass respective estimated ET 0 is obtained at the output layer for all the test patterns. These ET 0 estimates are then compared with corresponding target FAO56-PM ET 0 values. Performance evaluation measures, MAPE (%), RMSE (mm week −1 ), R 2, and NSE as described in the previous section are then computed using desired and estimated output of different types of soft computing models and equivalent empirical approaches for comparison of model performance, which ultimately leads to model selection.
The computed values of performance evaluation measures for different types of soft computing models and equivalent empirical approaches considered are listed in Tables 5 and 6   which is very high as compared to type III soft computing models. iv. Type IV soft computing models yield better results as compared to all other types of soft computing and empirical models. MAPE ranges between a low of 1.1 to a high of 3.9 at Raipur, followed by 3.7 to 4.4 at Jagdalpur and 2.2 to 4.6 at Ambikapur with RBFDE4 and MLANN4 respectively. MAPE with the Blaney-Criddle method is again quite inferior as compared to type IV soft computing approaches and ranges from 15.5 (at Jagdalpur) to 22.6 (at Raipur). v. Type V models also produced good results, as reasonably fair estimates of ET 0 can be obtained between a low MAPE of 1.9 with RBFDE5 (at Raipur) to 5.2 with MLANN5 (at Jagdalpur and Ambikapur), which is very much comparable to that of type IV models, even without taking humidity data as one of the input features.  ing models produce better estimates of FAO56-PM ET 0 than empirical models.
Results of the performance evaluation analysis indicate that the evolutionary optimized hybrid soft computing models considered for the investigation (RBFDE and RBFPSO) performed consistently better than other conventional soft computing techniques (RBFNN and MLANN) and empirical approaches in all the objectives. From the inferences, it is also evident that when a complete set of the climatic variable is involved in the computation of ET 0 using these models, it looks very difficult to choose between RBFDE and RBFPSO as they look statistically similar in some cases. However, the proposed RBFDE is recommended because of its preciseness and generalization performance in estimating ET 0 in all the stations considered for the study.

Conclusions ET 0
The present investigation is carried out to examine the generalized potential of evolutionary optimized hybrid soft computing techniques RBFDE and RBFPSO for the estimation of ET 0 in different ACZs. The ET 0 estimates obtained with proposed RBFDE and RBFPSO models are compared to the conventional neural network (RBFNN, MLANN) and existing empirical approaches. Looking to the scarcity of complete datasets required for computation of FAO-PM ET 0 , four variants of each category of soft computing models (RBFDE, RBFPSO, RBFNN, and MLANN) equivalent (in terms of input feature combination) to empirical approaches (Hargreaves, Turc, Open Pan and Blaney-Criddle) is examined. It can be concluded that different soft computing models considered in this investigation, have resulted in improved and more consistent FAO56-PM ET 0 estimates as compared to equivalent empirical approaches. Among the soft computing models, evolutionary models RBFDE and RBFPSO produced a more precise estimation of FAO56-PM ET 0 than conventional RBFNN and MLANN as proposed RBFDE and RBFPSO models resulted in low MAPE and RMSE and high R 2 and NSE close to 1 in most of the cases. However, ET 0 estimates obtained with the proposed RBFDE seems to be slightly better than RBFPSO. Hence, appropriate soft computing models may be recommended for the estimation of ET 0 in other stations of respective ACZs of the study area. The proposed soft computing models may be embedded in crop weather simulation models as subroutines for precise estimation ET 0 with available input features. However, recalibration and re-validation of these data-driven models are essentially required for their effective implantation in other parts of the world.

Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of interest.
Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors.
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