Machine learning and conventional method for reference evapotranspiration estimation using limited climatic data scenarios

doi:10.21203/rs.3.rs-2002124/v1

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Machine learning and conventional method for reference evapotranspiration estimation using limited climatic data scenarios

https://doi.org/10.21203/rs.3.rs-2002124/v1

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Reference evapotranspiration (ET₀) is one important agrometeorological parameter for hydrological studies and agricultural water management. The ET₀ calculated by the Penman-Monteith - FAO method requires several input data. However, in the Minas Gerais region, the meteorological data are limited. The aim of this study was to evaluate the performance of Artificial Neural Network (ANN), Random Forest (RF), Support Vector Machine (SVM) and Multiple Linear Regression (MLR) to estimate the monthly mean ET₀ with different input data combinations and scenarios. Three scenarios were evaluated: at the state level, where all climatological stations were used (Scenario I - SI) ; and at regional level, where the Minas Gerais state was divided according to the climatic classification of Thornthwaite (Scenario II - SII) and by Köppen (Scenario III - SIII). ANN and RF performed better in ET₀ estimating among the models evaluated in the SI, SII and SIII scenarios with the following data combination: i) latitude, longitude, altitude, month, mean, maximum and minimum temperature, and relative humidity; and ii) latitude, longitude, altitude, month, mean temperature, and relative humidity. Also, the SVM and MLR models are recommended for all scenarios in situations with limited climatic data, where only air temperature and relative humidity data are available. Although dividing into scenarios results in less input data for models training, SII and SIII showed a slightly better result in the southern areas of the Minas Gerais state.

Artificial Neural Network

Random Forest

Support Vector Machine

Multiple Linear Regression

crop water requirements

meteorological data

Evapotranspiration is essential information in agriculture. The information based on evapotranspiration are important in order to estimate crop water requirements, irrigation water requirements and control several hydrological processes (Mattar 2016; Souza et al. 2022; Wen et al. 2015; Yassin et al. 2016). Evapotranspiration (ET) is an agrometeorological parameter that can be measured using the lysimeter or water balance approach. These methods for measuring ET are not always possible to use. The lysimeter and water balance approach are a time-consuming method and needs precisely and carefully planned experiments (Kumar et al. 2002). Therefore, the use of evapotranspiration estimation methods are very important, and for that, an adequate meteorological database is necessary to perform good estimates.

Evapotranspiration concept is related to the transfer rate of water from the soil-plant system to the atmosphere. In this study, we focus on the use of reference evapotranspiration (ET₀), which is related to the rate of water consumption from a reference crop surface (grass or alfafa). The ET₀ can be used on a large area, e.g., climatic classification of a region (Almorox et al. 2015; Yang et al. 2017), or small areas, e.g., obtaining crop water requirements or crop evapotranspiration (ETc) (Ewaid et al. 2019; Xiang et al. 2020). The standard model used today for the reference evapotranspiration estimation is the Penman-Monteith evapotranspiration model. This model is considered more realistic physically, but it requires some additional meteorological variables when compared to other methods (Yang et al. 2017). This dependence on several meteorological variables associated with the limited of weather stations network and interruptions and errors in the database makes it difficult to measure ET₀. Thus, some models are used to estimate ET₀. These models seek less dependence on many weathers inputs and high predictive power.

Among the models used in the literature, this study focused on the following models: artificial neural network (ANN), random forest (RF), support vector machine (SVM), and multiple linear regression (MLR). These models showed a different levels predictive capacity of different meteorological variables and in other fields of science. The ANN, RF, and SVM models can capture complex relationships between input and output data, which makes them powerful models in modeling. These machine learning models have been successfully used to estimate ET₀ with fewer input meteorological data (Ferreira et al. 2019; Ferreira and da Cunha, 2020; Yin et al. 2017). Although the inability of MLR to handle non-linear relationships between dependent and independent variables is evident in some studies, the MLR have been successfully used to estimate ET₀ (Malik et al. 2019; Martí et al. 2011).

Considering the models, the ANN is a promising and effective tool for non-linear modeling and complex time-series. The ANNs architecture is composed of three layers: input, hidden, and output layers and each layer include an array of processing elements (Ferreira et al. 2019; Kumar et al. 2002; Yin et al. 2017). Several papers showed the excellent predictive capacity of the ANN model with different architectures in studies with ET₀ (Ferreira and da Cunha, 2020; Nourani et al. 2019; Sattari et al. 2021). Also, the RF is non-parametric statistical data modeling methods that is decision tree-based. RF is a classification and regression technique that also has been adopted to predict agrometeorological parameter such as ET₀ (Ferreira and Da Cunha, 2020; Feng et al. 2017; Wang et al. 2019b). RF has been found to be a more efficient predicting tool compared to other tools like ANN (Benali et al. 2019; Zhou et al. 2016). The SVM is a supervised machine learning algorithm developed by (Vapnik 2013). The SVM is used for regression, classification, pattern recognition and forecasting. This model has been used in the meteorological variables estimation and shown a high predictive power (Mohammadrezapour et al. 2019; Shiri et al. 2014). Also, the MLR aim at explaining the collinearity between a dependent variable and independent by means of a linear combination of predictors independent variables (more than one). This regression technique has been adopted in several fields of science, including climatology, hydrology, and irrigation, with different performances (Martí et al. 2011).

This paper focuses on ET₀ estimation in the Minas Gerais state, Brazil, using different models. Agriculture is an important role in the Minas Gerais state and the ET₀ is a fundamental variable in the irrigation management, study of water springs and in the agricultural development, because through this variable it is possible to measure the volume of water transferred from a hydrological basin or from a cultivated area to the atmosphere. The presence of gaps or discontinuities in the data series can delay the state development. In this context, this study proposes to analyze the use of different combinations of input data and climate scenarios for the accurate estimation of ET₀, and especially, with the minimum possible use of input data in the models, this can facilitate the estimation of ET₀.

The ET₀ calculated by the FAO Penman-Monteith method requires several input data. This amount of input data makes it difficult to use this method. New technologies can make it easier to obtain ET₀ reliably. In this context, the aim of this study was to develop, evaluate and compare the performance of ANN, RF, SVM and MLR models in estimating ET₀ with four different combinations of input data in three climate scenarios.

2.1 Study area and data sources

The Minas Gerais state is the fourth largest in Brazil, with a territorial extension with 586,513.993 km² (IBGE 2022). The study was performed with data base of Minas Gerais state, Brazil, at the parallels of 14° 13' 58" and 22° 54' 00", of south latitude, and the meridians of 39° 51' 32" and 51° 02' 35" a west of Greenwich. Monthly data from 56 climatological stations of the Brazilian National Institute of Meteorology (INMET) were used. The respective geographical coordinates, altitude and climatic classification have been presented in Table 1.

Table 1

Principal climatological station of the INMET used to estimate ET₀
ID	Local	Lat/Lon/Alt (°/°/m)		Tho
1	Aimorés	-19.49 / -41.07 / 79.93	Aw	D
2	Araçuaí	-16.84 / -42.06 / 317.67	As	D
3	Araxá	-19.6 / -46.94 / 1018.28	Cwb	B2
4	Arinos	-15.91 / -46.1 / 523	Aw	C1
5	Bambuí	-20.03 / -46 / 684.43	Cwa	B2
6	Barbacena	-21.23 / -43.78 / 1128.8	Cwb	B3
7	Belo Horizonte	-19.93 / -43.95 / 915.47	Cwb	B2
8	Bocaiúva	-17.1 / -43.8 / 633	Cwa	C1
9	Bom Despacho	-19.72 / -45.36 / 695	Cwa	B1
10	Caparaó	-20.52 / -41.9 / 836.25	Cwb	B2
11	Capinópolis	-18.72 / -49.56 / 608.98	Aw	C2
12	Caratinga	-19.73 / -42.13 / 609.56	Cwa	C2
13	Conceição do Mato Dentro	-19.02 / -43.43 / 663.02	Cwa	B1
14	Coronel Pacheco	-21.54 / -43.26 / 411.03	Cwa	B2
15	Curvelo	-18.74 / -44.45 / 668.26	Cwa	C1
16	Diamantina	-18.23 / -43.61 / 1318.05	Cwb	B2
17	Divinópolis	-20.17 / -44.87 / 787.42	Cwa	B1
18	Espinosa	-14.91 / -42.8 / 565.52	Cwb	D
19	Florestal	-19.88 / -44.41 / 753.51	Cwa	B2
20	Formoso	-14.94 / -46.23 / 854.6	Aw	C2
21	Frutal	-20.03 / -48.93 / 547.09	Aw	C2
22	Governador Valadares	-18.84 / -41.9 / 156.54	Aw	C1
24	Itamarandiba	-17.85 / -42.85 / 919.37	Cwb	C2
25	Ituiutaba	-18.95 / -49.52 / 540.09	Aw	C2
26	Jaíba	-15.08 / -44.01 / 453.62	As	D
26	Jaíba	-19.49 / -42.54 / 298	As	C2
27	Janaúba	-15.8 / -43.29 / 534.61	As	D
28	Januária	-15.44 / -44.36 / 480	Aw	C1
29	João Monlevade	-19.82 / -43.14 / 859.84	Cwb	B2
30	João Pinheiro	-17.74 / -46.17 / 759.62	Aw	C2
31	Juiz de Fora	-21.77 / -43.36 / 936.9	Cwb	B3
32	Juramento	-16.77 / -43.66 / 655.59	Cwb	C1
33	Lambari	-21.94 / -45.31 / 884.56	Cwb	B3
34	Lavras	-21.22 / -44.97 / 916.19	Cwb	B2
35	Machado	-21.68 / -45.94 / 892.44	Cfb	B2
36	Maria da Fé	-22.31 / -45.37 / 1281.36	Cwb	A
37	Monte azul	-15.16 / -42.86 / 623.22	As	D
38	Montes Claros	-16.68 / -43.84 / 645.87	Cwa	C1
39	Paracatu	-17.24 / -46.88 / 711.41	Aw	C2
40	Patos de Minas	-18.52 / -46.44 / 947.68	Cwa	B1
41	Pedra Azul	-16 / -41.28 / 647.97	As	C1
42	Pirapora	-17.34 / -44.92 / 509.52	Aw	C1
43	Poços de Caldas	-21.91 / -46.38 / 1077.08	Cwb	B3
44	Pompéu	-19.22 / -45 / 692.21	Cwa	C2
45	Salinas	-16.15 / -42.28 / 476.07	As	D
46	São João Del Rei	-21.3 / -44.27 / 991	Cwb	B3
47	São Lourenço	-22.12 / -45.04 / 930.65	Cwb	B3
48	São Sebastião do Paraíso	-20.9 / -47.11 / 820	Cwb	B3
49	Serra Azul de Minas	-20.02 / -44.35 / 765	Cwa	B2
50	Serra dos Aimorés	-17.79 / -40.25 / 211.92	Aw	C1
51	Sete Lagoas	-19.48 / -44.17 / 753.68	Cwa	B1
52	Teófilo Otoni	-17.86 / -41.5 / 349.11	Aw	C1
53	Uberaba	-19.73 / -47.95 / 753.41	Cwa	B2
54	Uberlândia	-18.91 / -48.25 / 874.6	Cwa	B2
55	Unaí	-16.36 / -46.88 / 595.59	Aw	C1
56	Viçosa	-20.76 / -42.86 / 697.53	Cwa	B1
Where: K: Köppen climatic classification; Tho: Thornthwaite climatic classification; Cwb - Humid subtropical with dry winter and temperate summer, Cwa - Humid subtropical with dry winter and hot summer, Cfb - Humid subtropical with oceanic climate without dry season and temperate summer, As - Tropical with dry summer and Aw - Tropical with dry winter; A – super humid, B4 - humid, B3 - humid, B2 - humid, B1 - humid, C2 – sub humid, C3 - dry sub humid, and D - semiarid. Source: The authors (2022).

The average monthly data mean, maximum, and minimum air temperatures (Tmean, Tmax, Tmin), relative humidity (RH), atmospheric pressure (P), wind speed (U₂), and insolation (n) of climatological stations with at least 10 years of flawless data (no missing or faulty data) from a period between 1989 to 2019 (30 years) were used for the studies. Wind speed, measured at 10 m height, was converted to 2 m (Allen et al. 1998). Days with missing or faulty data were removed. Faulty data were identified when Tmin was higher than Tmax or Tmean; Tmean was higher than Tmax; RH out of the range 0–100%; P was higher than 101.4 kPa and U₂ and n were negative.

The reasons for using these variables were: the latitude and longitude are the variables related to the position. The solar radiation intensity changes as the position changes on the terrestrial globe. The altitude variable is regarded as the surface component. It can be stated that the higher the altitude, the lower the temperature. Temperature is the availability of energy in the system, and the relative humidity is the difference in gradient, the lower the humidity, the greater the capacity of the environment to absorb humidity. All these factors can influence evapotranspiration.

The models were developed in three different scenarios (SI, SII and SIII) build seeks to achieve the maximum predictive capacity of each model: SI - At State Level: the models were trained and tested with data from the 56 climatological stations. The resulting model estimates evapotranspiration in any location within the Minas Gerais state; SII - At Regional Level: The Minas Gerais state was divided into two regions according Climatic Classification Systems proposed by Thornthwaite (1948). The region with climate classification A, B4, B3, B2 and B1 (Tho1–27 climatological stations), and the region with climate classification C2, C1 and D (Tho2–29 climatological stations). The models were trained and tested with data from climatological stations of each climatic region (Fig. 1); and SIII - At Regional Level: The Minas Gerais state was divided into two regions. The region with climate classification Cwb, Cwa and Cfb (K1–35 climatological stations) and the region with climate classification Aw and As (K2–21 climatological stations) Climatic Classification Systems proposed by Köppen. The models were trained and tested with data from climatological stations of each climatic region (Fig. 1).

2.2 Penman–Monteith FAO model

The FAO Penman–Monteith equation (FPM) was used to estimate average monthly ET₀. This method is described by Allen et al. (1998). It is a common practice to use ET₀ estimated by the FPM equation as reference data. Climatological stations used in the study do not provide net solar radiation (Rn) data. The Rn data were obtained by means of insolation, latitude, day of the year and other variables. They are calculated using the equations detailed by Allen et al. (1998).

2.3 Model development and statistical tests

In this study, different input combinations of the average monthly data were used as inputs to estimate ET₀. The input data were geographic coordinates, altitude, month, Tmean, Tmax, Tmin and RH. In the search for better performance, the four input combinations (I_n: n is the amount of input data) evaluated in this paper were: (I₈) latitude, longitude, altitude, month, Tmean, Tmax, Tmin, and RH; (I₆) latitude, longitude, altitude, month, Tmean and RH; (I₃) month, Tmean and RH; (I₂) Tmean and RH. The ANN, RF, SVM and MLR model were trained for each combination. The models were developed using data from each climate scenarios. These combinations were compared to each other in each model.

The predictive quality of each models in terms of variation, precision, accuracy, and performance were evaluated by four statistical criteria. The statistical criteria were: the mean absolute error (MAE), root mean square error (RMSE), coefficient of determination (R²) and Pearson’s correlation coefficient (r). The MAE and RMSE indicates revealed how close the predicted values were to the observed value. Thus, the accuracy of each model could be predicted. The R² represents the percentage of the variation of the dependent variable explained by the independent variable. The r indicates the degree of dispersion of the data obtained in terms of the mean.

2.4 Artificial neural networks (ANN)

The ANN has performance characteristics resembling biology of the human brain. ANNs, in general, have an architecture with connections between nodes (neural networks) and methods to determine the connections weight. In this study, the ANN of the feed-forward multilayer perceptron (MLP) type was used (Fausett 1994). The training in this ANN involves two phases. In the first phase or forward pass, the input sign spreads layer-by-layer forward. In the second phase or reverse pass, the sign is backpropagated for the correction of the error.

ANN was implemented using the Waikato Environment for Knowledge Analysis (WEKA; version 3.8.2 © 1999–2017) developed by the University of Waikato, Hamilton, New Zealand. The input data consisted of different combinations of the latitude, longitude, altitude, month, Tmean, Tmax, Tmin and RH of each evaluated location, using ET₀ as the output variable.

All adjustments were by cross-validation. According to Sattari et al. (2021), the cross-validation approach enables successful results. This technique separates the data into two categories, where the first is used to train the model and the second part is processed as test data to determine the model’s performance. After preliminary tests, the architecture with the best performance for each data input combination was obtained (Fig. 2). The ANN different configurations and number of folds in cross-validation are shown in Table 2.

Table 2 WEKA configuration in the ANN implementation

	ANN
	I8	I6	I3	I2
Learning rate	0.3	0.3	0.3	0.3
Momentum	0.2	0.2	0.2	0.2
Number of training epochs	1000	1000	1000	1000
Number of input data	8	6	3	2
Number of hidden layers	2	2	1	1
Number of neurons into the hidden layer	7,7	7,7	7,7	7
Number of folds in cross-validation	18	18	8	8

In italics WEKA default values.

2.5 Support vector machine (SVM)

In this study, it was applied SVM equations based on Vapnik’s theory (Vapnik 2013). SVM are separated into two main categories: (i) the classifier model and (ii) the regression model (SVR). SVR is used to take a hyperplane suitable for the data used. The distance to any point in this hyperplane shows the error of that point (Sattari et al., 2021). SVR can be translated into the following equation:

$$\text{y}=\text{f}\left(\text{x}\right)={\omega } {\phi } \left({\text{x}}_{\text{i}}\right)+\text{b}$$

Where x is the input data; φ(x) represents a function, which can transfer the x into the high-dimensional feature spaces; ω (weight vector) and b are coefficients which are estimated by minimizing the regularized risk function. The error function in SVM model is minimized based on mentioned constraints in equation below. Further details on the application of SVM can be found in Chang and Lin (2012).

$$\text{min }\frac{\text{1}}{\text{2}}\text{ }{‖\text{w}‖}^{\text{2}}\text{+C}\sum _{\text{i=1}}^{\text{N}}\text{(}{\text{ξ}}_{\text{1}}\text{+}{\text{ξ}}_{\text{1}}^{\text{*}}\text{)}$$

$$\text{Subject to }\left\{\begin{array}{c}{\text{y}}_{\text{i}}\text{-}\text{b}\text{ }\left[\text{ω}\text{ }\text{φ}\left({\text{x}}_{\text{i}}\right)\right]\text{ }\text{≤}\text{ }\text{Ɛ}\text{+}\text{ }{\text{ξ}}_{\text{1}}\\ \left[\text{ω}\text{ }\text{φ}\left({\text{X}}_{\text{i}}\right)\right]\text{+}\text{b}\text{-}\text{ }{\text{y}}_{\text{i}}\text{ }\text{≤}\text{ }\text{Ɛ}\text{+}{\text{ξ}}_{\text{1}}^{\text{*}}\\ {\text{ξ}}_{\text{1}}{\text{ξ}}_{\text{1 }}^{\text{*}}\text{≥}\text{0}\end{array}\right.$$

Where, C is the capacity or penalty parameter, y_i is the estimated output by SVM, ξ_i and ξ^*_i are slack variables which must satisfy the function constraints. The SVM model changes the scale of the problem by using Kernel functions to solve non-linear problems. SVM provides four different Kernel functions: sigmoid, linear, polynomial, and radial basis functions. In this study, during SVM modelling, all Kernel functions have been tested. The linear Kernel function proved to be more efficient in estimating ET₀. The linear Kernel function is as follows:

$$\text{K}\left({\text{x}}_{\text{i}}\text{, }{\text{y}}_{\text{i}}\right)\text{= }{\text{x}}_{\text{i}}\text{ }{\text{y}}_{\text{i}}$$

Where x_i and x_j are vectors in the input space. The SVM was implemented by WEKA. The input data consisted of different combinations (I₈, I₆, I₄ and I₂) and were evaluated in the three different scenarios. WEKA configuration parameters in the SVM implementation were: SVM Type, ϵ-SVR and cost parameter C, 0.01. The other WEKA configuration parameters were kept as standard. Eighteen folds of the sample set were used in cross-assessed. The same WEKA configuration parameters were used in all input data combinations and in all scenarios.

2.6 Random forest (RF)

RF is an ensemble learning technique based on a collection of tree predictors (Xu et al. 2014). It is a combination of many predictor trees (forest), in which each tree is generated from a random vector, sampled independently and with the same distribution for all trees in the forest. The operating steps of the RF model were presented in Fig. 3. According to Wang et al. (2019b), there are three simple steps to building an RF model: (i) Build n bootstrap samples from the original data; (ii) build an unpruned regression tree; (iii) and predict new data by aggregating the predictions of the n.

Further details can be found in Wang et al. (2019a) and Wang et al. (2019b).

RF was implemented by the WEKA. The WEKA configuration that resulted in the greatest predictive capacity was: 100 is the size bag, as percentage of the training set size and 500 iterations. The other WEKA configuration parameters were kept as standard. All adjustments were by cross-validation and twenty folds of the sample set were used. The same WEKA configuration parameters were used in all input data combinations and in all scenarios.

2.7 Multiple linear regression (MLR)

MLR was developed to estimate the ET₀ based on different combinations of the independent variables. The base regression equation can be expressed as:

$${\text{Y}}_{\text{i}}\text{=}\text{ }{\text{β}}_{\text{0}}\text{+}\text{ }{\text{β}}_{\text{1}}\text{lat}\text{ }\text{+}\text{ }{\text{β}}_{\text{2}}\text{lon}\text{ }\text{+}\text{ }{\text{β}}_{\text{3}}\text{alt}\text{ }\text{+}{\text{ }\text{β}}_{\text{4}}\text{month}\text{ }\text{+}{\text{ }\text{β}}_{\text{5}}\text{Tmax}\text{ }\text{+}\text{ }{\text{β}}_{\text{6}}\text{Tmean}\text{ }\text{+}{\text{ }\text{β}}_{\text{7}}\text{Tmin}\text{ }\text{+}{\text{ }\text{β}}_{\text{8}}\text{RH}$$

Where Y_i is the dependent variable (ET₀); lat, lon, alt, month, Tmean, Tmax, Tmin and RH are independent variables; and β₀, β₁, β₂, β₃, β₄, β₅, β₆, β₇ and β₈ are the regression coefficients.

MLR was implemented using the WEKA. The attribute selection method used in the WEKA configuration was M5 method. This method initially builds the MLR model with all independent variables. Then, independent variables with the smallest standardized coefficient are step-wisely removed until no improvement is observed in the estimate of the error given by the Akaike information criterion (AIC). The AIC seeks the best model in terms of complexity and performance. This technique evaluates different models relative to each other, therefore, by adding the more parameters, the AIC of the model may present inadequate performance (Samadianfard et al. 2018). The other WEKA configuration parameters were kept as standard. Eighteen folds of the sample set were used in cross-assessed. The same WEKA configuration parameters were used in all input data combinations and in all scenarios.

The results presented in this study are essential for a more adequate water management, since the accurate estimation of ET₀ is fundamental for the water demand quantification. Furthermore, the use of different estimation techniques and the combination of input data in the models allowed us to obtain important results applied at different spatial scales.

According the results, it was possible to observe linear correlation between the input data and the ET₀, with the variables Tmean, Tmax, and Tmin showing the best correlation (Fig. 4). The other variables have low (lat, alt and RH) or no (lon and month) correlation with ET₀. The inversely proportional behavior with ET₀was observed in the lat, alt and RH variable. High latitudes tend to be cooler regions, with less energy available for the ET₀process. Also, an increase in the altitude results in a decrease of the temperature according to the vertical thermal gradient in the troposphere. The increase in RH increases the potential gradient, increasing the water transfer rate from the soil-plant system to the atmosphere. However, a proportional behavior was observed between the Tmean, Tmax and Tmin variables with ET₀. The increase in Tmean, Tmax and Tmin results in more energy available for ET₀. Sattari et al. (2021) observed the same behavior of the variables Tmean, Tmax, Tmin and RH when estimating ET₀. The variables Tmean, Tmax, and Tmin were all highly correlated with ET₀and the RH mean was the least correlated variable.

In this way, the ability of machine learning approaches with the variables mentioned above in different conditions and scenarios was investigated. The ANN, RF, SVM and MLR statistical performance indicators for estimating ET₀in any location within the Minas Gerais state (SI: data from the 56 climatological stations - 100% of the input data available) is presented in Table 3.

Table 3 Statistical performance indicators of the ANN, RF, SVM and MLR models in SI

I₈

I₆

I₃

I₂

MAE

RMSE

MAE

RMSE

MAE

RMSE

MAE

RMSE

ANN

0.966

0.167

0.215

0.963

0.178

0.224

0.943

0.210

0.278

0.860

0.332

0.429

0.955

0.191

0.250

0.966

0.166

0.220

0.934

0.220

0.296

0.859

0.335

0.426

SVM

0.933

0.23

0.290

0.927

0.242

0.310

0.878

0.311

0.399

0.877

0.312

0.399

MLR

0.933

0.231

0.298

0.928

0.241

0.308

0.877

0.313

0.399

0.877

0.312

0.398

Value in bold indicates the best result within each model; value in italics indicates the best result within input data combination. Combination data: (I8) latitude, longitude, altitude, month, Tmean, Tmax, Tmin, and UR; (I6) latitude, longitude, altitude, month, Tmean and RH; (I3) month, Tmean and UR; and (I2) Tmean and RH.

All the models developed with the I₈ and I₆ input combination exhibited better performances than their versions developed with the I₃ and I₂. The lowest predictive capacity was observed when the RF model was used with the I₈ input combination. The greatest predictive capacity, in SI, was observed when the RF and ANN models was used with the I₆ and I₈ input combination, respectively. The models SVM and MLR exhibited better performances than ANN and RF when only Tmean and RHmean (I₂) were used as input data.

When comparing the combination I₈ with I₆, the average r, MAE and RMSE of all models does not show high variations. The removal of the geographic coordinate (I₆ to I₃) resulted in a more expressive performance reduction of the SVM and MLR models. The highest impact on performance was observed in the ANN and RF when the month variable was removed (I₃ to I₂). The average r decreased by 8%; MAE and RMSE increased by 52.2% and 43.9%, respectively. The removal of month did not impact the SVM and MLR models performance.

In the case of the SII, a scenario in which the state of Minas Gerais was divided into two areas (Tho1 and Tho2), the statistical performance indicators of the models used in the ET₀ estimation are shown in Table 4.

Table 4 Statistical performance indicators of the ANN, RF, SVM and MLR models in SII

	SII
	Tho.1 (A, B4, B3, B2 and B1)
	I₈			I₆			I₃			I₂
	r	MAE	RMSE	r	MAE	RMSE	r	MAE	RMSE	r	MAE	RMSE
ANN	*0.976*	*0.135*	*0.168*	0.965	0.156	0.196	0.959	0.169	0.212	0.904	0.266	0.322
RF	0.964	0.164	0.198	0.973	0.143	0.174	0.963	0.16	0.198	0.920	0.234	0.291
SVM	0.955	0.181	0.219	0.948	0.191	0.235	0.925	0.232	0.281	0.924	0.235	0.284
MLR	0.957	0.178	0.216	0.949	0.190	0.233	0.927	0.23	0.278	0.925	0.233	0.282
	Tho.2 (C2, C1 and D)
	I₈			I₆			I₃			I₂
	r	MAE	RMSE	r	MAE	RMSE	r	MAE	RMSE	r	MAE	RMSE
ANN	0.940	0.211	0.269	*0.944*	*0.210*	*0.260*	0.895	0.269	0.353	0.818	0.377	0.462
RF	0.925	0.227	0.302	0.943	0.200	0.267	0.879	0.29	0.374	0.817	0.350	0.453
SVM	0.893	0.276	0.353	0.898	0.271	0.346	0.840	0.342	0.427	0.840	0.340	0.427
MLR	0.898	0.269	0.345	0.899	0.267	0.342	0.839	0.339	0.427	0.839	0.339	0.427

Tho1 and Tho2 had 48.2% and 51.8%, respectively, of the data available as input data. The highest predictive capacity in Tho1 and Tho2 area was observed when the ANN model was used with the I₈ input combination and RF model was used with the I₆ input combination, respectively. The removal of the Tmax and Tmin input data (I₆) did not increase the model’s predictive capacity in Tho1 area, except for the RF model. This behavior is similar to that observed in SI. However, all models performed better when the I₆ input combination in Tho2 area was used (better results).

The removal of the month variable (I₃ to I₂) resulted in the highest impact on the ANN and RF models quality. When comparing the combination I₈ with I₃, the average r of the ANN and RF models decreased by 7.2% and 5.7%, respectively. The MAE of the ANN and RF models increased by 36.4% and 31.6%, respectively. However, no expressive variation was observed in the performance of SVM and MLR models.

In the case of SIII, the statistical performance indicators of models for the scenario presented by Table 5, were the Minas Gerais state was divided in K1 and K2 area, which was characterized as 62.5% and 37.5% of the climatological stations, respectively.

Table 5 Statistical performance indicators of the ANN, RF, SVM and MLR models in SIII

	SIII
	K1 (Cwa, Cwb and Cfb)
	I₈			I₆			I₃			I₂
	r	MAE	RMSE	r	MAE	RMSE	r	MAE	RMSE	r	MAE	RMSE
ANN	0.966	0.170	0.209	0.968	0.163	0.204	0.961	0.180	0.225	0.912	0.270	0.334
RF	0.963	0.175	0.221	*0.973*	*0.15*	*0.191*	0.962	0.174	0.222	0.920	0.261	0.318
SVM	0.949	0.199	0.256	0.944	0.209	0.267	0.927	0.247	0.305	0.926	0.248	0.306
MLR	0.950	0.204	0.253	0.945	0.212	0.266	0.928	0.245	0.303	0.917	0.247	0.305
	K2 (Am and Aw)
	I₈			I₆			I₃			I₂
	r	MAE	RMSE	r	MAE	RMSE	r	MAE	RMSE	r	MAE	RMSE
ANN	*0.964*	*0.16*	*0.201*	0.889	0.269	0.350	0.895	0.263	0.340	0.817	0.360	0.447
RF	0.924	0.23	0.294	0.943	0.203	0.258	0.885	0.285	0.352	0.826	0.347	0.429
SVM	0.889	0.270	0.347	0.890	0.274	0.347	0.846	0.329	0.405	0.847	0.326	0.403
MLR	0.892	0.269	0.343	0.894	0.269	0.340	0.846	0.325	0.404	0.848	0.323	0.403

In general, the ANN and RF models were higher than the SVM and RLM models with the input combinations I₈, I₆ and I₃. When the I₂ combination was used, the SVM and RLM models were superior. The model with highest predictive capacity in K1 area was the ANN with the I₈ input combination. The RF model with the I₆ input combination showed highest predictive capacity in K2 area.

In the K1 area, the removal of the month variable resulted in the highest impact on the ANN and RF models performance. The removal of the alt, lat and lon variable resulted in the highest impact on the SVM and MLR performance. In the K2 area, the behavior of the RF, SVM and MLR were similar to that observed in the K1 area. However, the withdrawal of the alt, lat and lon variable resulted in the highest impact ANN in the area K2.

The ANN and RF models showed greater predictive capacity in all scenarios when compared to the SVM and MLR models. This high capacity is achieved with the data input combination I₈ or I₆. Both models had similar performances, but on average the RF showed slight superiority. Ferreira et al. (2019) and Ferreira and Da Cunha (2020) conducted studies aimed at evaluating the performance of different machine learning in the ET₀estimation in Brazil. In these studies, it was observed that, in general, ANN performed slightly better than the other traditional machine learning models (i.e., RF and Extreme gradient boosting - XGBoost). However, in some studies the RF model performed slightly better than other models (i.e., Generalized Regression Neural Networks - GRNN) to estimating ET₀ (Feng et al. 2017; Wang et al. 2019b). There are papers suggesting better performance than other machine learning models in different situations and regions (Mehdizadeh et al. 2017; Shiri et al. 2014). Therefore, there is a need for studies that address more than one models.

The SVM and MLR models showed similar statistical indices and behavior in all scenarios. These results can be explained by the use of the linear Kernel function by SVM that probably presented behavior similar to an MLR. Tests with the nonlinear Kernel function did not result in improvements in prediction. According to Pisner and Schnyer (2020), this SVM is used to recognize patterns in complex databases. Possibly the data used does not present a complexity that justifies the use of SVM.

The SVM and MLR models showed a greater predictive capacity in all scenarios when the input data limited to only Tmean and RH (I₂). This result may indicate a low predictive capacity of the ANN and RF models in situations of low variability in the input data. This low variability may hinder the search for patterns that justify variations in ET₀.

In some scenarios the remotion of Tmax and Tmin improved the ET₀ estimation results. Sattari et al. (2021) observed an increase in the accuracy of the support vector regression (SVR) and Gaussian process regression (GPR) models with the removal of some input data, including Tmax and Tmin.

Although the Tmax and Tmin showed a good correlation with ET₀ (Fig. 4), the weight of Tmax and Tmin is diluted in the calculation of the Tmean used in the calculation of ET₀. Thus, adding Tmax and Tmin can make the ET₀estimate more complex or confusing. This fact can decrease the accuracy of the models, and the removal of this input data can improve the prediction. Determining the input data is critical to the success of the models. This selection can facilitate the training and testing processes, improving the understanding of the system (Bowden et al. 2005; Maier and Dandy 2000). However, this result shows that only linear regression is not enough to decide which input data should be removed in order to increase the predictive performance.

When the independent variables lat, lon and alt were removed (I₃), a reduction in the statistical indexes of all models was observed. These variables are related to the spatial location of the observed data. Although the correlation observed between these variables and ET₀is low (Fig. 4), the joint removal of these data negatively impacted the model’s performance. According to Mehdizadeh et al. (2017) and Souza et al. (2022), air temperature and solar radiation are one of the main impact data on ET₀. Several studies have indicated the influence of lat, lon and alt variables on air temperature and solar radiation (Alvares et al. 2013b; Ozgoren et al. 2012) Thus, variations in lat, lon and alt may indirectly impact ET₀. This can explain these observed results.

The division of the input data into two areas with climatic similarity aimed to increase the performance of the models. The division presented in SII and SIII managed to slightly increase the capacities of the models in relation to SI. However, this increase was only observed in the Tho1 and K1 areas. Thus, we can infer that, although the division into areas with climatic similarity can reduce the amount of input data for training, in some situations this division is valid, and the models can respond more accurately. Machine learning models developed for broader scenarios (e.g., SI) typically have reduced predictive capacity due to the high nonlinearity and low similarity between input data; however, these models have a greater ability to generalize (Shiri et al. 2014). According to Ferreira et al. (2019), although the models developed locally perform better, these models may have low predictive capacity when used in other regions, since they can be highly specific to the location.

Regarding the importance of each input variable in the response variable of the evaluated algorithms, WEKA was used to select the attributes (Figures 5, 6 and 7). Attributes were selected using the "ClassifierAttributeEval" tool associated with "Ranker" method. These tools rank attributes by their individual evaluations. The correlation coefficient was the measure used to evaluate the performance of attribute combinations in the Ranker configuration. The same ranking method of WEKA was used by Yadav et al. (2014), in order to verify the importance of each input variable in the solar radiation prediction.

Different ANN settings were used for each input data (Table 2). These ANN settings resulted in different weights for each input attribute (Fig. 5). However, a similar behavior was observed between the different configurations. In all scenarios, Tmean, Tmax and Tmin had a greater weight in the estimate. In SIII K2, the relative importance of Tmax surpassed Tmed (Fig. 5). This result may explain the decrease in ANN's performance in this scenario when Tmax and Tmin were removed (Table 5). The variables lat, and month had a similar weight in all scenarios. Although similar, the removal of the month variable resulted in a greater reduction in the ANN performance when compared to the removal of the variables lal, lon and alt.

Ranked value of each input variable of RF is show in Figure 6. The Tmean and month variable had a higher weight in the ET₀estimate. In SII Tho1 and SIII K2, the month variable was more important than Tmean variable. This result may explain the drop in the RF model performance when it removed the month variable (I₃ to I₂). The Tmax and Tmin variables also had a high weight in the ET₀estimate. However, the removal of these variables increased the capacity of the RF model as observed (Table 3, 4 and 5) and discussed previously.

The relative importance of each input variable of SVM is show in Figure 7. It was possible to observe that Tmean, Tmax and Tmin variables had a higher weight in the ET₀estimate. Followed by HR and lat. The month variable was of low importance in the ET₀estimate. In SI, the month showed a negative weight. Therefore, this input data can negatively impact the ET₀estimate. In the performance results of the SVM model (Table 3, 4 and 5), there was no significant variation in performance when the month variable was removed. Both results make it possible to highlight that, for this region, the month variable does not contribute to the performance of the SVM model.

Although each model has a different pattern in the ranking of the input variables (Fig. 5, 6 and 7), air temperature was the most important attribute. The observed correlation between air temperature and ET₀ (Fig. 4) may explain the importance of air temperature in the estimate. This behavior was not observed in SIII K2 and SII Tho2. However, in these scenarios, no significant difference was observed between the month and Tmean variables. Wang et al. (2019b) observed the rank of importance of meteorological variables based on the RF method. The three most important variables were: insolation (n), Tmax and RH. The high relative importance observed corroborates the results of the present study.

The other variables presented different weights according to each model applied. These results indicate a peculiarity of the models experienced. In this way, new research or applications can base on these results and choose the best method that suits the conditions of the input data. However, it is recommended that the models be previously experimented with different input data, as noted, some variables may have a relatively high weight in the ET₀estimate, but their use can decrease the predictive performance of the model. This behavior was observed when using the RF model. In this model, the removal of the variables Tmax and Tmin increased the predictive capacity, although these variables have shown high relative importance.

It is important to note that the month variable was highly important in estimating through the RF. However, a low importance was observed when the SVM model was used, since this variable was not correlated with ET₀ (Fig. 4). These results highlight the need for more techniques to select the meteorological variables used in the modeling. Linear regression alone is not sufficient to identify the relevance of the input data. Furthermore, different models may present different behaviors regarding the classification of the importance of the input variable and still present satisfactory results.

Differently from the evaluation of the importance of the ANN, RF and SVM attributes, for the MLR method, the attribute selection method was applied (M5 method), which indicates the importance of each input attribute in the generated model. The adjusted coefficients are shown in Table 6. It is observed that in some models the method used (M5 method) excluded the month variable. This behavior indicates a low importance of this variable in the MLR estimate. This result was similar to that observed in the analysis of the importance of the input variables in the SVM. The exclusion of lat and Tmax was also observed in some cases.

Table 6 Coefficients of the multiple linear regression models in SI, SII and SIII

		MLR method coefficients
		lat	lon	alt	month	Tmax	Tmean	Tmin	RH
		β₁	β₂	β₃	β₄	β₅	β₆	β₇	β₈	β₀
SI	I₈	-0.0208	0.0579	0.0016	0.0091	0.0758	0.2966	-0.0396	-0.02	-1.8209
	I₆	-0.0222	0.0402	0.0013	0.0065	-	0.2972	-	-0.0264	-0.4453
	I₃	-	-	-	ø	-	0.2262	-	-0.0234	0.4921
	I₂	-	-	-	-	-	0.2262	-	-0.0234	0.4921
		Tho.1 (A, B4, B3, B2 and B1)
SII	I₈	-0.0343	0.06	0.0012	0.0172	0.0633	0.3096	-0.0532	-0.0229	-1.2174
	I₆	-0.0523	0.0294	0.0008	0.0159	-	0.2807	-	-0.0278	-0.7404
	I₃	-	-	-	0.0159	-	0.2521	-	-0.0234	-0.0037
	I₂	-	-	-	-	-	0.2498	-	-0.0251	0.2709
		Tho.2 (C2, C1 and D)
	I₈	ø	0.0517	0.0017	ø	ø	0.3857	-0.0447	-0.0215	-1.344
	I₆	ø	0.0511	0.0016	ø	-	0.331	-	-0.0247	-0.6378
	I₃	-	-	-	ø	-	0.2858	-	-0.0297	-0.6149
	I₂	-	-	-	-	-	0.2858	-	-0.0297	-0.6149
		K1 (Cwa and Cwb)
SIII	I₈	ø	0.0713	0.0013	0.0149	0.091	0.2515	-0.0203	-0.0231	-0.1477
	I₆	-0.0166	0.0461	0.0009	0.0122	-	0.2861	-	-0.029	0.6759
	I₃	-	-	-	0.0128	-	0.256	-	-0.0243	-0.0213
	I₂	-	-	-	-	-	0.254	-	-0.0257	0.2013
		K2 (Am and Aw)
	I₈	-0.0428	0.0595	0.002	ø	0.066	0.3543	-0.0588	-0.0139	-3.4505
	I₆	ø	0.0316	0.0014	ø	-	0.3329	-	-0.0208	-1.7699
	I₃	-	-	-	ø	-	0.3172	-	-0.029	-1.5306
	I₂	-	-	-	-	-	0.3172	-	-0.029	-1.5306

Ø: input data excluded by the M5 method

The results presented revealed that, for locations in the Minas Gerais state, the models can be used safely. The ANN and RF models are recommended to estimate ET₀when considering a wider range of input data, as they have a better predictive capacity in this situation. The SVM and MLR models are recommended in situations where only temperature and relative humidity data are available. However, between these two models, MLR is recommended because it presents less computational effort. The models, although they have a high predictive capacity, cannot be perfect. Other meteorological variables not considered as input data (e.g., solar radiation, wind speed and vapour-pressure deficit) and other factors (e.g., data recorded in error) contributed to the decrease in the predictive capacity of these models.

No statistical method or machine learning can produce results that are the same as the observed and/or recorded data. There will always be some error, no matter how small. Therefore, it is important that the meteorological stations function continuously. These models developed in this study are expected to help decision-making by different professionals, mainly farmers. Agricultural companies are responsible for a considerable part of the Brazilian gross domestic product (Brugnaro and Bacha 2006) and the Minas Gerais state has the third largest Gross Domestic Product in Brazil of 2018 (IBGE 2020). The results of these models assist in the irrigation management, in climatic zoning, in the construction of productivity models among other applications. In addition, the approaches used in the present study have the potential to benefit the development of other types of models and studies from other regions.

The combination of input data I₆ (alt, lat, lon, month, Tmean and RH) for the scenarios SI, SII and SIII, in general, provided the best results in the ET₀ estimate between the evaluated models, so this data combination was recommended to be used. The RF and ANN models presented the highest predictive ability in the ET₀ estimate. Both models in best-case scenarios with input data I₆ or I₈ explains more than 96% of the variability of the variables estimated using the independent dataset. Also, the SVM and MLR models are recommended for all scenarios in situations with limited climatic data, where only air temperature and relative humidity data are available. Although dividing into scenarios results in less input data for models training, SII and SIII scenarios showed a slightly better result in the southern areas of the Minas Gerais state.

The air temperature was the input meteorological variable that presented the greatest relative importance while the month variable presented the greatest variation of importance in relation to the model used. Therefore, it is concluded that although the temperature is fundamental for the estimation of ET₀, other variables can present different levels of importance in the prediction of ET₀.

Funding

The authors would like to thank the Coordination for the Improvement of Higher Education Personnel (CAPES—Finance Code 001) for the research and study grants.

Author contribution

All authors contributed to the study conception, design and statistical design. Material preparation, data collection and analysis were performed by Pietros André Balbino dos Santos, Luiz Gonsaga de Carvalho, Victor Buono da Silva Baptista and Felipe Schwerz. The first draft of the manuscript was written by Pietros André Balbino dos Santos and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Data Availability Not applicable.

Code availability Not applicable.

Ethics approval Not applicable.

Consent to participate Not applicable.

Consent for publication Not applicable.

Competing interests The authors declare no competing interests.

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Machine learning and conventional method for reference evapotranspiration estimation using limited climatic data scenarios

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Abstract

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1 Introduction

2 Material And Methods

2.1 Study area and data sources

2.3 Model development and statistical tests

2.4 Artificial neural networks (ANN)

2.6 Random forest (RF)

2.7 Multiple linear regression (MLR)

3. Results And Discussions

4 Conclusion

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References

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