Generalized Daily Reference Evapotranspiration Models Based on a Hybrid Optimization Algorithm Tuned Fuzzy Tree Approach

Reference evapotranspiration (ET0) is an important driver in managing scarce water resources and making decisions on real-time and future irrigation scheduling. Therefore, accurate prediction of ET0 is crucial in the water resources management discipline. In this study, the prediction of ET0 was performed by employing several optimization algorithms tuned Fuzzy Inference System (FIS) and Fuzzy Tree (FT) models, for the first time, whose generalization capability was tested using data from other stations. The FISs and FTs were developed through parameters tuning using Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Pattern Search (PS), and their combinations. The FT was developed by combining several FIS objects that received ranked meteorological variables. A total of 50 FIS and FT models were developed and the model ranking was performed utilizing Shannon’s Entropy (SE). Evaluation outcomes revealed the superiority of the hybrid PSO-GA tuned Sugeno type 1 FT model (with R = 0.929, NRMSE = 0.169, accuracy = 0.999, NS = 0.856, and IOA = 0.985) over others. For evaluating the generalization capability of the best model, three different parts of datasets (all-inclusive, 1st half, and 2nd half) of the five test stations were evaluated. The proposed hybrid PSO-GA tuned Sugeno type 1 FT model performed similarly well, according to the findings, on the datasets of the test stations. The study concluded that the hybrid PSO-GA tuned Sugeno type 1 FT approach, which was composed of several standalone FIS objects, was suitable for predicting daily ET0 values.


Introduction
Evapotranspiration (ET) is considered as one of the most visible and important constituents of the hydrologic cycle and serves a crucial function in the reliable and careful management of water resources, especially in the field of irrigated agriculture where sustainable agricultural practices greatly rely on irrigation water (Ferreira and da Cunha 2020). Due to this vital importance, accurate estimation of ET demands special attention from planners and policymakers in drawing the right decisions on the allocation of precious water resources. Direct measuring of ET can be accomplished using a costly lysimetric method (Holmes 1984), which is associated with small area coverage, high setup and operational expenses and laborious work involved in data collection (Chia et al. 2020). Alternatively, ET can be measured along with other meteorological variables by means of the eddy covariance and Bowen ratio methods (Bowen 1926;Shoemaker et al. 2011), which are also costly and involve setup and operational complexities. Due to these shortfalls, indirect approaches to ET measurement, based solely on reference evapotranspiration (ET 0 ) estimation are gaining popularity in many different places around the globe (Allen et al. 1998;Ding et al. 2013). Estimation of ET 0 primarily depends on the climatic variables, and it can be predicted accurately once a direct association between ET 0 and climatic variables are established. Therefore, there is a growing demand for accurate prediction models that work on input-output data patterns, and can be generalized for other stations once developed for a particular station. Hence, this effort intends to propose an approach utilizing Machine Learning (ML) algorithms for predicting and generalizing ET 0 for six meteorological stations located in south Florida, USA. ET 0 model was developed for a station, and the generalization capability of this developed model was verified using data from the other five stations in the same hydrological setting.
Although numerous ML-based ET 0 models have recently been developed, the majority of the modelling tasks (i.e., train, test, and validate the models) were carried out using meteorological data from the same station. Few studies have been conducted to assess the model's performance using data from different stations (i.e., apart from stations for which the model was developed). For instance, Wang et al. (2019) used data from many weather stations within China to assess the potential generalization capacity of RF and GEP models to estimate ET 0 . In another study, Roy et al. (2021) utilized data from two weather stations 1 3 in Bangladesh: data from the first station was used to develop an ET 0 model based on a PSO tuned Hierarchical Fuzzy Systems (PSO-HFS) while the generalization capability of the developed PSO-HFS model was tested using data from the second station. The current study presents a method in which training, validating and testing of the model were being carried out for a particular meteorological station, then the created model was tested using meteorological data from surrounding stations (five stations) to demonstrate the proposed model's generalization capacity.
Apart from that, soft computing techniques tuned using numerous optimization algorithms were thought to improve ET 0 prediction accuracy substantially (Tao et al. 2018). As an example, Alizamir et al. (2020) demonstrated ET 0 modelling using Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) tuned ANFIS models, and compared the prediction precision of the optimization algorithm tuned ANFIS models with that of the classic ANFIS and ANN models. To optimize the ANFIS parameters for forecasting ET 0 in subtropical climatic zones, Roy et al. (2020) used Biogeography-based Optimization (BBO), Firefly Algorithm (FA), PSO, and Teaching-Learning-based Optimization (TLBO). They also proposed an ensemble of these ANFIS models that were tuned using evolutionary algorithms: the ensemble model outperformed the majority of the single models, according to the researchers. Roy et al. (2021) proposed a PSO-HFS to predict daily ET 0 and concluded that the proposed PSO-HFS can be used as an effective tool to model daily ET 0 with an acceptable level of accuracy. In three different climatic zones of Iran, Whale Optimization Algorithm (WOA) tuned Support Vector Regression (SVR) models outperformed classic SVR models in forecasting ET 0 (Mohammadi and Mehdizadeh 2020). In a different research work, Tikhamarine et al. (2020a) suggested optimized SVR models with WOA, Multi-verse Optimizer (MVO), and Ant Lion Optimizer (ALO) for forecasting monthly ET 0 at two climatological stations in Algeria's northwestern region using three different input scenarios. Tikhamarine et al. (2020b) developed a Grey Wolf Optimizer (GWO) tuned SVR model for calculating monthly ET 0 at three Algerian climatological stations. Wu et al. (2020) developed hybrid ELM models to estimate monthly pan evaporation by combining ELM with WOA and Flower Pollination Algorithm (FPA). They found that FPA considerably improved the accuracy of the ELM model. In a different study, the PSO tuned ELM model was found to be superior to standard ELM, ANN, and RF-based prediction models for projecting daily ET 0 utilizing fewer climatological data as inputs for a research work done in a dry area of China (Zhu et al. 2020). Consequently, it is evident that model performance improves substantially when optimization algorithms are employed to tune the parameters of the models; hence, in the present research, optimization algorithms (e.g., GA, PSO, Pattern Search (PS), and their combinations) were employed to tune the model parameters of the standalone Fuzzy Inference System (FIS) and Fuzzy Tree (FT) models.
There have been very few studies conducted on hydrological and agricultural fields using FTs. However, they have already been successfully applied in different fields (Barach et al. 2019). Recently, a few studies on water management and hydrology also centered on the application of Fuzzy Decision Trees (FDT). For example, while characterizing the magnitudes and frequencies in flood time series, Sikorska-Senoner and Seibert, (2020) used an FDT rather than a typical trend analysis. Another study by Sikorska et al. (2015) demonstrated the use of FDTs in flood classification to identify the patterns of flood at a catchment scale and concluded that FDTs provided better estimates of flood class than utilizing a crisp decision tree. To construct working conditions for a reservoir system, Wei and Hsu (2008) compared three forms of decision trees: conventional decision trees, neural decision trees, and FDTs. The findings of their comparison showed that FDTs outperformed the other two 1 3 forms of decision trees. Another study by Schärer et al. (2006) reported the utility of FDTs in modelling and predicting phosphorus export from agricultural catchments. They demonstrated the high potential of FDTs in the decision support system to provide guidances to the policy makers. Using fuzzy logic-based FDTs, Han et al. (2002) tackled the uncertainty in instantaneous flood projections. The findings of the previous studies clearly demonstrated the capability of tree-based modelling approaches in various research domains. Moreover, optimization algorithms can considerably improve the ET 0 prediction capabilities of the FT-based models. This is evident in a recent research by Roy et al. (2021), who improved the prediction capability of an HFS model by tuning its parameters using a PSO algorithm. However, they used only the PSO algorithm to tune both the input-output Membership Function (MF) parameters and the rule base of the HFS model. In contrast, the present study incorporates six different combinations of three optimization algorithms (GA, PSO, and PS) for the rule base learning and tuning of all parameters separately to develop different FIS and FT models. Moreover, a total of 50 models were developed and compared including the incorporation of both Sugeno and Mamdani type 1 and type 2 fuzzy objects. Therefore, this study proposes a hybrid optimization algorithm tuned hierarchical tree-based FT model composed of individual fuzzy objects to predict daily ET 0 . The performance of the proposed FT model was compared with optimization algorithm tuned FIS, Regression Tree (RT), and M5Tree models as benchmarks.
In addition, most applications (e.g., ET 0 modelling) include input variables that are positively or negatively correlated with the target variable, and a cascaded structure of FT models is better suited for such applications. Therefore, an FT model having a cascaded structure combining hierarchical fuzzy systems was developed in this study for ET 0 modelling. According to the most recent literature, hybrid optimization algorithm tuned FT models are yet to be applied in the research fields of agriculture and hydrology, particularly in the modelling of ET 0 . Furthermore, the generated ET 0 model's generalization capacity for the surrounding weather stations has not been assessed using the hybrid optimization algorithm tuned FT-based ET 0 modelling technique. The necessity to build and calibrate a model particular to each station is eliminated by using a generalized prediction model that can be geographically validated using data from several stations. As a result, it is possible to create a generalized prediction model that depicts a hydrogeological setting. Hence, the explicit objectives of the current study are as follows: (1) to explore the potentiality of an optimization algorithm tuned FT (based on both Sugeno and Mamdani type 1 and type 2 fuzzy objects) model for predicting daily ET 0 in the subtropical climate of South Florida, United States; (2) to compare the performance of the proposed FT model with a benchmark single FIS, RT, and M5Tree based modelling approaches; and (3) to evaluate the generalization capability in terms of prediction accuracy of the FT model using data from the other five neighboring stations located in the similar climatic area and hydrologic setting.

Study Area and Data
The study area, situated in south Florida, USA ( Fig. 1) has spatially varying ecosystems with different plant species and substrates. Daily meteorological data along with the measured ET 0 values from six spatially and ecologically varying meteorological stations were acquired from the website of the United States Geological Survey (USGS 2020).
Daily values of six meteorological variables, namely Latent Heat (LH), Solar Radiation (SR), Net Radiation (NR), Sensible Heat Flux (SHF), Relative Humidity (RH), and Air Temperature (AT) along with the quantified daily values of ET 0 used in this study were obtained from the following six meteorological stations:

Station 1-Blue Cypress Marsh Station
The station is located between 26° 54′ 06" North latitude and 80° 47′ 21" West longitude with the available meteorological and ET 0 data for around 5 years (from 11 December 2009 to 20 October 2014).

Station 2-Cypress Swamp Station
The vegetation covers and substrate of this station are recognized as Tall Cypress Strand, which locates between 25° 45′ 10" North latitude and 81° 06′ 01" West longitude. The Station has a data availability of nearly 3 years (from 25 April 2007 to 24 April 2010).

Station 3-Dwarf Cypress Station
The coordinates of this station lie between 25° 45′ 45" North latitude and 80° 54′ 27" West longitude. The station has an herbaceous vegetative cover, namely Dwarf Cypress and Sawgrass. Available data ranges between 18 April 2007 and 10 April 2010 (approximately 3 years).  Upland and Cypress Domes. The station has available data for around 3 years (from 23 April 2007 to 16 April 2010). The geographic location of this station lies between 25° 59′ 59" North latitude and 80° 55′ 29" West longitude.

Station 6-Wet Prairie Station
This station has available data for approximately 3 years (from 10 October 2007 to 11 October 2010). The geographic location of this station lies between 25° 44′ 41" North latitude and 80° 56′ 24" West longitude. The measurement of ET 0 values used in this study was performed using the eddy covariance energy-budget method (Shoemaker et al. 2011). The two main instruments used in the eddy covariance method are the sonic anemometers (measure latent heat fluxes) and krypton hygrometers (measure sensible heat fluxes).

Input-output Data for the Developed Models
Meteorological variables including LH, SR, NR, SHF, RH, and AT and corresponding ET 0 values constitute the input-output patterns for training of the proposed FT-based ET 0 prediction models. ET 0 prediction models were built using the data from the Blue Cypress Marsh station. This station was selected as a primary station for model development and validation purposes because of the availability of a wider range of data (from 11 December 2009 to 13 June 2018) and the minimum number of missing values. However, few values were missing, which were imputed with the RF algorithm (MathWorks 2022a). The missing data imputation was performed for both the training and test stations. The developed models at the training station (Blue Cypress Marsh) were then used to predict the ET 0 values for the other five neighboring stations, mainly to evaluate the generalization capability of the proposed ET 0 model. The entire dataset of the Blue Cypress Marsh station, as a representative station of the study area, was divided into the training (40%) (from 11 December 2009 to 03 May 2013), validation (40%) (from 04 May 2013 to 01 October 2016), and applied (20%) (from 02 October 2016 to 13 June 2018). As the dataset was sequential, for better model building, the first 80% of the sequential data was randomized and split into two datasets of equal size (the first 40% for training and the last 40% for the validation). The last 20% of the dataset was taken in sequence and was served into the applied dataset. This method of data splitting provides for a more accurate assessment of the generated model's performance on the applied data, which is usually a later part of the sequential data (Francone 2001). The trained and validated ET 0 models for the Blue Cypress Marsh station were then employed in predicting ET 0 values for the other five stations without developing models with the datasets at these stations. For assessing the generalization capability of the ET 0 models, three different arrangements of the test station datasets were evaluated: entire sequential datasets, the first half of the dataset, and the last half of the dataset. These scenarios of the dataset were used to test the ability of the developed ET 0 models to capture the input-output relationships contained in different parts of the dataset with variations in data ranges and numbers. Descriptive statistics of the input meteorological variables and the output ET 0 for the training (Blue Cypress Marsh station) and test stations are presented in Table 1.

Proposed Optimization Algorithm Tuned Fuzzy Tree (FT) Models
Fuzzy Inference Systems (FISs) are considered to be useful tools for modelling dynamic, complex and non-linear systems with multiple inputs and single outputs (Sugeno and Yasukawa 1993;Takagi and Sugeno 1985). A Sugeno-type FIS is often preferred against a Mamdani FIS since the Sugeno FIS holds the advantage of having a more computationally efficient defuzzification process. In general, however, the computing efficiency of a FIS differs widely depending on the number of inputs and the numeral value of rules, since the number of inputs grows exponentially as the quantity of rules increases. A large number of rulesets affects a FIS's computing performance and makes tweaking the rule base and membership function settings more challenging. Furthermore, a larger quantity of rule bases lowers the ability of tailored FISs to generalize, especially in cases where training data is scarce, as is the case in many hydrological applications. Rather than a single huge FIS object associated with numerous predictors, an FT made up of reduced intersected FIS objects may be used to solve these difficulties. The fuzzy systems in an FT are arranged in hierarchical tree configurations, with the low-level FIS outputs serving as inputs to the high-level FISs. When compared to a single FIS, an FT usually delivers more computing efficiency with an equal amount of input variables. Although several FT structures can be utilized, the following three are the most commonly used FT structures used in many application areas: (a) incremental, (b) aggregated, and (c) cascaded or combined, which is the combination of incremental and aggregated structures (Siddique and Adeli 2013). Because a cascaded structure is ideal for an application having correlated and uncorrelated variables together, and the meteorological variables used in this work are positively and negatively correlated with each other, this study used a cascaded or mixed (combined) FT framework.
This study utilizes an FT to predict daily ET 0 (output data attribute) by using six input data attributes (meteorological variables: LH, SR, NR, SHF, RH, and AT). The procedure of predicting daily ET 0 values using an FT structure consists of the following two major steps: (a) Constructing the FT This step is associated with the following three sub-steps: (i) Ranking of the input meteorological variables according to their Pearson correlations with the output (ET 0 ): Correlation coefficients between each of the input meteorological variables and the output variable (ET 0 values) for the training data were calculated. The correlation matrix is presented in Fig. 2. It is observed from Fig. 2 that RH showed a negative correlation whereas all the other five input variables showed positive correlations with the ET 0 . Ranking of the input attributes having positive correlations was performed in descending order using the absolute values of the attribute's correlation coefficients. (ii) Creating several FIS objects with the help of the ranked input variables: To incorporate both the negative and positive effects on the output (ET 0 ) for prediction, input attributes (meteorological variables) were combined based on the ranked attributes with both positive and negative correlation values. Finally, the input attributes were grouped with respect to their ranks as follows: • fis1: Relative Humidity and Latent Heat • fis2: Net Radiation and Air Temperature • fis3: Solar Radiation and Sensible Heat Flux (iii) Constructing an FT using the created FIS objects: The FT was constructed using five (fis1, fis2, fis3, fis4, and fis5) two-input and one-output FIS elements to decrease the total quantity of rule sets in the inference system. Three FISs (fis1, fis2, and fis3) receive the input variables directly and produce intermediate values of ET 0 , which were then integrated using the leftover two FISs (fis4 and fis5). Figure 3 represents the created FT structure. Tuning was accomplished in two phases: (i) Learning of the rule base only (input and output Membership Function (MF) parameters were kept constant), and (ii) Tuning of both the input/output MF parameters and the rule base. The first phase required less computation time compared to the second phase and was converged rapidly to a fuzzy rule base during the training step. In the second phase, the learnt new rules from the first phase along with the input and output MF parameters were tuned simultaneously. The parameter tuning process in the second phase was made faster (with convergence to near Global optima) through using the first phase's tuned rule base as the second phase's initial condition.

Tuning of the Proposed Models
A two-phase tuning process was carried out on the FT. During phase one, the rule base was learned while the MF parameters for the inputs and the output were kept constant. During the second phase, tuning of the input and output MF parameters as well as the rule base was performed simultaneously in which the first phase's tuned rule base was used as the second phase's initial condition (MathWorks 2022b). This enables speedy parameter adjustment and global optima convergence. To obtain optimal parameter values of the FT models, optimization algorithms such as GA, PSO, and PS as well as their combinations were used for both phases of a simulation. Various combinations of optimization algorithms for learning rules and tuning all parameters for the FT-based models are presented in Table 2.
Obtaining optimal parameter sets through tuning was completed before the userdefined highest number of iterations was reached. In the case of non-convergence, iterations continued till the optimal solution was found or until the user-specified total quantity of iterations had been reached. The optimum options (various optimization parameters) for different optimization algorithms employed in this study are presented in Table 3.

Single Fuzzy Inference System (FIS)
A single FIS object (both Sugeno and Mamdani type 1 and type 2 FIS objects) representing the individual fuzzy object in the proposed FT model was also developed for comparison purposes. Similar parameters, as in the case of fuzzy objects in the FT, as well as the similar MFs and tuning approaches, were used to provide a fair comparison. The main difference between the single FIS and the FT was the number of input variables fed to the models. Unlike FT modelling approach, the single FIS received all six meteorological variables as inputs in predicting the target variable, daily ET 0 . Details of Sugeno and Mamdani type 1 and type 2 FIS objects can be found in Sugeno and Yasukawa (1993), and are not repeated.

Regression Tree (RT)
RTs are variants of decision trees that use input-output training patterns to create simple, flexible, and easy-to-understand models. The notion of the 'Classification and Regression Tree (CART)' algorithm is connected with RTs (Breiman et al. 1984;Krzywinski and Altman 2017). To generate models, the CART method uses three primary sequential procedures: (a) creating a complicated tree, (b) pruning, and (c) picking an ideal subtree. At the initial stage, a binary split process is used to create a complex complete tree with multiple terminal nodes. In the second stage, the complex tree created in the initial step is trimmed to avoid or at best decrease the tendency of model overfitting issues. The CART algorithm selects an appropriate subtree for new sample prediction in the third stage. By tracing the choices from the root node to the leaf node inside the tree, the generated RTs offer an anticipated response. An RT's responses or outputs are stored in the leaf node. In the MATLAB environment, the RT-based ET 0 prediction models were created.

M5 Model Tree (M5Tree)
M5Tree was created using the M5 method's concepts (Quinlan 1992). This approach creates single trees using the 'Standard Deviation Reduction' criterion, similar to the 'M5' method. Model trees (MT) are made by combining standard regression trees at the leaf nodes with linear regression functions. The constituents of a leaf node of an MT, which is nothing more than an RT without these activities, are determined by pruning and smoothing processes. Machine learning algorithms such as MTs (Quinlan 1992) have proved their predictive skills in a variety of study disciplines (Bhattacharya and Solomatine 2005). The MTs are referred to as the 'inverted trees,' meaning that the root nodes are at the crown of the tree while the leaves are at the bottom.
MTs and RTs are Decision Tree (DT) variants that are especially suitable to solve regression problems (Quinlan 1992). The MTs, on the other hand, differ from RTs in that they produce linear type models in their leaf nodes, whereas RTs produce a constant value in their leaf nodes. The input-output linkages created at the leaves are consequently used in predicting responses for a particular dataset. When it comes to managing enormous datasets and providing reliable predictions, MTs outperform RTs. M5Tree employs the 'divideand-conquer' strategy, which allows for the division of a large domain of data into several little data sub-domains (Bhattacharya and Solomatine 2005). In this method, the input parameter domain is divided into numerous subspaces, each of which symbolizes a linear regression model. M5Tree can create a hierarchical tree structure with splitting rules' in its 'non-terminal nodes' and 'expert models' in its leaves by means of this unique data splitting technique.

Model Ranking: Calculation of Shannon's Entropy Weight
When alternative performance evaluation indices are used, machine learning-based prediction models consistently generate conflicting prediction results. For instance, one model may perform better than others when Index of Agreement (IOA) criterion is used. On the other hand, the better performer (based on the IOA criterion) may not perform equally well when another performance indicator, for instance, Nash-Sutcliffe efficiency coefficient is evaluated. As such, more than a few statistical indices need to be computed on the test dataset, and the derived indices should be used within a general framework based on a decision theory to choose the most effective model. In this study, the performance of the constructed prediction models was ranked and to establish the ranking, Shannon's entropy was applied. There were three benefit indices (the higher the better) and three cost indices (the lower the better), which were incorporated in a general context to compute ranking of the models.
Competing ET 0 models and their performance indicators were integrated into a decision matrix. The calculation steps of the Shannon's entropy can be found in Roy et al. 2020, and are not repeated here.

Statistical Indices for Performance Evaluation
To assess the performance of the FT-based ET 0 models as well as the benchmark ET 0 models, several performance indicators were used in this study as follows: Root Mean Squared Error (RMSE): Normalized RMSE: Accuracy: Mean Bias Error (MBE):

Performances of the FT, FIS, RT, and M5Tree Based Models During the Training, Validation, and Applied Phases
The ET 0 values, obtained from the optimization algorithm tuned FIS and FT models, RT, and M5Tree were compared with each other to obtain the best performing model. The values of the performance evaluation criteria for the training phase (training and validation sets of data) and testing phase (applied set of data) are presented in Figs. 4 and 5, respectively. Fig. 4 illustrates model performances during the steps of training and validation for the models created at the base station (Blue Cypress Marsh station) concerning R, RMSE, and NRMSE. For the ease of representation, the model names were shortened and the total 50 models were represented by M1 through M50 as shown in Table 4.
From Fig. 4, it is evident that all statistical indices indicate a considerably satisfactory prediction performance of the models during the training as well as the validation phases as indicated by the computed R, RMSE, and NRMSE values. All models produced higher values of R and smaller quantities of RMSE and NRMSE, and all models showed negligible differences in magnitudes of the evaluation indices between the training and validation phases. Although models M4, M13, M31, M32, and M33 provided slightly higher differences in the values of R between the training and validation phases, the models produced acceptable R values during both the training and validation periods. As far as the RMSE criterion is concerned, higher differences between the training and validation phases were observed for M2, M10, M13, and M21 models. On the other hand, the NRMSE criterion reveals that higher differences between the training and validation NRMSE values were observed in almost all models. Nevertheless, the extent of these differences is rather low and these NRMSE values also show that the generated models perform well when compared to the criteria established by Heinemann et al. (2012) and Li et al. (2013). Figure 5 presents radar (spider) plots of model performances during the test period (applied dataset) for the training station (Blue Cypress Marsh station) using R, NS, IOA, RMSE, MAE, and MAD criteria. It is perceived from Fig. 5a that the developed models provided reasonably accurate performance concerning IOA (> 0.9) and R (> 0.8) criteria. However, performances were relatively poor when the NS criterion was considered: Models M4 (NS criterion ranged between 0.7 and 0.8), M16 (NS criterion ranged between 0.7 and 0.8), M33 (NS criterion ranged between 0.5 and 0.6), and M45 (NS criterion ranged between 0.7 and 0.8) produced NS values less than 0.8. Figure 5b presents spider plots of the performances with respect to RMSE, MAE, and MAD criteria. It can be witnessed from Fig. 5b that the developed models produced lower values of RMSE, MAE, and MAD except for models M3, M31, M32, M33, M44, and M45 for which the values were slightly higher. However, these values were relatively smaller (< 1.0 mm d −1 ), which is acceptable for machine learning-based prediction modelling. As can be seen from Figs. 4 and 5, in general, the performances of the proposed FT and the other three benchmark models apparently showed a similar trend in prediction capabilities. However, a closer look at the performance indices clearly demonstrates that the models performed in a different way when various performance indices were employed. That is to say, the models showed a contradiction in the prediction performances pertaining to different performance evaluation indices computed from the actual and model-predicted outputs. It is really difficult to make decisions in this situation. Therefore, a decision theory based on Shannon's entropy was used to choose the bestperforming model among a group of various models. Instead of a single index, Shannon's entropy included a collection of multiple performance evaluation indices.
In this study, Shannon's entropy was computed using 6 (six) performance evaluation indicators (R, NS, IOA, RMSE, MAE, and MAD) of which three were benefit indices (R, NS, IOA) and the remaining three were cost indices (RMSE, MAE, MAD). These performance evaluation indices were the indices computed on the applied dataset during the model testing phase. Table 5 shows the results of model ranking using Shannon's entropy.
Based on the results presented in Table 5, the proposed FT model (FISTree_PSO_ GA_Sug_Type1) is deemed superior, and, therefore, the developed FT model is recommended to predict daily ET 0 values at the training station (Blue Cypress Marsh station).
The training, validation, and applied performances of the proposed best FT model developed at the training station (Blue Cypress Marsh) are presented using absolute error box plots (Fig. 6). It is revealed from Fig. 6 that the proposed FT model produced similar errors for both the training and validation phases indicating proper training without model over-or under-fitting. Similar trends in errors as with the training and validation errors were observed during the applied phase (with the applied set of data). This implies that the model's performance in the applied phase was comparable.
Although ET 0 's actual and anticipated values seem slightly higher for the applied dataset, the magnitude of errors was relatively small and thus is acceptable for prediction modelling aspects.
To further validate the performance of the proposed FT model (FISTree_PSO_GA_Sug_ Type1), it was employed to predict daily ET 0 values for the nearby stations located in the same climatic zone. The following paragraphs summarize the findings.

FT-based Model Performance During the Generalization Capability Validation Phase
The proposed best-performing FT (FISTree_PSO_GA_Sug_Type1) model developed at Blue Cypress Marsh station was further tested using climatic data from five other meteorological stations, the data of which were utilized neither to train the model nor to validate it, i.e., data from outside the training station was used to validate the model. The performance evaluation indices were calculated for the whole, 1 st half, and the 2 nd half of the dataset in all five test stations. The use of three sets of data from all test stations allows us to look into the model's capacity to generalize better. Tables 6, 7, and 8 provide the findings of the evaluation in terms of several statistical indices. As shown by the results, the model performance was equally good when compared to the training and validation performance for the training station, while the model performance was considerably superior to the training station's applied performance.

3
The model performance was acceptable with respect to the calculated performance indices: for each of the three datasets of all the test stations, the model yielded higher accuracy (higher values of NS, IOA, and R, as well as lower values of RMSE, NRMSE, and MBE). Table 6 presents the model performance on the entire dataset of the test stations. It is observed that with RMSE as the performance criterion, the testing performance was the best at Cypress Swamp station (0.051 mm d −1 ) followed by Wet Prairie (0.060 mm d −1 ), Dwarf Cypress (0.063 mm d −1 ), Marsh (0.108 mm d −1 ), and Pine Upland (0.997 mm d −1 ) stations. The numeric values of other statistical indices followed a similar trend. Table 7 presents the model performance on the first half of the entire dataset for test stations. Table 7 shows that the proposed best-performing FT model functioned likewise when the first half of the data for the test stations were used for testing. As in the case where the entire dataset was used, the FT model performed the best and the worst at Wet Prairie (RMSE = 0.030 mm d −1 ) and Marsh (RMSE = 0.994 mm d −1 ) stations, respectively. It is worth noting that the performance was exceptionally good at all stations as far as the prediction modelling context was considered (Table 7). Table 8 presents the model performance on the second half of the entire dataset for the test stations. For this dataset, the model performance was also incredibly good, which is consistent with the previous results when either the entire or the first half of the dataset was used for model testing. With RMSE as a performance criterion, the model performance was the best at Cypress Swamp (0.053 mm d −1 ) station whereas it was worst at Marsh (0.120 mm d −1 ) station. Therefore, it can be perceived that the testing performance of the developed best-performing FT model was consistent and followed similar trends for different datasets at each test station. The prediction results of the test stations are presented using absolute error boxplots. Figure 7 illustrates the absolute differences in ET 0 values between actual and expected for the nearby five stations. It is observed from Fig. 7 that the errors are relatively small and are acceptable for the prediction modelling context.

Conclusion
This study looked into the potentialities of a hybrid PSO-GA optimized Sugeno type 1 FT modelling approach for predicting ET 0 utilizing the meteorological factors. Six input attributes (climatic variables) such as Latent Heat, Solar Radiation, Net Radiation, Sensible Heat Flux, Relative Humidity, and Air temperature were used as inputs to compute the daily values of ET 0 . Five FIS objects with input features that are ranked (correlations of the inputs with the output, ET 0 ) were used to create the FTs. The created FTs' input-output MFs and rule bases were then optimized in two steps using several combinations of different optimization algorithms, resulting in quick convergence of the parameter optimization procedure pertaining to the train station's (Blue Cypress Marsh) training dataset. The train, validation, and testing (applied dataset) results using the train station's dataset revealed that the created FT model correctly mapped the train station dataset's input-output patterns. Therefore, the suggested FT (PSO-GA optimized Sugeno type 1 FT) can be used to forecast ET 0 using climatic data as inputs. Nonetheless, testing the produced FT model's effectiveness outside of the training station is critical. To see how well the FT model is at predicting ET 0 for the neighboring stations without developing individual models for those stations, the best FT (PSO-GA optimized Sugeno type 1 FT) model developed at Blue Cypress Marsh station was employed to predict ET 0 for the other five stations. Three different datasets (1 st half, 2 nd half, and the complete dataset) from each of the test stations were tested. According to the findings, a Fuzzy logic-based FT may accurately predict daily ET 0 , especially when PSO and GA are used in the development of Sugeno type 1 FIS to construct the FT model (PSO-GA optimized Sugeno type 1 FT). The FT model's accuracy and reliability in predicting ET 0 for test stations were demonstrated by the findings. In subtropical climates, the proposed modelling technique offers a potential method for ET 0 prediction.
This study used data from one meteorological station and the created model was tested for other stations located in the same sub-tropical climatic zone but with varying substrates and plantations. It is worthwhile to include weather stations from various climatic zones to improve the applicability of the suggested FT modelling approach. Other bio-inspired optimization techniques may be explored and compared in future research, for instance, the Firefly Algorithm, or Whale Optimization Algorithm for the FT's parameter tuning process. In addition, the present study evaluates the model performance to predict daily ET 0 values which can be extended in a future study by developing models with weekly, monthly, and seasonal scales.