## Abstract

In this study, a hybrid model of least square support vector machine-gamma test (LSSVM-GT) is proposed for estimating daily *ET _{o}* under arid conditions of Zahedan station, Iran. Gamma test was used for selecting the best input vectors for models. The estimated ET

_{o}by LSSVM-GT model with different kernels of RBF, linear and polynomial, were compared with other hybrid approaches including ANN-GT, ANFIS-GT, and empirical equations. The gamma test revealed that climate variables of minimum and maximum air temperature and wind speed are the most important parameters. The LSSVM model performed better than the ANFIS and ANN models when similar meteorological input variables are used. Also, the performance of the three models of LSSVM, ANFIS, and ANN were better than the empirical equations such as Blaney–Criddle and Hargreaves–Samani. The RMSE, MAE, and R

^{2}for the best input vector by LSSVM were 0.1 mm day

^{−1}, 0.13 mm day

^{−1}, and 0.99, respectively. The threshold of relative absolute error of 95% predicted values by LSSVM, ANN, and ANFIS models were about 8.4%, 9.4%, and 24%, respectively. Based on the comparison of the overall performances, the developed LSSVM-GT approach is greatly capable of providing favorable predictions with high precision in arid regions of Iran.

## NOMENCLATURE

- ET
Evapotranspiration

- ET
_{o}Reference evapotranspiration

- ET
_{c}Crop evapotranspiration

- SVMs
Support vector machines

- LSSVM
Least square support vector machine

- GT
Gamma test

- RBF
Radial basis function

- ANN
Artificial neural network

- ANFIS
Adaptive neuro fuzzy inference system

- FAO
Food and Agriculture Organization

- ET
_{o(PM)}Calculated ET

_{o}by Penman–Monteith equation - ET
_{o(HS)}Calculated ET

_{o}by Hargreaves–Samani equation - ET
_{o(BC)}Calculated ET

_{o}by Blaney–Criddle equation - RMSE
Root mean square error

- MAE
Mean absolute error

- R
^{2}Coefficient of determination

- AARE
Average absolute relative error

- TS
Threshold statistics

- KKT
Karush–Kuhn–Tucker

*R*_{n}Net radiation

*G*Soil heat flux density

*T*Mean air temperature

*U*_{2}Average wind speed at 2 m height

*T*_{mean}Mean air temperature

*e*_{s}Saturation vapor pressure

*e*_{a}Actual vapor pressure

*P*Mean annual percentage of daytime hours

*R*_{a}Water equivalent of extraterrestrial radiation

- T
_{min}Minimum temperature

*T*_{dew}Dew point temperature

*T*_{max}Maximum temperature

*N*Number of data

*O*_{i}Observed value

Average of observed values

*P*_{i}Predicted value

Average of predicted values

- RE
_{t}Relative error in predicted values at time

*t* *Y*_{x}Number of computed

*ET*_{o}*a*and*b*Equation parameters in Blaney–Criddle equation

- x
Input data

- y
Output data

- |…|
Euclidean distance

*p*Number of near neighbors

p

^{th}nearest neighbors to x_{i}value for each vector x

_{i}Output value related with

- Γ
Gamma statistic value

*A*Line gradient

Output variance

- r
Noise

- Var(r)
Variance of r parameter

*δ*(p)Mean square distance to the p

^{th}nearest neighbors of the input vectors*ф*(*x*)Non-linear function

*w*The m-dimensional weight vector

- MSE
Minimum mean square error

*φ*Mapping function that maps

*x*into the m-dimensional feature vectorLagrange multipliers

*b*Bias

*e*_{i}Slack variables

*i*Input layer

*O*_{k}Output at the node

*k*of the output layer*w*_{ji}Controller for the strength connection between the input nodes i and the hidden node

*j**I*_{i}Input value

*V*_{j}Hidden value to node node j of the hidden layer

*g*_{2}Activation function for the output layer

*w*_{jk}Controller for the strength connection between the hidden node

*j*and the output node*k*.*t*_{pk}Target output

- E
_{p}Total error in ANN network

*z*_{pk}Output of ANN

*v*_{pk}Error of output unit

*k*of data pattern pMembership degree of

*x*in*A*set_{i}Normalized membership degree of

*i*rule*c*,_{i}*b*, and_{i}*a*_{i}Membership function of ANFIS

Membership degree of

*y*in*B*set_{i}*r*,_{i}*q*, and_{i}*p*_{i}Adaptive parameters of the ANFIS

*D*Slope of the saturation vapor pressure function

*c*Psychometric constant

## INTRODUCTION

Evapotranspiration (ET) calculation is an essential and important subject for quantifying crop water requirements (Lovelli *et al.* 2008) particularly in arid regions such as the southeast of Iran. Irrigation engineers need to calculate crop water requirement irrigation, especially in agricultural regions, to obtain a satisfactory yield and to estimate other components of the water balance and system design (Kisi 2008). The proper prediction of ET has an important role in the optimal utilization of water resources. Measuring ET_{o} using a lysimeter is a direct and relatively accurate method, but it is expensive and time-consuming and has application limitations. Therefore, ET is usually determined by means of reference evapotranspiration (ET_{o}) in the agricultural sector (Mehdizadeh *et al.* 2017).

The Food and Agriculture Organization (FAO) introduced the combination equation of Penman–Monteith for estimating ET_{o} modified by Allen *et al.* (1998) (FAO-56 PM) as the reference method for ET_{o} estimates. This approach is commonly used throughout the world, and has been proven to precisely estimate the ET_{o} in different climates (Kişi & Öztürk 2007; Jain *et al.* 2008; Kisi 2008; Doğan 2009; Marti *et al.* 2010; Traore *et al.* 2010). It is very difficult to formulate a simple equation that can create accurate estimates under different climate conditions (Tabari *et al.* 2013).

During the last decades, several models such as artificial neural network (ANN), adaptive neuron fuzzy inference system (ANFIS), and genetic programing (GP) have studied the reliability for estimating ET_{o} as a function of climate variables. Recently, a new simulation model, the support vector machine (SVM), has emerged as a data-driven computation in complex and practical studies (Liong & Sivapragasam 2002). The SVM is a powerful model for solving non-linear classiﬁcation problems, function estimation, and density evaluation. This model solves convex optimization problems (Yu *et al.* 2006). SVMs are advanced machine learning models based on structural risk minimization (SRM), which minimizes the expected error of a learning model and decreases the problem of overﬁtting (Yu *et al.* 2006). SVM is a very good model for solving pattern recognition and classiﬁcation problems that can be applied to regression problems by introducing an unusual loss function. However, finding the final SVM model can be computationally very difficult because it needs to solve a set of non-linear equations (Niazi *et al.* 2008). Thus, the least square support vector machine (LSSVM) is recommended as a modified statement of SVM, which offers a set of linear equations instead of a non-linear programming problem (Niazi *et al.* 2008).

Many researchers have studied the ANN and ANFIS models for estimating ET_{o} (e.g., Kumar *et al.* 2002; Trajkovic 2005; Kişi & Öztürk 2007; Kim & Kim 2008; Traore *et al.* 2010). Recently, wide application of the SVM model has been reported in hydrological engineering (Pai & Hong 2007; Behzad *et al.* 2009; Khemchandani & Chandra 2009; Misra *et al.* 2009; Kalteh 2013). A few past studies have used the LSSVM model for estimating ET_{o}. Kisi & Cimen (2009), Torres *et al.* (2011), Kisi (2013), Samui & Dixon (2012), and Tezel & Buyukyildiz (2015) investigated the accuracy of SVM and ANN models for predicting ET_{o} and evaporation, and their comparison results revealed that the SVM could be successfully used in modeling the ET_{o} process. Samui (2011) used the regression model of LSSVM for prediction evaporation losses in reservoirs. The results showed the LSSVM as a robust model for evaporation prediction from a reservoir. Tabari *et al.* (2013) compared the ANFIS and SVM models and empirical equations including Blaney–Criddle, Makkink, Turc, Priestley–Taylor, Hargreaves & Ritchie for estimating crop evapotranspiration (ET_{c}) of potato when weather or lysimeter data were not complete for applying the FAO method. The results confirmed that the SVM and ANFIS models could provide more accurate ET_{c} estimates. Shrestha & Shukla (2015) used SVM and ANN models for predicting crop coefficient (K_{c}) and ET_{c} of bell pepper and watermelon using lysimeter dataset. The SVM model was superior to ANN and the improved accuracy of the SVM model makes it useful for deriving K_{c} and ET_{c} using available hydro-climatic data. Mehdizadeh *et al.* (2017) investigated the SVM for estimating ET_{o} in Iran. The inputs' selection for the model was conducted based on the parameters used in 16 empirical equations. The performance of the SVM was better than the used empirical equations.

ET is a non-linear and complex process because the estimation and calculation of this parameter requires a large number of meteorological variables, such as maximum air temperature, minimum air temperature, dew point temperature, relative humidity, solar radiation, wind speed, etc. As selecting false variables can prevent achieving the optimal solution in the simulation model, the proper selection of input variables is a challenging and vital problem. There are several methods for reducing the number of input variables and selecting effective variables. In this study, the gamma test (GT) was used as an advanced method for optimal selection of input variables. The GT estimates the minimum mean square error (MSE) that can be acquired when modeling unseen data using any continuous non-linear models. The ability of GT was evaluated for selecting variables to make suitable non-linear models for estimating radiation. The number of data needed to build a reliable model was determined by M-test (Remesan *et al.* 2008). In this research, the GT model was used for selection of the best input combination from climate variables that have the most effect on daily ET_{o}.

Basically, the LSSVM model is a type of soft computing technique that has recently obtained importance in different applications such as ET_{o} estimation. The application of hybrid models for ET_{o} estimation has gained great popularity but input selection techniques hybridizing with machine learning is missing. As a consequence, in this research, a new model is developed to estimate daily ET_{o} by hybridizing the LSSVM and GT models. Thus, the main objectives of this study are: (1) to examine Blaney–Criddle and Hargreaves–Samani equations against the FAO-56 PM as the reference equation using weather data from Zahedan synoptic station located in an arid climate in Iran; (2) selecting the best and optimal input combinations for the LSSVM, ANN, and ANFIS models using the GT method; (3) to investigate the capability of the LSSVM model with different kernels including RBF, linear, and polynomial for modeling ET_{o}; (4) to evaluate the performance of ANFIS and ANN models to predict daily ET_{o} in an arid area; and (5) to compare the performances of LSSVM, ANFIS, ANN, and climate-based models.

## MATERIALS AND METHODS

### Model parameter selection using gamma test

*f*is a smooth function with restricted gradient, and

*r*is a stochastic variable that represents the noise with mean zero. Even though

*f*is unknown, the gamma test computes an approximation for the var(

*r*) from the dataset (Jones

*et al.*2002).

The gamma test estimates what percentage of the *y* variance is caused by the stochastic variable *r*, and what percentage is caused by unknown function *f*. The gamma test is described based on the distance between two points. If two points are close in input space, then their corresponding outputs should be close together in the output space. Otherwise, the difference between output distances is considered as a noise (Karimaldini *et al.* 2012).

*p*nearest neighbors to

^{th}*x*, then delta function can be written (Durrant 2001) as: where

_{i}*δ*(

*p*) is the mean square distance to the

*p*nearest neighbors of the input vectors, is value for each vector

^{th}*x*, |…| denotes Euclidean distance, and

_{i}*p*is number of near neighbors that is fixed and restricted, usually

*p*≈ 10 (Jones 2004).

*δ*(

_{M}*k*),

*γ*(

_{M}*k*)) (Jones

*et al.*2002):

*δ*is zero) describe the gamma statistic value denoted by Γ (Moghaddamnia

*et al.*2009). If Γ is small, the function

*f*exists and the output value

*y*is largely determined by the input variables and which demonstrates a powerful relation between inputs and output. If Γ is large,

*y*is primarily the effect of stochastic variation and the inputs are irrelevant to the output. This means the ability for prediction is limited by one of four conditions: (1) not measuring some important input variable, (2) noise is being created due to measurement error, (3) there are not enough data to model a complex curve, or (4) there are discontinuities in the underlying causal function. The line gradient (

*A*) proposes a simple estimate from the model's complexity (Jones

*et al.*2002). The results can be standardized by considering

*V*that is defined as: where is the output variance of

_{ratio}*y*. A

*V*close to zero shows there is a high degree of predictability of the given output

_{ratio}*y*and

*V*close to 1 indicates the output y is stochastic. The standard error in regression line can be useful as an indicator of the reliability of gamma statistic for estimating Var(

_{ratio}*r*) (Jones

*et al.*2002).

### Least square support vector machine

The LSSVM was introduced by Suykens & Vandewalle (1999). The LSSVM formulation has the same constraints as the SVM model but it performs better than the SVM model computationally. In this case, training needs to solve a set of linear functions instead of solving the quadratic programming problem of the classical SVM model (Khemchandani & Chandra 2009). The LSSVM model effectively reduces the complexity of the algorithm and uses all training data for solving the optimization problem (Suykens & Vandewalle 2000).

*x*(climate variables) and output

_{i}*y*(ET

_{i}_{o}) time series. Vapnik (1995) showed that input vector

*x*can be mapped into a feature space with higher dimension by a non-linear function

*ф*(

*x*) and inner products. According to the LSSVM model, the non-linear function can be represented as: where

*f*indicates the relationship between the climatic variables and ET

_{o},

*w*is the m-dimensional weight vector,

*φ*is the mapping function that maps

*x*into the m-dimensional feature vector, and

*b*is the bias term (Shu-gang

*et al.*2008).

*e*is error vector for

_{i}*x*and

_{i}*γ*is positive constant which determines the degree of penalized losses when a training error occurs (regularization constant parameter) (Yu

*et al.*2006).

The procedure of the LSSVM regression algorithm is illustrated in Figure 1.

*γ*

*>*0, the LSSVM function is given as follows (Khemchandani & Chandra 2009): where

*k*(

*x*,

*x*) is a kernel function. In this case, the non-linear RBF, polynomial, and linear kernels were used and are defined as: where

_{i}*σ*,

*c*, and

*d*are the kernel function parameters for RBF and polynomial kernels.

### Artificial neural networks

*et al.*2011). During the learning process, the weights and the neural biases are iteratively adjusted to minimize errors (Kisi

*et al.*2012). In this study, a model based on a feed forward neural network with a single hidden layer was used for designing ANN networks. A typical three-layer feed forward ANN is given in Figure 2 and Equation (16) as: where

*I*is the input value to node

_{i}*i*of the input layer,

*V*is the hidden value to node

_{j}*j*of the hidden layer,

*O*is the output at the node

_{k}*k*of the output layer, and

*g*

_{2}is the activation function for the output layer. The

*w*controls the strength of the connection between the input nodes

_{ji}*i*and the hidden node

*j*, and the

*w*controls the strength of the connection between the hidden node

_{jk}*j*and the output node

*k*.

*w*,

_{ji}*w*) of the networks are learned through training in which large numbers of input–output pairs are introduced to the network. In a repetitive process, the weights are corrected, the input pattern is introduced to the network, the output

_{kj}*z*is made as prediction, and then predicted output is compared to the target output

_{pk}*t*. The total error

_{pk}*E*, based on the squared difference between predicted and target outputs for pattern

_{p}*p*, is computed as (Riahi-Madvar

*et al.*2011): where

*v*is the error of

_{pk}*k*output from

*p*data pattern. In this study, input vector is climatology parameters optimized by GT and the output vector is the ET

_{o}predictions.

### Adaptive neuro fuzzy inference system

*et al.*2009). Figure 3 illustrates a Sugeno fuzzy and an ANFIS system with two inputs (

*x*), one output (

*y*), and two rules as:

*x*is input value to

*i*node, and

*c*,

_{i}*b*, and

_{i}*a*are membership function parameters of this set which are usually called ‘if’ parameters.

_{i}In this model, the main training algorithm is error back-propagation. By using a gradient descent algorithm, error signals are propagated towards the input layers and nodes and model parameters adopted (Riahi-Madvar *et al.* 2009). Gaussian membership function was used for designing ANFIS networks. The number of membership functions for each variable was determined through trial and error.

### Empirical equations of ET_{o} estimation

_{o}estimation (Allen

*et al.*1998). In addition, numerous researchers in the world have accepted this equation as the most precise model for calculating ET

_{o}. The ET

_{o}values of Zahedan station were calculated using the FAO-56 PM method with the following equation: where

*ET*is reference evapotranspiration (mm day

_{o}^{−1}),

*D*is slope of the saturation vapor pressure function (kPa °C

^{−1}),

*R*is net radiation (MJ m

_{n}^{−2}day

^{−1}),

*G*is soil heat flux density (MJ m

^{−2}day

^{−1}),

*c*is psychometric constant (kPa °C

^{−1}),

*T*is mean air temperature (°C

^{−1}),

*U*

_{2}is average wind speed at 2 m height (m s

^{−1}),

*e*is the saturation vapor pressure (kPa), and

_{s}*e*is the actual vapor pressure (kPa).

_{a}_{o}in this study. The Blaney–Criddle equation, as described in the FAO 24 manual (Doorenbos & Pruitt 1977) is: where

*T*is the mean air temperature (°C),

_{mean}*P*is the mean annual percentage of daytime hours that can be obtained from Doorenbos & Pruitt (1977), and

*a*and

*b*are the equation parameters.

*R*is water equivalent of extraterrestrial radiation (mm day

_{a}^{−1}).

_{o}computed by FAO-56 PM and empirical equations as Equation (28). Empirical equations were calibrated using this regression equation: where

*Y*is ET

_{o}computed by FAO-56 PM (mm day

^{−1}),

*X*is ET

_{o}computed by empirical equations (mm day

^{−1}),

*a*is slope, and

*b*is intercept.

### Evaluation criteria

*R*

^{2}), mean absolute error (MAE), and root mean square error (RMSE). The MAE and RMSE measure the degree of fitness at the average ET

_{o}. where

*N*is the number of data,

*P*is predicted value,

_{i}*O*is observed value, is the average of predicted values, and is the average of observed values.

_{i}*et al.*2001), were used. The AARE and TS not only provide the performance index in terms of predicting values but also these parameters show the distribution of the prediction errors (Nayak

*et al.*2005). These parameters can be computed as: where

*RE*is the relative error in predicted values at time t(%). The better performance of models produces the smaller values of AARE.

_{t}*x*% is a measure of the consistency in predicting errors from a particular model. The threshold statistics are represented as the percentage. It is computed for

*x*% level as (Nayak

*et al.*2005): where

*Y*is the number of computed ET

_{x}_{o}for which the absolute relative error is less than

*x*% of the model.

### Case study

_{o}estimation models have stated that the effective variables on ET

_{o}are minimum temperature (

*T*), maximum temperature (

_{min}*T*), dew point temperature (

_{max}*T*), wind speed (

_{dew}*U*

_{2}), mean relative humidity (

*RH*), sunshine hours (

*n*), and precipitation (

*P*). Initial survey and preprocessing of analysis showed that there were data gaps before 1982, so the data from 1982 to 2003 were used. Before developing the models, the original dataset is normalized so that all of the data are distributed in the range of [min, max] = [0.1, 0.9] (Equation (35)). This can be done by rescaling the values in such a way that the smallest and largest values become 0.9 and 0.1, respectively (Wang

*et al.*2014; Lee

*et al.*2016). The construction function of normalization is: where

*x*is normalized data,

_{in}*x*is original data,

_{i}*x*is the minimum value of original data, and

_{min}*x*is the maximum value of original data.

_{max}A precise measurement of meteorological variables is an important issue in ET_{o} studies. Thus, it is necessary to investigate the accuracy of meteorological data. In this paper, evaluation of meteorological data was conducted using recommendations in guidelines of FAO-56 PM (Allen *et al.* 1998) and ASCE reports (Allen 1996).

The FAO-56 PM method was proposed as the standard method for estimating of ET_{o} at an international level. This model is used for evaluating the results of other models when lysimeter measured data are not available (Terzi *et al.* 2006; Kişi & Öztürk 2007; Zanetti *et al.* 2007; Jain *et al.* 2008; Doğan 2009; Marti *et al.* 2010; Traore *et al.* 2010; Kisi 2013). Similarly, the FAO-56 PM model was used for evaluating the results of LSSVM, ANN, and ANFIS models in this study.

### Modeling framework

In this study, six climate variables that have been measured at the Meteorological Organization of Iran, including *T _{min}*,

*T*,

_{max}*T*,

_{dew}*RH*,

_{mean}*n*, and

*U*

_{2}are used for training and testing of LSSVM, ANN, and ANFIS models. Combining these variables creates 63 different combinations of input variables, and 30 additional combinations are also created by using solar radiation (

*R*) instead of sunshine hours, for example, I

_{s}_{1}:

*ET*

_{o}*=*

*f*(

*T*),

_{min}*…*,

*I*

_{63}:

*ET*

_{o}*=*

*f*(

*T*,

_{min}*T*,

_{max}*RH*,

_{mean}*n*,

*u*

_{2},

*T*).

_{dew}In previous studies on ET_{o}, a trial and error approach was used for input variable selection. Using trial and error approach for modeling these 93 combinations is time-consuming. On the other hand, there is not any practical guidance in the literature about input vectors that must be used to develop robust expert models for ET_{o} predictions. Due to this shortcoming in input vector selection of ET_{o}, in this study, the GT technique was used to determine the best input vector required for developing non-linear models for ET_{o} estimation, and to find the most important variables that effect ET_{o} estimation. Indeed, in this study, the new method of GT is combined with expert models and hybrid prediction models are developed. MATLAB software was used for developing computational programs and learning and simulating algorithms.

In this study, three kernel functions of linear, radial basic function (RBF), and polynomial were used for the LSSVM model. These functions have *γ* and *σ*^{2} calibration parameters, where their values should be determined during the calibration of the model to achieve the maximum performance of the LSSVM model. The *γ* parameter specifies the trade-off between the fitting error minimization and the smoothness of the estimated function. These parameters do not have specific values predetermined and therefore should be determined separately for each combination. For this purpose, an exponential sequences series of these parameters including and were used for each factor. The ten-fold grid search algorithm was used for finding the best ratio between these values. Also, a large number of trials were applied for determining the best parameters of *c* and *d* for polynomial kernel.

The models were trained using all available combinations of coefficients and the combination that causes the least amount of error was selected. The values of regulatory parameters related to optimization problem and kernel functions were introduced with a matrix of input (combinations of meteorological variables) and output (ET_{o} values calculated from FAO-56 PM function) training data and then the bias values were determined. Modeling was performed using selected parameters in the previous stage and input matrix from selected training data to predict the desired output values. By this way, a k-fold algorithm is used not only for parameter optimization of LSSVM, but also for training of the expert models. The flowchart of modeling and methodology is shown in Figure 5.

## RESULTS AND DISCUSSION

The whole dataset was divided into two parts and 75 and 25% of the dataset were selected for training and testing, respectively. The first dataset of 1982–1997 was used for training the models and the remaining data (1998–2003) was utilized to test the models.

### Empirical equations result

In order to evaluate the performance of the climate-based models, the computed ET_{o} values using Blaney–Criddle and Hargreaves–Samani equations are compared by the FAO-56 PM model. The results of the statistical analysis of these empirical equations versus the FAO-56 PM model are given in Table 1. Based on the results, the ET_{o} predicted by Blaney–Criddle (ET_{o(BC)}) is better matched to the calculated ET_{o} by the FAO-56 PM model with lower errors rates (RMSE = 2.03 mm/day and MAE = 1.78 mm/day) than the Hargreaves–Samani (ET_{o(HS)}) model. Thus, the estimation error of the Hargreaves–Samani equation was higher than the Blaney–Criddle equation at this station.

Performance statistics . | Empirical equation . | ||
---|---|---|---|

Blaney–Criddle . | Hargreaves–Samani . | ||

Calibration | RMSE | 2.03 | 2.58 |

MAE | 1.78 | 1.79 | |

Equation | |||

Test | RMSE | 0.49 | 1.23 |

MAE | 0.38 | 1.02 |

Performance statistics . | Empirical equation . | ||
---|---|---|---|

Blaney–Criddle . | Hargreaves–Samani . | ||

Calibration | RMSE | 2.03 | 2.58 |

MAE | 1.78 | 1.79 | |

Equation | |||

Test | RMSE | 0.49 | 1.23 |

MAE | 0.38 | 1.02 |

Valipour *et al.* (2017) showed that the Blaney–Criddle is the best model for estimating the ET_{o} based on FAO-56 PM in arid regions. It can be clearly seen from the statistics given in Table 1 that the accuracy of Blaney–Criddle and Hargreaves–Samani equations increase using calibration with the FAO-56 PM equation. The result of ET_{o} values computed by empirical equations and their calibration equations are given in Figure 6. Also, Figure 7 shows the comparison plots, between the daily estimated ET_{o} values by empirical models and those obtained from the FAO-56 PM model. As seen from Figure 7, two empirical models have a tendency to underestimate the ET_{o(PM)} values in the arid climate of Zahedan. This result has been approved by many researchers such as as Mohawesh (2010), Sabziparvar & Tabari (2010), Raziei & Pereira (2013), and Ngongondo *et al.* (2013) in arid areas.

### GT input selection results

The Spearman correlation matrix between the input variables and output is given in Table 2. The results showed that ET_{o(PM)} was strongly correlated with minimum, maximum, and dew point air temperature and solar radiation variables. When the plants are in a hot air condition, water is released from their open stomata, and transpiration will increase (Crawford *et al.* 2012). The ET_{o} has a very strong correlation with radiation parameters at Zahedan station. Mean relative humidity was negatively and strongly correlated with ET_{o(PM)}. This negative correlation demonstrates that relative humidity has inverse relationships with ET_{o}. It is noticeable that the maximum temperature is negatively correlated with RH_{mean}. According to this, if the maximum temperature increases, the mean relative humidity variable would decrease. This is demonstrated, because at higher temperatures more water is lost from the Earth's surface and from plant cells to the atmosphere due to low humidity in the atmosphere (Edoga & Suzzy 2008). Thus, a low rate of ET_{o} is obtained when the air is cool, cloudy, and humid while the rate of ET_{o} is low in hot, sunny, and dry conditions. The effect of *RH _{mean}* for estimating ET

_{o(PM)}was greater than

*RH*and

_{min}*RH*. Wind speed was also positively but moderately correlated with daily ET

_{max}_{o}. This result may be due to the non-linear effect of wind speed on the daily ET

_{o}and the complex nature of the aerodynamic effects in relation to ET (Vanderlinden

*et al.*2004).

Parameter . | T
. _{min} | T
. _{max} | RH
. _{min} | RH
. _{max} | T
. _{dew} | RH
. _{mean} | U
. _{2} | R
. _{s} | n
. | ET_{o(PM)}
. |
---|---|---|---|---|---|---|---|---|---|---|

T _{min} | 1 | 0.77 | 0.25 | 0.45 | 0.99 | −0.2 | 0.33 | 0.48 | 0.18 | 0.74 |

T _{max} | 0.77 | 1 | −0.16 | 0.27 | 0.75 | −0.51 | 0.01 | 0.69 | 0.48 | 0.84 |

RH _{min} | 0.25 | −0.16 | 1 | 0.8 | 0.28 | 0.34 | 0.17 | −0.17 | −0.32 | −0.08 |

RH _{max} | 0.45 | 0.27 | 0.8 | 1 | 0.49 | 0.14 | 0.02 | 0.1 | −0.1 | 0.21 |

T _{dew} | 0.99 | 0.75 | 0.28 | 0.49 | 1 | −0.16 | 0.33 | 0.45 | 0.15 | 0.72 |

RH _{mean} | −0.2 | −0.51 | 0.34 | 0.14 | −0.16 | 1 | −0.04 | −0.7 | −0.63 | −0.62 |

U_{2} | 0.33 | 0.01 | 0.17 | 0.02 | 0.33 | −0.04 | 1 | 0.05 | −0.07 | 0.41 |

R _{s} | 0.48 | 0.69 | −0.17 | 0.1 | 0.45 | −0.7 | 0.05 | 1 | 0.88 | 0.83 |

n | 0.18 | 0.48 | −0.32 | −0.1 | 0.15 | −0.63 | −0.07 | 0.88 | 1 | 0.6 |

ET_{o(PM)} | 0.74 | 0.84 | −0.08 | 0.21 | 0.72 | −0.62 | 0.41 | 0.83 | 0.6 | 1 |

Parameter . | T
. _{min} | T
. _{max} | RH
. _{min} | RH
. _{max} | T
. _{dew} | RH
. _{mean} | U
. _{2} | R
. _{s} | n
. | ET_{o(PM)}
. |
---|---|---|---|---|---|---|---|---|---|---|

T _{min} | 1 | 0.77 | 0.25 | 0.45 | 0.99 | −0.2 | 0.33 | 0.48 | 0.18 | 0.74 |

T _{max} | 0.77 | 1 | −0.16 | 0.27 | 0.75 | −0.51 | 0.01 | 0.69 | 0.48 | 0.84 |

RH _{min} | 0.25 | −0.16 | 1 | 0.8 | 0.28 | 0.34 | 0.17 | −0.17 | −0.32 | −0.08 |

RH _{max} | 0.45 | 0.27 | 0.8 | 1 | 0.49 | 0.14 | 0.02 | 0.1 | −0.1 | 0.21 |

T _{dew} | 0.99 | 0.75 | 0.28 | 0.49 | 1 | −0.16 | 0.33 | 0.45 | 0.15 | 0.72 |

RH _{mean} | −0.2 | −0.51 | 0.34 | 0.14 | −0.16 | 1 | −0.04 | −0.7 | −0.63 | −0.62 |

U_{2} | 0.33 | 0.01 | 0.17 | 0.02 | 0.33 | −0.04 | 1 | 0.05 | −0.07 | 0.41 |

R _{s} | 0.48 | 0.69 | −0.17 | 0.1 | 0.45 | −0.7 | 0.05 | 1 | 0.88 | 0.83 |

n | 0.18 | 0.48 | −0.32 | −0.1 | 0.15 | −0.63 | −0.07 | 0.88 | 1 | 0.6 |

ET_{o(PM)} | 0.74 | 0.84 | −0.08 | 0.21 | 0.72 | −0.62 | 0.41 | 0.83 | 0.6 | 1 |

Therefore, the results of correlation indicate that the radiation parameters and air temperature are the most important factors influencing the daily ET_{o} at Zahedan station. In addition, the high correlation between climate variables such as (1) *T _{min}*,

*T*,

_{max}*T*, (2)

_{dew}*R*,

_{s}*n*, etc., show that a combination of these parameters can offer a good estimation for ET

_{o}.

In this study, the best combinations of the input dataset were determined with GT to assess their influence on the ET_{o} modeling in an arid area. Different combinations (93 combinations) of input variables were examined and the best combination determined by the lowest of the gamma values. The best combinations that had the smallest gamma values are given in Table 3. These input combinations were evaluated by LSSVM, ANFIS, and ANN models in the present study. In Table 3, the input vectors of *I*_{11} and *I*_{12} include the same climate inputs required for the Blaney–Criddle and Hargreaves–Samani equations, respectively. As shown, the *I*_{1} input vector with parameters of *T _{max}*,

*T*,

_{dew}*RH*,

_{mean}*R*,

_{s}*U*

_{2}had the least values of gamma and

*V*, although its result was similar and near to those of the

_{ratio}*I*

_{2}, …,

*I*

_{7}input vectors. The input vectors

*I*

_{11}and

*I*

_{12}had higher values of gamma and

*V*than the other input vectors. The

_{ratio}*V*, gradient, and standard error of Γ represent the accuracy and complexity of the model that should be developed (Karimaldini

_{ratio}*et al.*2012).

Input vectors . | Γ . | V_{ratio}
. | Gradient . | Standard error . |
---|---|---|---|---|

(I_{1}): T, _{max}T, _{dew}RH, _{mean}R, and _{s}U_{2} | 1.42 × 10^{−3} | 5.69 × 10^{−3} | 0.08 | 1.09 × 10^{−4} |

(I_{2}): T, _{min}T, _{max}T, _{dew}R, and _{s}U_{2} | 1.43 × 10^{−3} | 5.73 × 10^{−3} | 0.08 | 1.18 × 10^{−4} |

(I_{3}): T, _{min}T, _{max}RH, _{mean}R, and _{s}U_{2} | 1.55 × 10^{−3} | 6.19 × 10^{−3} | 0.07 | 0.99 × 10^{−4} |

(I_{4}): T, _{min}T, _{max}T, _{dew}RH, _{mean}R, and _{s}U_{2} | 1.56 × 10^{−3} | 6.34 × 10^{−3} | 0.1 | 1.04 × 10^{−4} |

(I_{5}): T, _{max}T, _{dew}R, and _{s}U_{2} | 1.73 × 10^{−3} | 6.93 × 10^{−3} | 0.11 | 0.7 × 10^{−4} |

(I_{6}): T, _{max}T, _{min}R, and _{s}U_{2} | 1.9 × 10^{−3} | 7.6 × 10^{−3} | 0.11 | 0.7 × 10^{−4} |

(I_{7}): T, _{max}RH, _{mean}R, and _{s}U_{2} | 2.18 × 10^{−3} | 8.72 × 10^{−3} | 0.09 | 0.55 × 10^{−4} |

(I_{8}): T, _{max}R, and _{s}U_{2} | 3.03 × 10^{−3} | 12.12 × 10^{−3} | 0.13 | 0.65 × 10^{−4} |

(I_{9}): T, _{min}T, _{max}RH, _{mean}U_{2}, and n | 7.52 × 10^{−3} | 30.1 × 10^{−3} | 0.08 | 2.93 × 10^{−4} |

(I_{10}): T, _{max}T, _{dew}RH, _{mean}U_{2}, and n | 7.45 × 10^{−3} | 29.8 × 10^{−3} | 0.08 | 2.49 × 10^{−4} |

(I_{11}): T, _{min}T, _{max}R, and _{a}n | 26.15 × 10^{−4} | 104.62 × 10^{−3} | 0.77 | 4.33 × 10^{−4} |

(I_{12}): T, _{min}T, and _{max}R _{a} | 26.34 × 10^{−4} | 105.35 × 10^{−3} | 0.73 | 5.07 × 10^{−4} |

Input vectors . | Γ . | V_{ratio}
. | Gradient . | Standard error . |
---|---|---|---|---|

(I_{1}): T, _{max}T, _{dew}RH, _{mean}R, and _{s}U_{2} | 1.42 × 10^{−3} | 5.69 × 10^{−3} | 0.08 | 1.09 × 10^{−4} |

(I_{2}): T, _{min}T, _{max}T, _{dew}R, and _{s}U_{2} | 1.43 × 10^{−3} | 5.73 × 10^{−3} | 0.08 | 1.18 × 10^{−4} |

(I_{3}): T, _{min}T, _{max}RH, _{mean}R, and _{s}U_{2} | 1.55 × 10^{−3} | 6.19 × 10^{−3} | 0.07 | 0.99 × 10^{−4} |

(I_{4}): T, _{min}T, _{max}T, _{dew}RH, _{mean}R, and _{s}U_{2} | 1.56 × 10^{−3} | 6.34 × 10^{−3} | 0.1 | 1.04 × 10^{−4} |

(I_{5}): T, _{max}T, _{dew}R, and _{s}U_{2} | 1.73 × 10^{−3} | 6.93 × 10^{−3} | 0.11 | 0.7 × 10^{−4} |

(I_{6}): T, _{max}T, _{min}R, and _{s}U_{2} | 1.9 × 10^{−3} | 7.6 × 10^{−3} | 0.11 | 0.7 × 10^{−4} |

(I_{7}): T, _{max}RH, _{mean}R, and _{s}U_{2} | 2.18 × 10^{−3} | 8.72 × 10^{−3} | 0.09 | 0.55 × 10^{−4} |

(I_{8}): T, _{max}R, and _{s}U_{2} | 3.03 × 10^{−3} | 12.12 × 10^{−3} | 0.13 | 0.65 × 10^{−4} |

(I_{9}): T, _{min}T, _{max}RH, _{mean}U_{2}, and n | 7.52 × 10^{−3} | 30.1 × 10^{−3} | 0.08 | 2.93 × 10^{−4} |

(I_{10}): T, _{max}T, _{dew}RH, _{mean}U_{2}, and n | 7.45 × 10^{−3} | 29.8 × 10^{−3} | 0.08 | 2.49 × 10^{−4} |

(I_{11}): T, _{min}T, _{max}R, and _{a}n | 26.15 × 10^{−4} | 104.62 × 10^{−3} | 0.77 | 4.33 × 10^{−4} |

(I_{12}): T, _{min}T, and _{max}R _{a} | 26.34 × 10^{−4} | 105.35 × 10^{−3} | 0.73 | 5.07 × 10^{−4} |

The results of GT confirmed that the *RH _{mean}* variable was more effective than

*RH*and

_{min}*RH*parameters for estimating daily ET

_{max}_{o}at Zahedan station in an arid environment. Although, based on Table 2, the wind speed had moderate correlation with ET

_{o}, but from Table 3, the wind speed is the most effective variable after temperature (without elimination of any

*RH*and

*R*variables) that should be considered in the input vectors of models. Hence, from the GT, the best results could be attained when wind speed and temperature variables are considered in the combination of input vectors. The difference between results of Tables 2 and 3 may be because of the robust non-linear nature of relations among climate variables. Also, by eliminating solar radiation variable from input vectors, the best results of GT were obtained using combinations that include the sunshine hour variable. This is because of the significant relations between solar radiation and sunshine hours variables with correlation coefficient equal to 0.88 (Table 2).

_{s}### LSSVM, ANN, ANFIS hybrid with GT model results

The LSSVM model with three kernels including RBF, polynomial, and linear were implemented by three different program codes written in MATLAB for predicting daily ET_{o}. Also, the ANFIS and ANN models were developed using the same training and test dataset used for the LSSVM model. The architectures of input vectors that were selected from gamma test results (*I*_{1}, *I*_{2}, *…*, *I*_{10}) as the best combinations and two other input vectors with the same variables as Blaney–Criddle and Hargreaves–Samani equations (*I*_{11} and *I*_{12}) were employed using these codes, then the proper model structure was determined for each input vector. The performance criteria summary of all models for estimating ET_{o} in training and testing phases is given in Table 4. Results show that the LSSVM model using RBF kernel (RBF-LSSVM) consistently outperformed the polynomial and linear kernels. The regularization parameter (*γ* = 22.78) and the kernel parameter of RBF (*σ*^{2} = 6.25) were determined by 10 k-fold. The RBF is a robust kernel that is very suitable for limiting the computational training process and modifying the generalization of estimation. The RBF kernel can successfully be used by the LSSVM model for estimating climate-oriented issues (such as evapotranspiration) that are naturally non-linear phenomena (Mirzavand *et al.* 2015).

Input vector . | Model . | Training phase . | Testing phase . | Structure . | ||||
---|---|---|---|---|---|---|---|---|

R^{2}
. | RMSE . | MAE . | R^{2}
. | RMSE . | MAE . | |||

I_{1} | RBF-LSSVM1 | 0.99 | 0.12 | 0.09 | 0.99 | 0.13 | 0.10 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM1 | 0.99 | 0.15 | 0.11 | 0.99 | 0.14 | 0.11 | γ = 8.06 | |

Linear-LSSVM1 | 0.95 | 1.11 | 1.20 | 0.95 | 1.14 | 1.20 | γ = 8 | |

ANFIS1 | 0.99 | 0.49 | 0.52 | 0.99 | 0.26 | 0.30 | NMF = 10 | |

ANN1 | 0.99 | 0.14 | 0.17 | 0.99 | 0.10 | 0.13 | NNHL = 10 | |

I_{2} | RBF-LSSVM2 | 0.99 | 0.12 | 0.09 | 0.99 | 0.13 | 0.10 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM2 | 0.99 | 0.15 | 0.11 | 0.99 | 0.15 | 0.12 | γ = 8.06 | |

Linear-LSSVM2 | 0.95 | 1.08 | 1.17 | 0.95 | 1.14 | 1.19 | γ = 8 | |

ANFIS2 | 0.99 | 0.24 | 0.29 | 0.99 | 0.38 | 0.42 | NMF = 10 | |

ANN2 | 0.99 | 0.09 | 0.13 | 0.99 | 0.10 | 0.13 | NNHL = 12 | |

I_{3} | RBF-LSSVM3 | 0.99 | 0.12 | 0.09 | 0.99 | 0.13 | 0.10 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM3 | 0.99 | 0.15 | 0.11 | 0.99 | 0.14 | 0.11 | γ = 8.06 | |

Linear-LSSVM3 | 0.95 | 1.13 | 1.22 | 0.95 | 1.14 | 1.19 | γ = 8 | |

ANFIS3 | 0.99 | 0.48 | 0.52 | 0.99 | 0.23 | 0.27 | NMF = 10 | |

ANN3 | 0.99 | 0.20 | 0.23 | 0.99 | 0.14 | 0.16 | NNHL = 5 | |

I_{4} | RBF-LSSVM4 | 0.99 | 0.12 | 0.08 | 0.99 | 0.13 | 0.10 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM4 | 0.99 | 0.15 | 0.11 | 0.99 | 0.14 | 0.11 | γ = 8.06 | |

Linear-LSSVM4 | 0.95 | 1.11 | 1.20 | 0.95 | 1.14 | 1.20 | γ = 0.09 | |

ANFIS4 | 0.98 | 0.64 | 0.69 | 0.99 | 0.34 | 0.37 | NMF = 10 | |

ANN4 | 0.99 | 0.12 | 0.15 | 0.99 | 0.09 | 0.12 | NNHL = 14 | |

I_{5} | RBF-LSSVM5 | 0.99 | 0.13 | 0.09 | 0.99 | 0.13 | 0.10 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM5 | 0.99 | 0.16 | 0.12 | 0.99 | 0.15 | 0.12 | γ = 8.06 | |

Linear-LSSVM5 | 0.95 | 1.07 | 1.17 | 0.95 | 1.14 | 1.20 | γ = 8 | |

ANFIS5 | 0.99 | 0.44 | 0.47 | 0.99 | 0.43 | 0.46 | NMF = 10 | |

ANN5 | 0.99 | 0.12 | 0.16 | 0.99 | 0.12 | 0.15 | NNHL = 7 | |

I_{6} | RBF-LSSVM6 | 0.99 | 0.14 | 0.09 | 0.99 | 0.14 | 0.10 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM6 | 0.99 | 0.16 | 0.12 | 0.99 | 0.16 | 0.12 | γ = 8.06 | |

Linear-LSSVM6 | 0.95 | 1.10 | 1.19 | 0.95 | 1.13 | 1.19 | γ = 8 | |

ANFIS6 | 0.99 | 0.19 | 0.25 | 0.99 | 0.17 | 0.21 | NMF = 10 | |

ANN6 | 0.99 | 0.11 | 0.15 | 0.99 | 0.13 | 0.16 | NNHL = 12 | |

I_{7} | RBF-LSSVM7 | 0.99 | 0.16 | 0.11 | 0.99 | 0.17 | 0.13 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM7 | 0.99 | 0.19 | 0.14 | 0.99 | 0.18 | 0.14 | γ = 1.01 | |

Linear-LSSVM7 | 0.95 | 1.10 | 1.20 | 0.95 | 1.20 | 1.26 | γ = 8 | |

ANFIS7 | 0.99 | 0.31 | 0.36 | 0.99 | 0.21 | 0.26 | NMF = 11 | |

ANN7 | 0.99 | 0.27 | 0.31 | 0.99 | 0.18 | 0.21 | NNHL = 15 | |

I_{8} | RBF-LSSVM8 | 0.99 | 0.18 | 0.13 | 0.99 | 0.19 | 0.14 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM8 | 0.98 | 0.21 | 0.15 | 0.99 | 0.20 | 0.14 | γ = 1.01 | |

Linear-LSSVM8 | 0.94 | 1.14 | 1.24 | 0.95 | 1.22 | 1.28 | γ = 8 | |

ANFIS8 | 0.99 | 0.22 | 0.27 | 0.99 | 0.20 | 0.25 | NMF = 10 | |

ANN8 | 0.99 | 0.15 | 0.20 | 0.99 | 0.25 | 0.29 | NNHL = 9 | |

I_{9} | RBF-LSSVM9 | 0.98 | 0.20 | 0.13 | 0.98 | 0.21 | 0.14 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM9 | 0.98 | 0.23 | 0.15 | 0.98 | 0.23 | 0.17 | γ = 1.01 | |

Linear-LSSVM9 | 0.94 | 1.08 | 1.16 | 0.95 | 0.97 | 1.07 | γ = 8 | |

ANFIS9 | 0.97 | 0.28 | 0.35 | 0.98 | 0.19 | 0.28 | NMF = 10 | |

ANN9 | 0.98 | 0.19 | 0.28 | 0.98 | 0.16 | 0.23 | NNHL = 11 | |

I_{10} | RBF-LSSVM10 | 0.98 | 0.20 | 0.13 | 0.98 | 0.21 | 0.14 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM10 | 0.98 | 0.22 | 0.15 | 0.98 | 0.23 | 0.16 | γ = 1.01 | |

Linear-LSSVM10 | 0.94 | 1.12 | 1.19 | 0.95 | 0.95 | 1.05 | γ = 8 | |

ANFIS10 | 0.97 | 0.29 | 0.36 | 0.98 | 0.19 | 0.29 | NMF = 10 | |

ANN10 | 0.98 | 0.25 | 0.32 | 0.98 | 0.14 | 0.23 | NNHL = 9 | |

I_{11} | RBF-LSSVM11 | 0.87 | 0.58 | 0.44 | 0.89 | 0.63 | 0.49 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM11 | 0.86 | 0.60 | 0.47 | 0.89 | 0.63 | 0.50 | γ = 1.01 | |

Linear-LSSVM11 | 0.85 | 1.12 | 1.31 | 0.89 | 1.21 | 1.39 | γ = 0.008 | |

ANFIS11 | 0.87 | 0.97 | 1.17 | 0.89 | 1.17 | 1.39 | NMF = 11 | |

ANN11 | 0.85 | 0.90 | 1.12 | 0.89 | 0.97 | 1.19 | NNHL = 1 | |

I_{12} | RBF-LSSVM12 | 0.86 | 0.58 | 0.45 | 0.89 | 0.63 | 0.49 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM12 | 0.85 | 0.60 | 0.47 | 0.89 | 0.63 | 0.50 | γ = 1.01 | |

Linear-LSSVM12 | 0.84 | 1.21 | 1.40 | 0.88 | 1.29 | 1.46 | γ = 0.02 | |

ANFIS12 | 0.86 | 0.94 | 1.14 | 0.89 | 1.15 | 1.36 | NMF = 10 | |

ANN12 | 0.85 | 0.85 | 1.07 | 0.89 | 0.90 | 1.10 | NNHL = 10 |

Input vector . | Model . | Training phase . | Testing phase . | Structure . | ||||
---|---|---|---|---|---|---|---|---|

R^{2}
. | RMSE . | MAE . | R^{2}
. | RMSE . | MAE . | |||

I_{1} | RBF-LSSVM1 | 0.99 | 0.12 | 0.09 | 0.99 | 0.13 | 0.10 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM1 | 0.99 | 0.15 | 0.11 | 0.99 | 0.14 | 0.11 | γ = 8.06 | |

Linear-LSSVM1 | 0.95 | 1.11 | 1.20 | 0.95 | 1.14 | 1.20 | γ = 8 | |

ANFIS1 | 0.99 | 0.49 | 0.52 | 0.99 | 0.26 | 0.30 | NMF = 10 | |

ANN1 | 0.99 | 0.14 | 0.17 | 0.99 | 0.10 | 0.13 | NNHL = 10 | |

I_{2} | RBF-LSSVM2 | 0.99 | 0.12 | 0.09 | 0.99 | 0.13 | 0.10 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM2 | 0.99 | 0.15 | 0.11 | 0.99 | 0.15 | 0.12 | γ = 8.06 | |

Linear-LSSVM2 | 0.95 | 1.08 | 1.17 | 0.95 | 1.14 | 1.19 | γ = 8 | |

ANFIS2 | 0.99 | 0.24 | 0.29 | 0.99 | 0.38 | 0.42 | NMF = 10 | |

ANN2 | 0.99 | 0.09 | 0.13 | 0.99 | 0.10 | 0.13 | NNHL = 12 | |

I_{3} | RBF-LSSVM3 | 0.99 | 0.12 | 0.09 | 0.99 | 0.13 | 0.10 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM3 | 0.99 | 0.15 | 0.11 | 0.99 | 0.14 | 0.11 | γ = 8.06 | |

Linear-LSSVM3 | 0.95 | 1.13 | 1.22 | 0.95 | 1.14 | 1.19 | γ = 8 | |

ANFIS3 | 0.99 | 0.48 | 0.52 | 0.99 | 0.23 | 0.27 | NMF = 10 | |

ANN3 | 0.99 | 0.20 | 0.23 | 0.99 | 0.14 | 0.16 | NNHL = 5 | |

I_{4} | RBF-LSSVM4 | 0.99 | 0.12 | 0.08 | 0.99 | 0.13 | 0.10 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM4 | 0.99 | 0.15 | 0.11 | 0.99 | 0.14 | 0.11 | γ = 8.06 | |

Linear-LSSVM4 | 0.95 | 1.11 | 1.20 | 0.95 | 1.14 | 1.20 | γ = 0.09 | |

ANFIS4 | 0.98 | 0.64 | 0.69 | 0.99 | 0.34 | 0.37 | NMF = 10 | |

ANN4 | 0.99 | 0.12 | 0.15 | 0.99 | 0.09 | 0.12 | NNHL = 14 | |

I_{5} | RBF-LSSVM5 | 0.99 | 0.13 | 0.09 | 0.99 | 0.13 | 0.10 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM5 | 0.99 | 0.16 | 0.12 | 0.99 | 0.15 | 0.12 | γ = 8.06 | |

Linear-LSSVM5 | 0.95 | 1.07 | 1.17 | 0.95 | 1.14 | 1.20 | γ = 8 | |

ANFIS5 | 0.99 | 0.44 | 0.47 | 0.99 | 0.43 | 0.46 | NMF = 10 | |

ANN5 | 0.99 | 0.12 | 0.16 | 0.99 | 0.12 | 0.15 | NNHL = 7 | |

I_{6} | RBF-LSSVM6 | 0.99 | 0.14 | 0.09 | 0.99 | 0.14 | 0.10 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM6 | 0.99 | 0.16 | 0.12 | 0.99 | 0.16 | 0.12 | γ = 8.06 | |

Linear-LSSVM6 | 0.95 | 1.10 | 1.19 | 0.95 | 1.13 | 1.19 | γ = 8 | |

ANFIS6 | 0.99 | 0.19 | 0.25 | 0.99 | 0.17 | 0.21 | NMF = 10 | |

ANN6 | 0.99 | 0.11 | 0.15 | 0.99 | 0.13 | 0.16 | NNHL = 12 | |

I_{7} | RBF-LSSVM7 | 0.99 | 0.16 | 0.11 | 0.99 | 0.17 | 0.13 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM7 | 0.99 | 0.19 | 0.14 | 0.99 | 0.18 | 0.14 | γ = 1.01 | |

Linear-LSSVM7 | 0.95 | 1.10 | 1.20 | 0.95 | 1.20 | 1.26 | γ = 8 | |

ANFIS7 | 0.99 | 0.31 | 0.36 | 0.99 | 0.21 | 0.26 | NMF = 11 | |

ANN7 | 0.99 | 0.27 | 0.31 | 0.99 | 0.18 | 0.21 | NNHL = 15 | |

I_{8} | RBF-LSSVM8 | 0.99 | 0.18 | 0.13 | 0.99 | 0.19 | 0.14 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM8 | 0.98 | 0.21 | 0.15 | 0.99 | 0.20 | 0.14 | γ = 1.01 | |

Linear-LSSVM8 | 0.94 | 1.14 | 1.24 | 0.95 | 1.22 | 1.28 | γ = 8 | |

ANFIS8 | 0.99 | 0.22 | 0.27 | 0.99 | 0.20 | 0.25 | NMF = 10 | |

ANN8 | 0.99 | 0.15 | 0.20 | 0.99 | 0.25 | 0.29 | NNHL = 9 | |

I_{9} | RBF-LSSVM9 | 0.98 | 0.20 | 0.13 | 0.98 | 0.21 | 0.14 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM9 | 0.98 | 0.23 | 0.15 | 0.98 | 0.23 | 0.17 | γ = 1.01 | |

Linear-LSSVM9 | 0.94 | 1.08 | 1.16 | 0.95 | 0.97 | 1.07 | γ = 8 | |

ANFIS9 | 0.97 | 0.28 | 0.35 | 0.98 | 0.19 | 0.28 | NMF = 10 | |

ANN9 | 0.98 | 0.19 | 0.28 | 0.98 | 0.16 | 0.23 | NNHL = 11 | |

I_{10} | RBF-LSSVM10 | 0.98 | 0.20 | 0.13 | 0.98 | 0.21 | 0.14 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM10 | 0.98 | 0.22 | 0.15 | 0.98 | 0.23 | 0.16 | γ = 1.01 | |

Linear-LSSVM10 | 0.94 | 1.12 | 1.19 | 0.95 | 0.95 | 1.05 | γ = 8 | |

ANFIS10 | 0.97 | 0.29 | 0.36 | 0.98 | 0.19 | 0.29 | NMF = 10 | |

ANN10 | 0.98 | 0.25 | 0.32 | 0.98 | 0.14 | 0.23 | NNHL = 9 | |

I_{11} | RBF-LSSVM11 | 0.87 | 0.58 | 0.44 | 0.89 | 0.63 | 0.49 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM11 | 0.86 | 0.60 | 0.47 | 0.89 | 0.63 | 0.50 | γ = 1.01 | |

Linear-LSSVM11 | 0.85 | 1.12 | 1.31 | 0.89 | 1.21 | 1.39 | γ = 0.008 | |

ANFIS11 | 0.87 | 0.97 | 1.17 | 0.89 | 1.17 | 1.39 | NMF = 11 | |

ANN11 | 0.85 | 0.90 | 1.12 | 0.89 | 0.97 | 1.19 | NNHL = 1 | |

I_{12} | RBF-LSSVM12 | 0.86 | 0.58 | 0.45 | 0.89 | 0.63 | 0.49 | γ = 22.78, σ^{2} = 6.25 |

Polynomial-LSSVM12 | 0.85 | 0.60 | 0.47 | 0.89 | 0.63 | 0.50 | γ = 1.01 | |

Linear-LSSVM12 | 0.84 | 1.21 | 1.40 | 0.88 | 1.29 | 1.46 | γ = 0.02 | |

ANFIS12 | 0.86 | 0.94 | 1.14 | 0.89 | 1.15 | 1.36 | NMF = 10 | |

ANN12 | 0.85 | 0.85 | 1.07 | 0.89 | 0.90 | 1.10 | NNHL = 10 |

NMF, number of member functions; NNHL, number of neurons in hidden layer.

As shown in Table 4, the RBF-LSSVM4 model with input variables of maximum, minimum, and dew point air temperature, mean relative humidity, solar radiation, and wind speed had the best performance (RMSE = 0.1 mm day^{−1}, MAE = 0.13 mm day^{−1}, and R^{2} = 0.99) among the LSSVM models. Also, the RBF-LSSVM1, RBF-LSSVM2, RBF-LSSVM3, and RBF-LSSVM5 provided the same results with RBF-LSSVM4 in testing phase. Among the ANFIS models, ANFIS6 with input parameters of maximum and minimum air temperature, solar radiation, and wind speed had the best accuracy with RMSE = 0.17 mm day^{−1}, MAE = 0.21 mm day^{−1}, and R^{2} = 0.99. After that, the ANFIS8 model with RMSE = 0.2 mm day^{−1}, MAE = 0.25 mm day^{−1}, and R^{2} = 0.99 was ranked as the second best model for estimating daily ET_{o}. ANFIS8 includes the three climate variables of maximum air temperature, solar radiation, and wind speed. Also, ANN4 had the best performance (RMSE = 0.09 mm day^{−1}, MAE = 0.12 mm day^{−1}, and R^{2} = 0.99) among ANN models for daily ET_{o} estimation and after that ANN2 had better accuracy than the other ANN models (RMSE = 0.1 mm day^{−1}, MAE = 0.13 mm day^{−1}, and R^{2} = 0.99). In total, the RBF-LSSVM4 showed the best performance among the other input vectors and among the other models in training and testing phases. The differences between the results of RBF-LSSVM4 and ANN4 models were small. Therefore, the best estimation of ET_{o} is achieved by all measured climate variables. The daily ET_{o} estimates from RBF-LSSVM4, ANFIS4, and ANN4 models in the testing phase are given in Figure 8 in the form of graph line. It is remarkable that the ET_{o} estimates obtained by the RBF-LSSVM4 model closely follow the corresponding ET_{o(PM)} values. Some underestimates of the ET_{o} values are clearly seen from the ANFIS4 and ANN4 models, but, overall the difference between the models for estimating ET_{o(PM)} by *I*_{4} input vector was small.

Comparison of the ET_{o} values estimated by RBF-LSSVM, ANFIS, and ANN models and the ET_{o} computed by the FAO-56 PM are given in Figures 9–11. Based on the performance statistics, the RBF-LSSVM mostly provided better accuracy for estimating ET_{o} than other models with respect to RMSE, MAE, and R^{2}. All of the RBF-LSSVM models overestimated the daily ET_{o} of FAO-56 PM. This result was reported by Tabari *et al.* (2012) in the semi-arid area of Iran. It is clear from the scatter plots that the four input vectors including RBF-LSSVM4 (RMSE = 0.13 mm day^{−1}, MAE = 0.1 mm day^{−1}, and R^{2} = 0.99), RBF-LSSVM1 (RMSE = 0.1 mm day^{−1}, MAE = 0.13 mm day^{−1}, and R^{2} = 0.99), RBF-LSSVM3 (RMSE = 0.1 mm day^{−1}, MAE = 0.13 mm day^{−1}, and R^{2} = 0.99), and RBF-LSSVM2 (RMSE = 0.1 mm day^{−1}, MAE = 0.13 mm day^{−1}, and R^{2} = 0.99) were better than the other models for ET_{o} estimations. The ET_{o} estimations from these input vectors are closer to the ET_{o} from corresponding FAO-56 PM values than those of the other models and followed the same trend. Thus, elimination of the relative humidity variable from input vectors does not decrease the precision of daily ET_{o} estimation and the performance of models.

Although the best result was attained from RBF-LSSVM4 with all required climate variables for FAO-56 PM, by eliminating mean relative humidity and minimum temperature from the two models RBF-LSSVM6 and RBF-LSSVM8, the performance still remained good. Thus, if all climate variables are not available, the daily ET_{o} can be estimated using the three parameters of *T _{max}*,

*R*, and

_{s}*U*

_{2}with high accuracy in arid areas.

Based on the results, the ANN model has better accuracy than the ANFIS in both training and testing phases for the all input vectors except for *I*_{8} input vector. The results of ANN, ANFIS, polynomial-LSSVM, and linear-LSSVM models are also in good agreement with ET_{o(PM)}, and the performance of models was similar. Therefore, the selection of one model over the others depends upon the available climatic data. Although the linear-LSSVM12 (input climate variables same as Hargreaves–Samani) had the highest error with RMSE = 1.46 mm day^{−1} and MAE = 1.29 mm day^{−1}, and lower R^{2} = 0.88, its performance was good. Overall, the RBF-LSSVM model showed superior performance over the ANN and ANFIS models for predicting daily ET_{o}. Therefore, it seems that the RBF-LSSVM model is very appropriate for modeling non-linear processes like evapotranspiration. The obtained results from this study were in line with studies carried out by Ramedani *et al.* (2014), Kisi (2013), Okkan & Serbes (2012), and Deng *et al.* (2011). It is noticeable that, by using a suitable choice of kernel, the dataset can be separable in the feature space to obtain non-linear algorithms, while the dataset is non-separable in the original input space (Bray & Han 2004).

After evaluation of the overall accuracy of the models, applied models could be ranked as RBF-LSSVM, ANN, polynomial-LSSVM, ANFIS, and linear-LSSVM models, respectively. Comparison of Tables 2 and 4 indicates that the LSSVM, ANFIS, and ANN models with the same input variables with empirical equations (*I*_{11} and *I*_{12}) performed better than the corresponding calibrated Blaney–Criddle and Hargreaves–Samani equations. Therefore, Hargreaves–Samani has the worse accuracy for daily ET_{o} estimation at Zahedan station.

### Error distribution analysis

In addition to calculating the average estimation error, evaluating the distribution of estimation error is important to find the applicability of any model to predict the daily ET_{o}. The error distribution in training and testing phases at different threshold levels for the RBF-LSSVM4, ANFIS4, and ANN4 models are shown in Figure 12. It is clear from Figure 12(a) that about 95% of the predicted values for the best combination of RBF-LSSVM4, ANN4, and ANFIS4 models for training phase had estimation errors of 9.2%, 11.1%, and 64.5%, respectively. Also, Figure 12(b) shows that about 98% of the estimated values for the best input combination of the RBF-LSSVM4, ANN4, and ANFIS4 models for the testing phase had estimation errors of 8.4%, 9.4%, and 24%, respectively. Thus, the AARE statistic shows the potential of RBF-LSSVM4 and ANN4 models in comparison with the ANFIS4 model for estimating ET_{o} from the error distribution viewpoint. The performance of the RBF-LSSVM model obviously shows that this model was more reliable and advantageous for simulating daily ET_{o} than the ANN and ANFIS models. Therefore, the RBF-LSSVM model can be effectively used to modify ET_{o} estimates and so it can improve irrigation water requirement predictions.

Considering all of the predictive models' results in this paper, it is declared that the three models of LSSVM, ANFIS, and ANN can be employed successfully for estimating ET_{o}. The ANNs are universal approximates that are beneficial for finding irregularities within a set of patterns. The ANNs provide an analytical alternative to conventional approaches when the diversity of the dataset is very large or relationships between variables are difficult to explain adequately with conventional techniques (Chang *et al.* 2004; Riahi-Madvar *et al.* 2011). The major capability of the ANFIS model is that it combines the fuzzy logic system power with the numerical power of ANNs in various modeling processes using several rules (Sayed *et al.* 2003). The LSSVM has major flexibility in non-linear relationships of the model, and able to remove non-support vectors from the model which lead to fast training and low computational cost. Therefore, the LSSVM is able to choose the main support vectors from the model in the training process. This allows the model to avoid over-fitting and lead to better generalization of LSSVM performance than ANFIS and ANN models (Zhou *et al.* 2009). In addition, the LSSVM will always find a global minimum by solving a convex optimization problem (Quej *et al.* 2017). Hence, it seems the LSSVM model has a robust theoretical background making it more certain than the ANN and ANFIS models (Noori *et al.* 2015).

## CONCLUSIONS

In this study, the potential of the LSSVM model in comparison with ANFIS and ANN models' hybrid with gamma test for estimating FAO-56 PM reference evapotranspiration was evaluated. Three models were trained and tested using climatic variables' dataset of Zahedan synoptic station, Iran, to estimate ET_{o(PM)}. The results of estimator models were compared with Blaney–Criddle and Hargreaves–Samani equation results. The gamma test was used to select proper combinations of input vectors in non-linear estimator models. The gamma test results specified the main variables that affect ET_{o} and provide them with the best input vectors. These input vectors are consistent with the physical basis of ET_{o} processes.

The results of the gamma test showed that the four input vectors of (1): *T _{min}*,

*T*,

_{max}*T*,

_{dew}*RH*,

_{mean}*R*,

_{s}*U*

_{2}; (2):

*T*,

_{max}*T*,

_{dew}*RH*,

_{mean}*R*,

_{s}*U*

_{2}; (3):

*T*,

_{min}*T*,

_{max}*RH*,

_{mean}*R*,

_{s}*U*

_{2}; and (4):

*T*,

_{min}*T*,

_{max}*T*,

_{dew}*R*,

_{s}*U*

_{2}were better than the other input combinations for estimating ET

_{o}at Zahedan station. These input vectors had smaller gamma values than the other combinations. Based on gamma test results, the air temperature and wind speed are the main variables affecting ET

_{o}processes, where elimination of one of these variables increased the gamma value for input vectors. Thus, adding these variables into input vectors increases the accuracy models in estimating daily ET

_{o}.

The performance of the LSSVM model using radial basis function (RBF) kernel and ANN was superior to the polynomial kernels of LSSVM and ANFIS models. Comparison results of RBF-LSSVM, ANFIS, and ANN models with the empirical equations results showed the superiority of these models for ET_{o} prediction in arid areas. None of the empirical equations have good results and show considerable error in comparison with RBF-LSSVM, ANFIS, and ANN models. In 95% of predicted values by RBF-LSSVM and ANN models, the threshold of relative absolute error was very low – about 8.4% and 9.4%, respectively, but it was high for predicted values by ANFIS – about 24%. The RBF-LSSVM model had 30% error for 100% of predicted values, but ANN and ANFIS models had 39% and 117% errors, respectively. The performance of the RBF-LSSVM model was superior to the ANN and ANFIS models. Since the RBF-LSSVM model worked well in an arid area, it is expected to work well for other climates.

The LSSVM model developed in this study will be useful to accurately quantify the crop water use and develop effective irrigation plans for the conservation of water resources at a farm as well as on a regional scale.

The application of estimator models like ANN, ANFIS, and LSSVM is not common for irrigation scheduling in Iran. The LSSVM model that was developed in this study will be useful to precisely calculate the daily ET_{o} at a farm and regional scale. Hence, this model can be used to develop effective irrigation plans for the protection of water resources.