This study proposes a novel downscaling technique based on stacking ensemble machine learning (SEML) to predict rainfall under climate change. The SEML consists of two levels. Rainfall time series predicted by level 1 algorithms MLR, MNLR, MARS, M5, RF, LSBoost, LSSVM-GS, and a novel hybrid algorithm namely LSSVM-RUN) are used as inputs to the level 2 machine learning algorithm (MARS and LSSVM_RUN). Then, meta-algorithms of SEML predict rainfall based on eight predicted rainfall in level 1. This approach boosts prediction accuracy by utilizing the strong points of different machine learning (ML) algorithms. Results showed that MARS and LSSVM-RUN could be employed to improve the modeling results as meta-algorithms (level 2 of the SEML). Three global climate models (GCMs) in the historical period (1985–2014) and three SSP scenarios in the future period (2021–2050) were considered for downscaling and predicting rainfall at Lake Urmia and Sefidrood basins. Using meta-algorithms, the prediction results showed that rainfall in all scenarios and stations decreased between 0.02 and 0.20% (except Takab station in model CanESM5 scenarios). Hence, the proposed stacking ensemble ML has the potential for modeling and predicting precipitation with good accuracy and high reliability.

  • Proposing a novel downscaling technique based on the stacking ensemble machine learning (SEML).

  • Using the SEML for rainfall prediction under climate change.

  • Utilization of a novel hybrid algorithm namely LSSVM-RUN for the first time.

  • Investigation of rainfall prediction uncertainty in climate change conditions by Bootstrapping method (BM).

  • The proposed technique has the potential to analyze other engineering problems.

A1

ACCESS-ESM1-5

ANN

Artificial neural network

BFs

Basic functions

BM

Bootstrapping method

C5

CanESM5

CMIP

Coupled Model Intercomparison project

CO2

Carbon dioxide

COVID-19

Coronavirus disease 2019

CS

Coefficient of skewness

CV

Coefficient of variation

ESQ

Enhanced solution quality

GCMs

Global climate models

GS

Grid search

GP

Genetic programming

KNN

k-nearest neighbors

LSBoost

Least-squares boost

LSSVM

Least-squares support vector machine

M2

MRI-ESM2-0

M5

M5 model tree

MAE

Mean absolute error

MARS

Multivariate adaptive regression splines

MCDM

Multi-criteria decision-making

ML

Machine learning

MLR

Multiple linear regression

MNLR

Multiple nonlinear regression

MSE

Mean squared error

NIS

Negative ideal solution

PIS

Positive ideal solution

R

Person correlation coefficient

RBF

Radial basis function

RCM

regional climate models

RCPs

Representative concentration pathways

RF

Random forest

RK

Runge Kutta

RMSE

Root mean square error

RRMSE

Relative root mean square error

RUN

RUNge Kutta optimizer

SCA

Sine Cosine algorithm

SCF

SSP245-cov-fossil

SCM

SSP245-cov-modgreen

SCS

SSP245-cov-strgreen

SDR

Standard deviation reduction

SEML

Stacking ensemble machine learning

SSPs

Shared socioeconomic pathways

SSVM

Smooth support vector machine

Std

Standard deviation

SVM

Support vector machine

TOPSIS

Technique for order of preference by similarity to the ideal solution

Rainfall is a major factor in the agricultural and industrial water supply, and sustainable water resource management requires predicting rainfall under various scenarios (Azari et al. 2021). Human activity is responsible for global warming by emitting tons of CO2 emissions from fossil fuels, which is one of the primary causes of climate change. In recent decades, climate change has impacted human health and ecological systems and increased extreme events (Sisco et al. 2017; Bennedsen 2021; Limaye 2021). Decreasing rainfall patterns, including those linked to global warming, significantly, have strained surface water resources (Abbas & Mayo 2021).

Predicting the future effects of climate change is done by using global climate models (GCMs). GCMs are tools that operate on a global scale. Predicting the impact of climate change in the future requires the use of GCMs converted to smaller scales. Dynamic and statistical downscaling methods are employed to obtain data that are downscaled for this reason. High-resolution RCM and physical laws are used in dynamic downscaling methods. Unlike dynamic methods, statistical downscaling methods use empirical relationships and have many advantages, including high accuracy and low computational costs, and for this reason, they are preferred over dynamic methods (Wilby et al. 1998; Langousis & Veneziano 2007; Stoll et al. 2011; Lafon et al. 2013; Huang et al. 2020). ML algorithms are among the best and newest tools for statistically downscaling data.

Recently, ML algorithms have been widely used in downscaling meteorological variables (Chen et al. 2010; Kundu et al. 2017; Sachindra et al. 2018; Vandal et al. 2018; Trinh et al. 2021). However, the accuracy of most ML was not satisfactory in rainfall downscaling. The accuracy of ML in rainfall downscaling can be improved by optimizing them using an optimization algorithm. There are some efforts in this task such as using a hybrid of ML with a whale optimization algorithm (Anaraki et al. 2020), and a hybrid of ML with a sine cosine algorithm (Farrokhi et al. 2021). Nevertheless, there is a need for more accurate methods for rainfall downscaling. Because, by increasing the accuracy of downscale and as a result of more accurate prediction of the amount of precipitation in the future, it is possibly better to manage water resources in climate change conditions.

ML algorithms have different structures that give them unique advantages. ML algorithms have many advantages in downscaling hydrological variables, but their structure has defects that impact their performance. SEML attempts to resolve these defects by combining several algorithms with various structures. Using multiple algorithms makes it possible to simultaneously use the benefits of different algorithms. For this reason, SEML can greatly help increase the precision of predicting hydrological variables. Morshed-Bozorgdel et al. (2022) modeled wind speed using SEML. The results showed that SEML increased the modeling accuracy by >43% compared to the single algorithms.

The advantages and disadvantages of previous rainfall downscaling studies are presented in Table 1. As seen, previous studies had drawbacks such as low accuracy, not using hybrid algorithms, lack of uncertainty analysis, and not using new GCMs and climate change scenarios. Moreover, some previous studies were conducted on limited case studies. Hence, the present study aims to cover these disadvantages by introducing a stacking ensemble machine learning algorithm (SEML) for rainfall downscaling. Although this study uses the machine learning (ML) algorithms, similar to previous studies, using SEML structure increases the accuracy of ML algorithms by using the advantages of different ML algorithms simultaneously. The new contributions of this study and its difference from other carried out studies are listed as follows:

  • In the present study, a novel modeling technique based on the stacking ensemble of regression-based, tree-based, curve-based, kernel-based, and hybrid algorithm-based has been designed for predicting rainfall under climate change in the northern basins of Iran.

  • The precision of rainfall modeling can be modified by the algorithms mentioned in this modeling technique at two levels.

  • Using a new hybrid algorithm (LSSVM_RUN) as a meta-model.

  • Predicting rainfall under climate change conditions (CMIP6 models and SSP-representative concentration pathways (RCP) scenarios) in the future.

  • The uncertainty of this modeling technique has been investigated by the Bootstrapping method (BM) as a reliable and robust uncertainty approach.

Table 1

Statistical criteria of rainfall from the investigated stations

Station nameCSCVStdAverage
Tabriz 20.90 19.58 0.94 1.57 
Sahand 19.16 18.50 0.97 1.47 
Urmia 26.28 28.07 1.07 1.55 
Maragheh 25.13 26.84 1.07 1.30 
Mianeh 23.49 22.13 0.94 1.20 
Mahabad 33.61 34.63 1.03 1.10 
Saqez 38.37 40.96 1.07 1.68 
Takab 25.92 24.20 0.93 1.58 
Zanjan 32.02 31.15 0.97 1.68 
Station nameCSCVStdAverage
Tabriz 20.90 19.58 0.94 1.57 
Sahand 19.16 18.50 0.97 1.47 
Urmia 26.28 28.07 1.07 1.55 
Maragheh 25.13 26.84 1.07 1.30 
Mianeh 23.49 22.13 0.94 1.20 
Mahabad 33.61 34.63 1.03 1.10 
Saqez 38.37 40.96 1.07 1.68 
Takab 25.92 24.20 0.93 1.58 
Zanjan 32.02 31.15 0.97 1.68 

The proposed approach has the potential to provide highly accurate and reliable rainfall modeling and prediction. The remaining work is structured as follows:

Section 2 presents the used data and presented approach for downscaling rainfall, Section 3 presents the obtained results and discussion, and section 4 presents the conclusion.

Study area and data

Iran is located in the south-west of Asia in the area known as the Middle East between 44° to 64°E longitude and 25° to 40°N latitude (FAO 2008). Lake Urmia is one of the most critical lakes globally, located in the northwest of Iran. Lake Urmia has an area of about 51,460 km2 and is 140 km long, and many rivers such as Sufi Chai and Zarrin River, flow into this lake. Sefidrood, with a length of 750 km, is one of the largest rivers in the Caspian Sea basin, the second-longest river in Iran, and flows through Gilan province to flow into the Caspian Sea. Sefidrood has several branches with an area of more than 57,000 km2. Iran Meteorological Organization provided the statistical rainfall data of these two basins for a period of 30 years, from 1985 to 2014. Monthly rainfall records collected from nine weather stations during a period of 30 years (1985–2014) were used to evaluate the proposed methodologies (a total of 360 patterns). According to Figure 1, nine stations of Tabriz, Sahand, Urmia, Maragheh, Mianeh, Mahabad, Saqez, Takab, and Zanjan are located in this study. The statistical criteria of rainfall in the nine investigated stations are stated in Table 1.
Figure 1

Case studies locations.

Figure 1

Case studies locations.

Close modal

GCMs and future climate change scenarios

In this study, three COVID-19 post-pandemic recovery scenarios (SCF, SCM, and SCS) of CMIP6 are used, which assumes that the current climate change trend remains constant, resulting in a representative concentration pathway of 4.5 W/m2 by 2100 and the effects of economic growth, population changes and urbanization (Forster et al. 2020; Lamboll et al. 2021) and COVID-19.

SSP2 depicts a moderate growth route in which future socioeconomic development patterns follow existing development tendencies. Research has shown that changes in human activity due to the COVID-19 epidemic have had a significant effect on the composition of the atmospheric composition (D'Souza et al. 2021), particularly in the amount of solar radiation reaching the planet's surface and aerosol optical depth across southern and eastern Asia. We investigated rainfall for historical and future periods using three GCM outputs (A1, C5, and M2) and three SSP-RCP scenarios from the most recent CMIP6 (Jones et al. 2021). Three assumptions were used for COVID-19 post-pandemic recovery scenarios, with the SSP245 scenario as the baseline. The selected models and scenarios details are presented in Tables 2 and 3 (for more information, see (Riahi et al. 2017, Swart et al. 2019, Yukimoto et al. 2019, Oshima et al. 2020, Ziehn et al. 2020 and Kadkhodazadeh et al. 2022).

Table 2

The considered CMIP6 GCMs

ModelInstitutionAtmosphere resolutiona
ACCESS-ESM1–5 Commonwealth Scientific and Industrial Research Organization (Australia) 250 km (N96), L38 
CanESM5 Canadian Centre for Climate Modelling and Analysis (Canada) 500 km (T63), L49 
MRI-ESM2-0 Meteorological Research Institute (Japan) 100 km (TL159, 1.125°), L80 
ModelInstitutionAtmosphere resolutiona
ACCESS-ESM1–5 Commonwealth Scientific and Industrial Research Organization (Australia) 250 km (N96), L38 
CanESM5 Canadian Centre for Climate Modelling and Analysis (Canada) 500 km (T63), L49 
MRI-ESM2-0 Meteorological Research Institute (Japan) 100 km (TL159, 1.125°), L80 

aShown as CMIP ‘nominal resolution’ in km, ‘L’ indicates number of vertical levels.

Table 3

SSP scenarios

Experiment-idActivity-idDescription
Baseline ScenarioMIP SSP-based RCP scenario with medium radiative forcing by the end of the century. Following approximately RCP4.5 global forcing pathway with SSP2 socioeconomic conditions. Radiative forcing reaches a level of 4.5 W/m2 in 2100 
Two year blip – Data are modified for all of 2020 and 2021 in accordance with observed activity levels in the sectors of different countries. This is projected to continue at 2/3 of the activity reduction value for the latest month available for the rest of the 2-year period. Activity is interpolated, month for month, back towards baseline over 2022 and is equal to baseline thereafter 
SSP245-cov-fossila DAMIP Follows 2-year blip until 2023. Thereafter, the effects of additional investment in fossil fuels during recovery are included in a globally uniform way. Financial modeling produced estimated global Kyoto gas emissions totals consistent with 10% higher emissions than the path met if countries meet their nationally determined contributions (NDCs). We used the open-source package Silicone to find a linear combination of MESSAGE-GLOBIOM SSP2 scenarios that gave the same total Kyoto emissions. We use the global relative emissions level of each aerosol and precursor in this composite scenario to rescale the 2D emissions maps. The relative concentration change arising from this scenario is used to rescale global greenhouse gas concentrations 
SSP245-cov-modgreena DAMIP Follows 2-year blip until 2023. Thereafter, the effects of small additional investment in green technology are included in a globally uniform way. Financial considerations as to what emissions change is plausible with moderate ambition (in keeping with results in McCollum et al. (2018)) produced a Kyoto emissions total in 2030 of 35% lower than the NDCs, which we resolve into a linear combination of MESSAGE-GLOBIOM SSP2 scenarios. We then set a global net zero CO2 trajectory for 2060, and resolve this CO2 total into a linear combination of MESAGE-GLOBIOM SSP2 scenarios again using Silicone. The relative difference between this scenario and the baseline is used to rescale emissions and concentrations as in the fossil fuel case 
SSP245-cov-strgreena DAMIP Follows 2-year blip until 2023. Thereafter, the effects of large additional investment in green technology are assumed to push the scenario towards an IMAGE SSP1 world. In 2030, we are assumed to reach the emissions rate of SSP1-19, around 52% lower than following current NDCs, and thereafter follow a global net zero CO2 target for 2050. The other emissions are formed by a linear combination of IMAGE SSP1 scenarios that give the closest total CO2 match to this pathway (this composite pathway is always close to the SSP1-19 pathway after 2023) 
Experiment-idActivity-idDescription
Baseline ScenarioMIP SSP-based RCP scenario with medium radiative forcing by the end of the century. Following approximately RCP4.5 global forcing pathway with SSP2 socioeconomic conditions. Radiative forcing reaches a level of 4.5 W/m2 in 2100 
Two year blip – Data are modified for all of 2020 and 2021 in accordance with observed activity levels in the sectors of different countries. This is projected to continue at 2/3 of the activity reduction value for the latest month available for the rest of the 2-year period. Activity is interpolated, month for month, back towards baseline over 2022 and is equal to baseline thereafter 
SSP245-cov-fossila DAMIP Follows 2-year blip until 2023. Thereafter, the effects of additional investment in fossil fuels during recovery are included in a globally uniform way. Financial modeling produced estimated global Kyoto gas emissions totals consistent with 10% higher emissions than the path met if countries meet their nationally determined contributions (NDCs). We used the open-source package Silicone to find a linear combination of MESSAGE-GLOBIOM SSP2 scenarios that gave the same total Kyoto emissions. We use the global relative emissions level of each aerosol and precursor in this composite scenario to rescale the 2D emissions maps. The relative concentration change arising from this scenario is used to rescale global greenhouse gas concentrations 
SSP245-cov-modgreena DAMIP Follows 2-year blip until 2023. Thereafter, the effects of small additional investment in green technology are included in a globally uniform way. Financial considerations as to what emissions change is plausible with moderate ambition (in keeping with results in McCollum et al. (2018)) produced a Kyoto emissions total in 2030 of 35% lower than the NDCs, which we resolve into a linear combination of MESSAGE-GLOBIOM SSP2 scenarios. We then set a global net zero CO2 trajectory for 2060, and resolve this CO2 total into a linear combination of MESAGE-GLOBIOM SSP2 scenarios again using Silicone. The relative difference between this scenario and the baseline is used to rescale emissions and concentrations as in the fossil fuel case 
SSP245-cov-strgreena DAMIP Follows 2-year blip until 2023. Thereafter, the effects of large additional investment in green technology are assumed to push the scenario towards an IMAGE SSP1 world. In 2030, we are assumed to reach the emissions rate of SSP1-19, around 52% lower than following current NDCs, and thereafter follow a global net zero CO2 target for 2050. The other emissions are formed by a linear combination of IMAGE SSP1 scenarios that give the closest total CO2 match to this pathway (this composite pathway is always close to the SSP1-19 pathway after 2023) 

aFor more details on how these were constructed, see Forster et al. (2020).

Present work steps

The SEML, composed of two levels, tries to increase the precision of the modeling by using several algorithms in its structure. The SEML combines the advantages of different algorithms to create a framework that works better than any single algorithm in the set (Morshed-Bozorgdel et al. 2022). For this purpose, in this paper, eight ML algorithms (MLR and MNLR as regression-based, M5, RF, and LSBoost as tree-based, MARS as curve-based, LSSVM-GS as kernel-based, and LSSVM-RUN as hybrid algorithms-based) have utilized. Figure 2 shows the SEML structure in this study. The steps of the rainfall prediction under climate change and uncertainty analysis are illustrated in Figure 3.
  1. Downscaling large-scale rainfall data by the SEML (level 1): Downscaling large-scale rainfall data using eight ML algorithms including MLR, MNLR, MARS, M5, RF, LSBoost, LSSVM-GS, and LSSVM-RUN under three GCMs, including A1, C5, M2 is performed at Tabriz, Sahand, Urmia, Maragheh, Mianeh, Mahabad, Saqez, Takab, and Zanjan stations.

  2. Select the best algorithms by the TOPSIS method in level 1: Since it is difficult to compare algorithms and select the best algorithm in level 1, the TOPSIS method is used to select the best algorithm. This process is as follows: (1) MAE, RMSE, RRMSE, and R are considered criteria of the TOPSIS method. (2) The algorithms are considered alternatives, and the lambda weight is the same for all assessment criteria. (3) The algorithm with the highest score is selected as the best algorithm.

  3. Downscaling large-scale rainfall data by the SEML (level 2): After selecting the best algorithm using the TOPSIS method, level 2 of downscaling large-scale rainfall data is performed. For this purpose, the output of the base algorithms is used as the input of the meta-algorithms.

  4. Prediction rainfall under climate change by meta-algorithms (2021–2050): Rainfall prediction for 2021–2050 under climate change in three GCMs, including A1, C5, and M2 and three SCF, SCM, and SCS scenarios by downscaled rainfall data and the meta-algorithms in the nine stations done.

  5. Uncertainty analysis: Uncertainty analysis of models and scenarios is performed using BM in nine stations.

  • The descriptions for MLR, MNLR, M5, RF, LSBoost and TOPSIS are given in Supplementary material, section S.1.

Figure 2

Structure of the SEML.

Figure 2

Structure of the SEML.

Close modal
Figure 3

Framework for rainfall prediction and uncertainty analysis.

Figure 3

Framework for rainfall prediction and uncertainty analysis.

Close modal

Multivariate adaptive regression splines

Multivariate adaptive regression splines (MARS), which was first proposed by Friedman (1991), is based on nonlinear methods. Different weights are assigned to the input variables in this algorithm to define nonlinear relationships. The MARS is made up of basic functions. Basic functions (BFs) that depend on spline functions estimate the output data by dividing the data into smaller sections. The relationship between inputs (x) and output (Y) is assumed to be as follows:
(1)
where wi is weight of the ith BF, n is number of BF, and i is the known counter of BF.

Least-squares support vector machine-grid search

Least-squares support vector machine-grid search (LSSVM), which is based on improving the performance of the support vector machine (SVM), uses linear relationships instead of nonlinear relationships to solve problems (Suykens et al. 2002). This algorithm is used in regression analysis, classification, and pattern recognition. The use of linear relationships in the structure of this algorithm has improved accuracy and speed compared to the SVM. The relationship between input (x) and output (Y) in LSSVM is as follows (Kadkhodazadeh & Farzin 2021):
(2)
where is ith Lagrange multiplier, K is the kernel, and b is biased. In LSSVM, the least square method obtains using optimization of the following objective function (Kadkhodazadeh et al. 2022):
(3)
(4)
where w is the weight of inputs, e is the empirical error, and C is the penalty coefficient. For more information about LSSVM, please refer to Zeng et al. (2019). Functions such as sigmoid, linear function, polynomial, and radial basis function (RBF) are used in nonlinear problems. In this paper, the RBF is used as a kernel function, which is expressed as follows (Ghadiri et al. 2021):
(5)
where σ is kernel width. The values of C and σ have an important effect on the precision of the SVM. The values of these coefficients can be found using a grid search. GS takes all values of C and σ and creates a network in a certain range.

RUNge Kutta optimizer

The RUNge Kutta optimizer (RUN) algorithm, first proposed by Ahmadianfar et al. (2021) is a population-based optimization method based on mathematical principles and the Runge kutta (RK) method. This algorithm can be used to solve optimization problems in a variety of areas. The RK method slope changes are used by this algorithm to search for optimal solutions. It also uses the ESQ mechanism to increase the speed of convergence and avoid falling into the trap of local optimization. For more information, see Ahmadianfar et al. (2021).

Novel hybrid algorithm (LSSVM-RUN)

The LSSVM only runs by defining two parameters including C and σ parameters. The mentioned parameters significantly impact the performance of LSSVM. However, there is no specific method for selecting these two parameters; their range is in (0, ), which is wide. The trial and error method for determining C and σ is inaccurate and requires a high computational cost. As a result, this study used the RUN algorithm to determine the optimal value of the LSSVM parameters. The values C and σ are considered as decision variables in the hybrid algorithm. First, the data are randomly divided into training and testing data. Then, the RUN parameters and the initial population of the optimization and simulation algorithm are determined. After training LSSVM, the test data are used to evaluate the performance of the simulation algorithm.

The objective function for selecting the optimal parameter of LSSVM is MSE in the testing period, which can be calculated as follows (Farzin et al. 2022):
(6)
where X is the observed values, Y is the predicted values, i is the index of data, and N is the number of data.

Bootstrapping method

Although there are many factors that affect rainfall prediction uncertainty, this study focuses on the uncertainties related to prediction GCMs and scenarios. Bootstrapping method (BM) is one of the most powerful and accessible methods for uncertainty analysis. The steps of this method are as follows:

  • Input data are resampled several times, and new output data are predicted.

  • Predicting new outputs, a 95% prediction confidence interval is obtained using the values generated for each observation to quantify the prediction uncertainties.

  • Sort upper and lower bands of 95% confidence interval for each time series.

  • The upper quartile (97.5%) and the lower quartile (2.5%) of the 95% band are determined.

  • The R-factor coefficient is calculated using the following formula. The lower the value of this coefficient, the less uncertainty.
    (7)
    (8)
    where is the standard deviation of the observed values, N is the number of observed data and and indicate the ith value of the upper quartile (97.5%) and the lower quartile (2.5%) of the 95% band.

Assessment criteria

In this study, assessment criteria including root mean square error (RMSE), MAE, RRMSE, and R have been used to compare the accuracy of different algorithms. The assessment criteria are as follows:
(9)
(10)
(11)
(12)
where N is the number of data, x is the observed values, y is the predicted values, is the mean observed values, and is the mean predicted values.

Downscaling large-scale rainfall data by the SEML (level 1)

The base algorithms in level 1 of the SEML downscale the large-scale rainfall in the nine stations. Table 4 and Supplementary material, Table S2 show the values of assessment criteria for base algorithms and three GCMs in the nine stations. According to this table, the accuracy of any algorithms that were investigated was not good at this level. Nevertheless, the MARS, LSBoost, and LSSVM-RUN had greater accuracy than other investigated algorithms during the testing period at different stations and three GCMs. In this condition, integrating the results of base algorithms with meta-algorithms can lead to good accuracy. The best algorithm in level 1 was used as a meta-algorithm. However, due to the differences in assessment criteria and the results obtained in different stations and algorithms, it is difficult to choose the best algorithm for use in level 2 of the SEML as a meta-algorithm. Hence, to select the best algorithm, the TOPSIS method was utilized.

Table 4

Level 1 results of downscaling large-scale rainfall with SEML (base algorithms)

ModelsStationsAlgorithmsAssessment criteria
Train
Test
MAERMSERRMSERMAERMSERRMSER
A1 Tabriz MLR 16.19 22.41 0.66 0.75 24.75 30.33 1.11 0.43 
MNLR 11.60 15.38 0.45 0.89 35.08 45.52 1.66 0.32 
M5 18.13 27.29 0.80 0.60 17.33 23.32 0.85 0.55 
MARS 18.28 27.62 0.81 0.59 15.88 20.50 0.75 0.66 
RF 9.05 14.28 0.42 0.94 16.62 21.70 0.79 0.61 
LSBoost 8.73 12.61 0.37 0.95 16.07 21.38 0.78 0.64 
LSSVM-GS 0.26 0.36 0.01 1.00 20.21 25.92 0.95 0.58 
LSSVM-RUN 15.27 23.27 0.68 0.78 16.61 21.66 0.79 0.61 
Urmia MLR 13.14 18.22 0.70 0.72 20.59 25.77 1.11 0.32 
MNLR 9.78 13.12 0.50 0.86 23.76 30.48 1.31 0.33 
M5 15.16 22.08 0.84 0.53 17.24 21.66 0.93 0.39 
MARS 15.23 22.14 0.85 0.53 15.77 19.55 0.84 0.53 
RF 7.70 12.02 0.46 0.93 17.52 21.47 0.93 0.39 
LSBoost 2.76 6.80 0.26 0.98 15.71 20.13 0.87 0.51 
LSSVM-GS 0.20 0.28 0.01 1.00 18.59 22.36 0.96 0.46 
LSSVM-RUN 13.44 19.37 0.74 0.73 16.67 20.09 0.87 0.50 
Zanjan MLR 9.46 12.33 0.65 0.75 12.30 16.17 0.86 0.53 
MNLR 7.65 9.49 0.50 0.86 24.09 28.70 1.53 0.18 
M5 10.86 14.60 0.78 0.63 11.45 15.95 0.85 0.53 
MARS 11.04 14.82 0.79 0.62 10.62 14.50 0.77 0.63 
RF 5.93 8.06 0.43 0.94 11.59 16.54 0.88 0.47 
LSBoost 8.18 10.38 0.55 0.87 11.02 15.67 0.83 0.55 
LSSVM-GS 0.15 0.20 0.01 1.00 12.18 18.14 0.96 0.47 
LSSVM-RUN 7.48 9.85 0.52 0.89 10.67 15.44 0.82 0.57 
C5 Tabriz MLR 14.38 18.87 0.55 0.83 29.50 37.52 1.37 0.35 
MNLR 0.00 0.00 0.00 1.00 107.05 131.80 4.81 0.18 
M5 18.24 27.98 0.82 0.57 16.80 23.54 0.86 0.53 
MARS 18.60 27.82 0.81 0.58 15.73 20.76 0.76 0.65 
RF 9.31 14.41 0.42 0.94 18.66 24.77 0.90 0.45 
LSBoost 17.17 26.60 0.78 0.66 16.67 23.16 0.84 0.53 
LSSVM-GS 0.26 0.35 0.01 1.00 19.66 25.28 0.92 0.65 
LSSVM-RUN 1.77 2.73 0.08 1.00 16.33 21.21 0.77 0.63 
Urmia MLR 10.55 13.94 0.53 0.85 24.35 32.01 1.38 0.21 
MNLR 0.00 0.00 0.00 1.00 76.67 98.96 4.27 0.17 
M5 15.32 21.82 0.83 0.55 17.53 22.20 0.96 0.36 
MARS 15.38 22.06 0.84 0.54 15.54 19.10 0.82 0.56 
RF 7.04 11.27 0.43 0.94 15.94 19.85 0.86 0.51 
LSBoost 14.43 20.80 0.79 0.67 16.39 20.05 0.86 0.50 
LSSVM-GS 0.20 0.27 0.01 1.00 18.26 21.92 0.94 0.56 
LSSVM-RUN 0.18 0.25 0.01 1.00 15.76 19.47 0.84 0.55 
Zanjan MLR 8.40 10.37 0.55 0.83 19.79 25.50 1.36 0.15 
MNLR 0.00 0.00 0.00 1.00 56.09 67.81 3.61 0.17 
M5 11.20 15.20 0.81 0.59 11.12 15.97 0.85 0.52 
MARS 11.70 15.68 0.83 0.55 10.16 14.48 0.77 0.65 
RF 5.97 9.02 0.48 0.89 12.19 16.59 0.88 0.47 
LSBoost 0.97 1.58 0.08 1.00 10.99 15.17 0.81 0.59 
LSSVM-GS 0.15 0.19 0.01 1.00 11.91 17.77 0.94 0.57 
LSSVM-RUN 3.74 4.91 0.26 0.98 10.56 15.37 0.82 0.58 
M2 Tabriz MLR 13.89 18.55 0.54 0.84 25.43 32.61 1.19 0.51 
MNLR 1.09 1.37 0.04 1.00 162.55 200.33 7.31 0.16 
M5 20.70 29.95 0.88 0.48 18.08 25.23 0.92 0.41 
MARS 19.85 28.73 0.84 0.54 17.40 24.06 0.88 0.52 
RF 9.81 15.12 0.44 0.95 17.86 23.61 0.86 0.50 
LSBoost 17.50 24.41 0.71 0.77 17.74 23.37 0.85 0.52 
LSSVM-GS 0.27 0.36 0.01 1.00 20.79 26.64 0.97 0.57 
LSSVM-RUN 10.35 15.35 0.45 0.95 16.25 21.90 0.80 0.60 
Urmia MLR 11.21 14.43 0.55 0.83 22.54 28.93 1.25 0.42 
MNLR 1.81 2.27 0.09 1.00 169.79 209.04 9.01 0.11 
M5 17.16 23.34 0.89 0.45 17.59 21.58 0.93 0.38 
MARS 16.23 23.03 0.88 0.47 17.31 21.58 0.93 0.40 
RF 8.15 12.32 0.47 0.92 17.01 20.63 0.89 0.45 
LSBoost 15.17 21.57 0.82 0.64 16.51 20.14 0.87 0.50 
LSSVM-GS 0.20 0.28 0.01 1.00 18.83 22.68 0.98 0.51 
LSSVM-RUN 0.21 0.30 0.01 1.00 15.74 19.66 0.85 0.53 
Zanjan MLR 7.75 9.63 0.51 0.86 18.08 22.12 1.18 0.39 
MNLR 1.35 1.70 0.09 1.00 75.34 97.81 5.20 0.17 
M5 11.28 14.98 0.79 0.60 11.57 16.11 0.86 0.52 
MARS 12.15 16.34 0.87 0.49 12.02 16.90 0.90 0.44 
RF 5.70 7.84 0.42 0.95 11.28 16.88 0.90 0.44 
LSBoost 1.91 2.38 0.13 0.99 11.56 15.88 0.84 0.53 
LSSVM-GS 0.16 0.20 0.01 1.00 12.43 18.42 0.98 0.46 
LSSVM-RUN 0.18 0.24 0.01 1.00 11.18 15.91 0.85 0.53 
ModelsStationsAlgorithmsAssessment criteria
Train
Test
MAERMSERRMSERMAERMSERRMSER
A1 Tabriz MLR 16.19 22.41 0.66 0.75 24.75 30.33 1.11 0.43 
MNLR 11.60 15.38 0.45 0.89 35.08 45.52 1.66 0.32 
M5 18.13 27.29 0.80 0.60 17.33 23.32 0.85 0.55 
MARS 18.28 27.62 0.81 0.59 15.88 20.50 0.75 0.66 
RF 9.05 14.28 0.42 0.94 16.62 21.70 0.79 0.61 
LSBoost 8.73 12.61 0.37 0.95 16.07 21.38 0.78 0.64 
LSSVM-GS 0.26 0.36 0.01 1.00 20.21 25.92 0.95 0.58 
LSSVM-RUN 15.27 23.27 0.68 0.78 16.61 21.66 0.79 0.61 
Urmia MLR 13.14 18.22 0.70 0.72 20.59 25.77 1.11 0.32 
MNLR 9.78 13.12 0.50 0.86 23.76 30.48 1.31 0.33 
M5 15.16 22.08 0.84 0.53 17.24 21.66 0.93 0.39 
MARS 15.23 22.14 0.85 0.53 15.77 19.55 0.84 0.53 
RF 7.70 12.02 0.46 0.93 17.52 21.47 0.93 0.39 
LSBoost 2.76 6.80 0.26 0.98 15.71 20.13 0.87 0.51 
LSSVM-GS 0.20 0.28 0.01 1.00 18.59 22.36 0.96 0.46 
LSSVM-RUN 13.44 19.37 0.74 0.73 16.67 20.09 0.87 0.50 
Zanjan MLR 9.46 12.33 0.65 0.75 12.30 16.17 0.86 0.53 
MNLR 7.65 9.49 0.50 0.86 24.09 28.70 1.53 0.18 
M5 10.86 14.60 0.78 0.63 11.45 15.95 0.85 0.53 
MARS 11.04 14.82 0.79 0.62 10.62 14.50 0.77 0.63 
RF 5.93 8.06 0.43 0.94 11.59 16.54 0.88 0.47 
LSBoost 8.18 10.38 0.55 0.87 11.02 15.67 0.83 0.55 
LSSVM-GS 0.15 0.20 0.01 1.00 12.18 18.14 0.96 0.47 
LSSVM-RUN 7.48 9.85 0.52 0.89 10.67 15.44 0.82 0.57 
C5 Tabriz MLR 14.38 18.87 0.55 0.83 29.50 37.52 1.37 0.35 
MNLR 0.00 0.00 0.00 1.00 107.05 131.80 4.81 0.18 
M5 18.24 27.98 0.82 0.57 16.80 23.54 0.86 0.53 
MARS 18.60 27.82 0.81 0.58 15.73 20.76 0.76 0.65 
RF 9.31 14.41 0.42 0.94 18.66 24.77 0.90 0.45 
LSBoost 17.17 26.60 0.78 0.66 16.67 23.16 0.84 0.53 
LSSVM-GS 0.26 0.35 0.01 1.00 19.66 25.28 0.92 0.65 
LSSVM-RUN 1.77 2.73 0.08 1.00 16.33 21.21 0.77 0.63 
Urmia MLR 10.55 13.94 0.53 0.85 24.35 32.01 1.38 0.21 
MNLR 0.00 0.00 0.00 1.00 76.67 98.96 4.27 0.17 
M5 15.32 21.82 0.83 0.55 17.53 22.20 0.96 0.36 
MARS 15.38 22.06 0.84 0.54 15.54 19.10 0.82 0.56 
RF 7.04 11.27 0.43 0.94 15.94 19.85 0.86 0.51 
LSBoost 14.43 20.80 0.79 0.67 16.39 20.05 0.86 0.50 
LSSVM-GS 0.20 0.27 0.01 1.00 18.26 21.92 0.94 0.56 
LSSVM-RUN 0.18 0.25 0.01 1.00 15.76 19.47 0.84 0.55 
Zanjan MLR 8.40 10.37 0.55 0.83 19.79 25.50 1.36 0.15 
MNLR 0.00 0.00 0.00 1.00 56.09 67.81 3.61 0.17 
M5 11.20 15.20 0.81 0.59 11.12 15.97 0.85 0.52 
MARS 11.70 15.68 0.83 0.55 10.16 14.48 0.77 0.65 
RF 5.97 9.02 0.48 0.89 12.19 16.59 0.88 0.47 
LSBoost 0.97 1.58 0.08 1.00 10.99 15.17 0.81 0.59 
LSSVM-GS 0.15 0.19 0.01 1.00 11.91 17.77 0.94 0.57 
LSSVM-RUN 3.74 4.91 0.26 0.98 10.56 15.37 0.82 0.58 
M2 Tabriz MLR 13.89 18.55 0.54 0.84 25.43 32.61 1.19 0.51 
MNLR 1.09 1.37 0.04 1.00 162.55 200.33 7.31 0.16 
M5 20.70 29.95 0.88 0.48 18.08 25.23 0.92 0.41 
MARS 19.85 28.73 0.84 0.54 17.40 24.06 0.88 0.52 
RF 9.81 15.12 0.44 0.95 17.86 23.61 0.86 0.50 
LSBoost 17.50 24.41 0.71 0.77 17.74 23.37 0.85 0.52 
LSSVM-GS 0.27 0.36 0.01 1.00 20.79 26.64 0.97 0.57 
LSSVM-RUN 10.35 15.35 0.45 0.95 16.25 21.90 0.80 0.60 
Urmia MLR 11.21 14.43 0.55 0.83 22.54 28.93 1.25 0.42 
MNLR 1.81 2.27 0.09 1.00 169.79 209.04 9.01 0.11 
M5 17.16 23.34 0.89 0.45 17.59 21.58 0.93 0.38 
MARS 16.23 23.03 0.88 0.47 17.31 21.58 0.93 0.40 
RF 8.15 12.32 0.47 0.92 17.01 20.63 0.89 0.45 
LSBoost 15.17 21.57 0.82 0.64 16.51 20.14 0.87 0.50 
LSSVM-GS 0.20 0.28 0.01 1.00 18.83 22.68 0.98 0.51 
LSSVM-RUN 0.21 0.30 0.01 1.00 15.74 19.66 0.85 0.53 
Zanjan MLR 7.75 9.63 0.51 0.86 18.08 22.12 1.18 0.39 
MNLR 1.35 1.70 0.09 1.00 75.34 97.81 5.20 0.17 
M5 11.28 14.98 0.79 0.60 11.57 16.11 0.86 0.52 
MARS 12.15 16.34 0.87 0.49 12.02 16.90 0.90 0.44 
RF 5.70 7.84 0.42 0.95 11.28 16.88 0.90 0.44 
LSBoost 1.91 2.38 0.13 0.99 11.56 15.88 0.84 0.53 
LSSVM-GS 0.16 0.20 0.01 1.00 12.43 18.42 0.98 0.46 
LSSVM-RUN 0.18 0.24 0.01 1.00 11.18 15.91 0.85 0.53 

Select the best algorithms by the TOPSIS method in the level 1

According to the values of the assessment criteria in Table 4 and Supplementary material, Table S1 and the weights assigned to each assessment criteria, each algorithm's score in different stations and algorithms was calculated. The assessment criteria have a lambda weight of 0.25, and the sum of all assessment criteria must be equal to one. In general, algorithms with the highest scores tend to have higher rankings. In Table 5, the results in different stations and algorithms showed that the two MARS and LSSVM-RUN algorithms with a mean score of 0.99 had better performance than other algorithms. The MARS is a nonlinear and non-parametric method that was highly effective in solving high-dimensional nonlinear problems and improving modeling accuracy with rapid convergence. Moreover, the RUN optimization algorithm worked well to find the optimal C and σ in the LSSVM, which makes the new hybrid LSSVM-RUN algorithm very precise.

Table 5

Scores of TOPSIS method for ranking of algorithms

ModelsStationsAlgorithms score
MLRMNLRM5MARSRFLSBoostLSSVM-GSLSSVM-RUN
A1 Tabriz 0.58 0.00 0.90 1.00 0.96 0.98 0.78 0.96 
Sahand 0.61 0.00 0.88 1.00 0.95 0.95 0.78 0.99 
Urmia 0.41 0.00 0.80 1.00 0.80 0.97 0.70 0.92 
Maragheh 0.71 0.00 0.91 1.00 0.95 0.96 0.78 0.97 
Mianeh 0.83 0.00 0.84 1.00 0.86 0.96 0.76 0.93 
Mahabad 0.60 0.00 0.89 0.99 0.96 0.99 0.84 0.99 
Saqez 0.83 0.00 0.92 0.93 0.94 1.00 0.88 0.99 
Takab 0.83 0.00 0.80 0.98 0.83 1.00 0.78 0.93 
Zanjan 0.88 0.00 0.91 1.00 0.89 0.94 0.81 0.96 
C5 Tabriz 0.85 0.00 0.98 1.00 0.96 0.98 0.96 0.99 
Sahand 0.85 0.00 0.99 0.98 0.98 0.98 0.96 1.00 
Urmia 0.84 0.00 0.96 1.00 0.99 0.99 0.96 1.00 
Maragheh 0.88 0.00 0.95 0.99 0.97 0.98 0.94 1.00 
Mianeh 0.85 0.00 0.96 1.00 0.98 0.98 0.95 1.00 
Mahabad 0.86 0.00 0.98 1.00 0.98 0.99 0.97 1.00 
Saqez 0.82 0.00 0.95 0.97 0.95 0.99 0.96 1.00 
Takab 0.80 0.00 0.96 0.96 0.98 1.00 0.95 1.00 
Zanjan 0.79 0.00 0.97 1.00 0.96 0.98 0.95 0.99 
M2 Tabriz 0.94 0.00 0.98 0.99 0.99 0.99 0.97 1.00 
Sahand 0.95 0.00 0.99 0.99 0.99 0.99 0.97 1.00 
Urmia 0.95 0.00 0.99 0.99 0.99 1.00 0.98 1.00 
Maragheh 0.96 0.00 0.98 0.99 0.99 0.99 0.97 1.00 
Mianeh 0.96 0.00 0.99 0.99 0.99 0.99 0.97 1.00 
Mahabad 0.96 0.00 0.99 1.00 0.98 1.00 0.98 1.00 
Saqez 0.93 0.00 0.98 0.99 0.98 1.00 0.98 1.00 
Takab 0.94 0.00 1.00 0.99 1.00 1.00 0.98 1.00 
Zanjan 0.91 0.00 1.00 0.99 0.99 1.00 0.97 1.00 
Mean score 0.83 0.00 0.94 0.99 0.95 0.98 0.91 0.99 
ModelsStationsAlgorithms score
MLRMNLRM5MARSRFLSBoostLSSVM-GSLSSVM-RUN
A1 Tabriz 0.58 0.00 0.90 1.00 0.96 0.98 0.78 0.96 
Sahand 0.61 0.00 0.88 1.00 0.95 0.95 0.78 0.99 
Urmia 0.41 0.00 0.80 1.00 0.80 0.97 0.70 0.92 
Maragheh 0.71 0.00 0.91 1.00 0.95 0.96 0.78 0.97 
Mianeh 0.83 0.00 0.84 1.00 0.86 0.96 0.76 0.93 
Mahabad 0.60 0.00 0.89 0.99 0.96 0.99 0.84 0.99 
Saqez 0.83 0.00 0.92 0.93 0.94 1.00 0.88 0.99 
Takab 0.83 0.00 0.80 0.98 0.83 1.00 0.78 0.93 
Zanjan 0.88 0.00 0.91 1.00 0.89 0.94 0.81 0.96 
C5 Tabriz 0.85 0.00 0.98 1.00 0.96 0.98 0.96 0.99 
Sahand 0.85 0.00 0.99 0.98 0.98 0.98 0.96 1.00 
Urmia 0.84 0.00 0.96 1.00 0.99 0.99 0.96 1.00 
Maragheh 0.88 0.00 0.95 0.99 0.97 0.98 0.94 1.00 
Mianeh 0.85 0.00 0.96 1.00 0.98 0.98 0.95 1.00 
Mahabad 0.86 0.00 0.98 1.00 0.98 0.99 0.97 1.00 
Saqez 0.82 0.00 0.95 0.97 0.95 0.99 0.96 1.00 
Takab 0.80 0.00 0.96 0.96 0.98 1.00 0.95 1.00 
Zanjan 0.79 0.00 0.97 1.00 0.96 0.98 0.95 0.99 
M2 Tabriz 0.94 0.00 0.98 0.99 0.99 0.99 0.97 1.00 
Sahand 0.95 0.00 0.99 0.99 0.99 0.99 0.97 1.00 
Urmia 0.95 0.00 0.99 0.99 0.99 1.00 0.98 1.00 
Maragheh 0.96 0.00 0.98 0.99 0.99 0.99 0.97 1.00 
Mianeh 0.96 0.00 0.99 0.99 0.99 0.99 0.97 1.00 
Mahabad 0.96 0.00 0.99 1.00 0.98 1.00 0.98 1.00 
Saqez 0.93 0.00 0.98 0.99 0.98 1.00 0.98 1.00 
Takab 0.94 0.00 1.00 0.99 1.00 1.00 0.98 1.00 
Zanjan 0.91 0.00 1.00 0.99 0.99 1.00 0.97 1.00 
Mean score 0.83 0.00 0.94 0.99 0.95 0.98 0.91 0.99 

Downscaling large-scale rainfall data by the SEML (level 2)

After selecting the best algorithms using the TOPSIS method, the two MARS and LSSVM-RUN algorithms were used in the level 2 SEML to modify the downscale large-scale rainfall data of three GCMs in the nine stations. To this end, as meta-algorithms, the MARS and LSSVM-RUN algorithms integrate the results of eight Level 1 algorithms to produce high-precision outputs. According to Table 6, the outputs of the base algorithms were applied as inputs to the MARS and LSSVM-RUN algorithms. The results of the two algorithms during the historical period were compared, and MARS performed slightly better than the LSSVM-RUN algorithm. In addition, taking into account the MAE, RMSE, RRMSE and R values, downscaled rainfall, in both algorithms, the C5 model showed a higher accuracy than the A1 and M2 models. This is due to the higher resolution of the C5 compared to other models.

Table 6

Level 2 results of downscaling large-scale rainfall with SEML (meta-algorithms)

AlgorithmsModelsStationsAssessment criteria
Train
Test
MAERMSERRMSERMAERMSERRMSER
MARS A1 Tabriz 6.76 10.38 0.33 0.94 8.48 12.01 0.35 0.94 
Sahand 9.04 15.13 0.37 0.93 10.27 15.64 0.39 0.92 
Urmia 6.49 11.20 0.41 0.91 6.74 10.65 0.53 0.85 
Maragheh 7.54 12.54 0.36 0.93 11.00 17.25 0.55 0.84 
Mianeh 7.36 12.24 0.46 0.89 6.51 10.03 0.41 0.91 
Mahabad 5.95 10.12 0.44 0.90 4.79 7.60 0.34 0.94 
Saqez 9.07 13.97 0.49 0.87 9.83 16.18 0.62 0.79 
Takab 4.67 7.43 0.40 0.92 4.37 6.20 0.40 0.92 
Zanjan 4.74 7.56 0.41 0.91 4.11 7.28 0.37 0.93 
C5 Tabriz 6.26 11.56 0.37 0.93 7.20 12.38 0.35 0.94 
Sahand 9.36 14.68 0.37 0.93 11.58 18.45 0.42 0.91 
Urmia 6.36 10.44 0.42 0.91 7.27 11.96 0.46 0.89 
Maragheh 8.77 14.23 0.41 0.91 10.72 17.16 0.53 0.84 
Mianeh 6.51 11.36 0.41 0.91 4.53 6.87 0.31 0.96 
Mahabad 5.27 8.47 0.39 0.92 6.50 11.89 0.47 0.88 
Saqez 8.56 14.45 0.53 0.85 8.86 15.45 0.54 0.85 
Takab 3.86 6.73 0.38 0.92 3.16 6.37 0.36 0.93 
Zanjan 4.44 7.27 0.36 0.93 4.91 8.15 0.52 0.85 
M2 Tabriz 6.22 9.73 0.31 0.95 8.12 12.92 0.37 0.93 
Sahand 9.76 16.07 0.38 0.93 7.96 12.71 0.35 0.94 
Urmia 6.45 10.77 0.44 0.90 6.25 9.73 0.36 0.93 
Maragheh 7.23 12.95 0.37 0.93 9.66 15.86 0.48 0.88 
Mianeh 6.54 11.54 0.44 0.90 7.38 13.78 0.53 0.85 
Mahabad 4.70 8.00 0.37 0.93 6.44 11.83 0.47 0.88 
Saqez 8.58 14.45 0.51 0.86 7.94 12.28 0.48 0.88 
Takab 4.48 6.87 0.39 0.92 5.27 9.58 0.52 0.87 
Zanjan 4.53 7.72 0.40 0.92 5.09 9.00 0.52 0.86 
LSSVM-RUN A1 Tabriz 1.40 2.32 0.07 1.00 8.75 15.03 0.47 0.88 
Sahand 7.36 12.85 0.31 0.95 9.86 17.15 0.44 0.90 
Urmia 5.64 9.26 0.39 0.92 6.51 10.93 0.39 0.92 
Maragheh 3.63 6.02 0.17 0.99 7.76 12.96 0.41 0.91 
Mianeh 5.94 10.40 0.39 0.92 7.89 12.90 0.53 0.85 
Mahabad 5.36 8.92 0.39 0.92 5.34 9.82 0.43 0.90 
Saqez 5.46 10.08 0.36 0.93 8.19 15.58 0.55 0.83 
Takab 3.91 6.07 0.34 0.94 4.90 7.95 0.46 0.89 
Zanjan 4.64 7.41 0.37 0.93 6.64 10.28 0.65 0.78 
C5 Tabriz 2.14 4.03 0.13 0.99 7.72 15.95 0.45 0.89 
Sahand 6.52 10.95 0.27 0.96 9.12 16.01 0.38 0.93 
Urmia 5.50 9.44 0.35 0.94 8.40 14.19 0.67 0.74 
Maragheh 6.30 11.35 0.33 0.94 8.34 15.64 0.45 0.89 
Mianeh 3.33 6.03 0.24 0.97 6.49 13.06 0.46 0.88 
Mahabad 2.38 4.26 0.18 0.98 5.36 8.30 0.39 0.92 
Saqez 4.75 9.14 0.34 0.94 8.17 14.43 0.48 0.88 
Takab 2.02 3.87 0.21 0.98 3.46 6.20 0.37 0.93 
Zanjan 1.79 3.46 0.18 0.98 3.83 7.23 0.41 0.91 
M2 Tabriz 6.51 11.43 0.36 0.94 8.59 13.71 0.42 0.91 
Sahand 8.21 13.73 0.33 0.95 9.13 16.65 0.41 0.92 
Urmia 5.53 9.44 0.37 0.93 7.15 12.07 0.50 0.87 
Maragheh 7.32 12.87 0.37 0.93 8.25 14.58 0.46 0.89 
Mianeh 6.49 11.57 0.43 0.91 6.58 10.62 0.44 0.90 
Mahabad 5.09 8.54 0.36 0.93 5.02 7.81 0.38 0.93 
Saqez 6.08 10.96 0.38 0.93 7.50 13.02 0.52 0.86 
Takab 3.19 5.52 0.30 0.96 4.71 7.26 0.48 0.88 
Zanjan 3.10 5.43 0.30 0.96 5.86 12.21 0.60 0.81 
AlgorithmsModelsStationsAssessment criteria
Train
Test
MAERMSERRMSERMAERMSERRMSER
MARS A1 Tabriz 6.76 10.38 0.33 0.94 8.48 12.01 0.35 0.94 
Sahand 9.04 15.13 0.37 0.93 10.27 15.64 0.39 0.92 
Urmia 6.49 11.20 0.41 0.91 6.74 10.65 0.53 0.85 
Maragheh 7.54 12.54 0.36 0.93 11.00 17.25 0.55 0.84 
Mianeh 7.36 12.24 0.46 0.89 6.51 10.03 0.41 0.91 
Mahabad 5.95 10.12 0.44 0.90 4.79 7.60 0.34 0.94 
Saqez 9.07 13.97 0.49 0.87 9.83 16.18 0.62 0.79 
Takab 4.67 7.43 0.40 0.92 4.37 6.20 0.40 0.92 
Zanjan 4.74 7.56 0.41 0.91 4.11 7.28 0.37 0.93 
C5 Tabriz 6.26 11.56 0.37 0.93 7.20 12.38 0.35 0.94 
Sahand 9.36 14.68 0.37 0.93 11.58 18.45 0.42 0.91 
Urmia 6.36 10.44 0.42 0.91 7.27 11.96 0.46 0.89 
Maragheh 8.77 14.23 0.41 0.91 10.72 17.16 0.53 0.84 
Mianeh 6.51 11.36 0.41 0.91 4.53 6.87 0.31 0.96 
Mahabad 5.27 8.47 0.39 0.92 6.50 11.89 0.47 0.88 
Saqez 8.56 14.45 0.53 0.85 8.86 15.45 0.54 0.85 
Takab 3.86 6.73 0.38 0.92 3.16 6.37 0.36 0.93 
Zanjan 4.44 7.27 0.36 0.93 4.91 8.15 0.52 0.85 
M2 Tabriz 6.22 9.73 0.31 0.95 8.12 12.92 0.37 0.93 
Sahand 9.76 16.07 0.38 0.93 7.96 12.71 0.35 0.94 
Urmia 6.45 10.77 0.44 0.90 6.25 9.73 0.36 0.93 
Maragheh 7.23 12.95 0.37 0.93 9.66 15.86 0.48 0.88 
Mianeh 6.54 11.54 0.44 0.90 7.38 13.78 0.53 0.85 
Mahabad 4.70 8.00 0.37 0.93 6.44 11.83 0.47 0.88 
Saqez 8.58 14.45 0.51 0.86 7.94 12.28 0.48 0.88 
Takab 4.48 6.87 0.39 0.92 5.27 9.58 0.52 0.87 
Zanjan 4.53 7.72 0.40 0.92 5.09 9.00 0.52 0.86 
LSSVM-RUN A1 Tabriz 1.40 2.32 0.07 1.00 8.75 15.03 0.47 0.88 
Sahand 7.36 12.85 0.31 0.95 9.86 17.15 0.44 0.90 
Urmia 5.64 9.26 0.39 0.92 6.51 10.93 0.39 0.92 
Maragheh 3.63 6.02 0.17 0.99 7.76 12.96 0.41 0.91 
Mianeh 5.94 10.40 0.39 0.92 7.89 12.90 0.53 0.85 
Mahabad 5.36 8.92 0.39 0.92 5.34 9.82 0.43 0.90 
Saqez 5.46 10.08 0.36 0.93 8.19 15.58 0.55 0.83 
Takab 3.91 6.07 0.34 0.94 4.90 7.95 0.46 0.89 
Zanjan 4.64 7.41 0.37 0.93 6.64 10.28 0.65 0.78 
C5 Tabriz 2.14 4.03 0.13 0.99 7.72 15.95 0.45 0.89 
Sahand 6.52 10.95 0.27 0.96 9.12 16.01 0.38 0.93 
Urmia 5.50 9.44 0.35 0.94 8.40 14.19 0.67 0.74 
Maragheh 6.30 11.35 0.33 0.94 8.34 15.64 0.45 0.89 
Mianeh 3.33 6.03 0.24 0.97 6.49 13.06 0.46 0.88 
Mahabad 2.38 4.26 0.18 0.98 5.36 8.30 0.39 0.92 
Saqez 4.75 9.14 0.34 0.94 8.17 14.43 0.48 0.88 
Takab 2.02 3.87 0.21 0.98 3.46 6.20 0.37 0.93 
Zanjan 1.79 3.46 0.18 0.98 3.83 7.23 0.41 0.91 
M2 Tabriz 6.51 11.43 0.36 0.94 8.59 13.71 0.42 0.91 
Sahand 8.21 13.73 0.33 0.95 9.13 16.65 0.41 0.92 
Urmia 5.53 9.44 0.37 0.93 7.15 12.07 0.50 0.87 
Maragheh 7.32 12.87 0.37 0.93 8.25 14.58 0.46 0.89 
Mianeh 6.49 11.57 0.43 0.91 6.58 10.62 0.44 0.90 
Mahabad 5.09 8.54 0.36 0.93 5.02 7.81 0.38 0.93 
Saqez 6.08 10.96 0.38 0.93 7.50 13.02 0.52 0.86 
Takab 3.19 5.52 0.30 0.96 4.71 7.26 0.48 0.88 
Zanjan 3.10 5.43 0.30 0.96 5.86 12.21 0.60 0.81 

In the Supplementary material, Figures S1 and S2 display both the observed rainfall time series and the downscaled rainfall time series of meta-algorithms during training and testing periods. As can be seen in Supplementary material, Figures S1 and S2, the difference between the values estimated by the MARS and LSSVM-RUN algorithms and the observed values was very small in most cases. The high R-value indicates the excellent performance of algorithms in downscaling rainfall data. The accuracy of the algorithms was significantly improved by applying the output of the base algorithms as inputs to the meta-algorithm, as shown in Supplementary material, Figures S1 and S2. According to the results, the SEML's downscaling accuracy was increased by over 45% because of the simultaneous use of the advantages of different algorithms in the SEML. The LSSVM-RUN performed well, as evidenced by the results. The proximity of the observed and downscaled values means that the RUN algorithm had a positive effect on improving the performance of LSSVM. According to the downscaling results with the MARS algorithm and C5 model, the highest downscaling accuracy was obtained in the Mianeh (MAE = 4.53, RMSE = 6.87, RRMSE = 0.31, R = 0.96), and Takab (MAE = 3.16, RMSE = 6.37, RRMSE = 0.36, R = 0.93) stations. The accuracy of downscaling using the LSSVM-RUN and C5 model in the Takab station was the highest among stations (MAE = 3.46, RMSE = 6.20, RRMSE = 0.37, R = 0.93).

Prediction rainfall under climate change by meta-algorithms (2021–2050)

After training and testing the SEML for downscaling the rainfall in the historical period, the rainfall in nine stations was predicted under three scenarios SCF, SCM, and SCS of different GCMs, including A1, C5, and M2. Tables 7 and Supplementary material, Table A3 show the results of predicting rainfall in investigated scenarios and stations using the MARS algorithm as a meta-algorithm of the SEML. In this table, the mean (Mean), minimum (Min), and maximum (Max) of observed and predicted rainfall values were compared. According to Tables 7 and Supplementary material, Table A3, by comparing the observed and predicted values in different GCMs and scenarios, in Tabriz station, the highest rate of rainfall changes (−0.13%) was related to the C5 and SCM, and the lowest rate (−0.06%) was for A1 and the SCF. In Sahand station, the most predicted change was −0.12% by A1 and SCS. However, the lowest rainfall change (−0.08%) was related to the C5 and the SCS. At Urmia station, the most change was predicted to be −0.06% by C5 and its three scenarios and the lowest rate was −0.05% by A1 and M2 and their three scenarios. A1 predicted the highest rainfall reduction (−0.16%) at Maragheh station under SCM and SCS. However, the smallest change (−0.05%) was in C5 in all scenarios. At the Mianeh station, the highest rainfall change was projected to be −0.09% by C5 and SCS, and the lowest rate (−0.07%) was related to M2 and its three scenarios. At Mahabad station, the C5 and SCM predicted a rainfall change rate of −0.10%, while M2 and its three scenarios predicted a rainfall change rate of −0.03%. At Saqez station, the highest rainfall change (−0.12%) was predicted by M2 under SCF and SCM scenarios. Also, the lowest change rate of rainfall (−0.04%) was predicted by model A1 and SCM scenarios. Takab station was the only station where the rainfall will increase in the future (0.06%). This increase was related to C5 under its three scenarios. In Zanjan station, the A1 predicted a rainfall change rate of −0.06% in all three scenarios. However, the lowest rate (−0.02%) was related to the C5 and its three scenarios.

Table 7

Results of prediction rainfall at the different stations by MARS algorithm

A1 SCFA1 SCMA1 SCSC5 SCFC5 SCMC5 SCSM2 SCFM2 SCMM2 SCS
 Tabriz 
Mean-Obs 31.88 31.88 31.88 31.88 31.88 31.88 31.88 31.88 31.88 
Mean-Pred 29.53 29.51 29.12 27.91 27.85 27.88 29.24 29.21 29.13 
Min-Obs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
Min-Pred 2.56 5.07 16.12 14.14 15.56 15.58 3.75 0.02 2.80 
Max-Obs 214.90 214.90 214.90 214.90 214.90 214.90 214.90 214.90 214.90 
Max-Pred 73.24 63.12 43.39 41.13 44.26 42.23 70.78 68.92 64.14 
Change mean (%) −0.06 −0.07 −0.09 −0.12 −0.13 −0.12 −0.08 −0.08 −0.09 
 Urmia 
Mean-Obs 26.02 26.02 26.02 26.02 26.02 26.02 26.02 26.02 26.02 
Mean-Pred 24.68 24.68 24.67 24.43 24.42 24.42 24.63 24.63 24.62 
Min-Obs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
Min-Pred 8.04 8.98 7.39 6.29 2.75 4.67 0.00 2.81 0.00 
Max-Obs 165.72 165.72 165.72 165.72 165.72 165.72 165.72 165.72 165.72 
Max-Pred 41.56 41.18 40.19 50.55 43.82 41.94 56.36 74.66 52.74 
Change mean (%) −0.05 −0.05 −0.05 −0.06 −0.06 −0.06 −0.05 −0.05 −0.05 
 Zanjan 
Mean-Obs 20.52 20.52 20.52 20.52 20.52 20.52 20.52 20.52 20.52 
Mean-Pred 19.35 19.29 19.28 20.08 20.08 20.08 19.87 19.90 19.88 
Min-Obs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
Min-Pred 5.30 4.19 5.10 5.01 4.81 7.46 4.28 3.78 1.66 
Max-Obs 114.84 114.84 114.84 114.84 114.84 114.84 114.84 114.84 114.84 
Max-Pred 51.31 60.46 66.41 39.73 35.81 38.89 34.24 35.27 42.28 
Change mean (%) −0.06 −0.06 −0.06 −0.02 −0.02 −0.02 −0.03 −0.03 −0.03 
A1 SCFA1 SCMA1 SCSC5 SCFC5 SCMC5 SCSM2 SCFM2 SCMM2 SCS
 Tabriz 
Mean-Obs 31.88 31.88 31.88 31.88 31.88 31.88 31.88 31.88 31.88 
Mean-Pred 29.53 29.51 29.12 27.91 27.85 27.88 29.24 29.21 29.13 
Min-Obs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
Min-Pred 2.56 5.07 16.12 14.14 15.56 15.58 3.75 0.02 2.80 
Max-Obs 214.90 214.90 214.90 214.90 214.90 214.90 214.90 214.90 214.90 
Max-Pred 73.24 63.12 43.39 41.13 44.26 42.23 70.78 68.92 64.14 
Change mean (%) −0.06 −0.07 −0.09 −0.12 −0.13 −0.12 −0.08 −0.08 −0.09 
 Urmia 
Mean-Obs 26.02 26.02 26.02 26.02 26.02 26.02 26.02 26.02 26.02 
Mean-Pred 24.68 24.68 24.67 24.43 24.42 24.42 24.63 24.63 24.62 
Min-Obs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
Min-Pred 8.04 8.98 7.39 6.29 2.75 4.67 0.00 2.81 0.00 
Max-Obs 165.72 165.72 165.72 165.72 165.72 165.72 165.72 165.72 165.72 
Max-Pred 41.56 41.18 40.19 50.55 43.82 41.94 56.36 74.66 52.74 
Change mean (%) −0.05 −0.05 −0.05 −0.06 −0.06 −0.06 −0.05 −0.05 −0.05 
 Zanjan 
Mean-Obs 20.52 20.52 20.52 20.52 20.52 20.52 20.52 20.52 20.52 
Mean-Pred 19.35 19.29 19.28 20.08 20.08 20.08 19.87 19.90 19.88 
Min-Obs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
Min-Pred 5.30 4.19 5.10 5.01 4.81 7.46 4.28 3.78 1.66 
Max-Obs 114.84 114.84 114.84 114.84 114.84 114.84 114.84 114.84 114.84 
Max-Pred 51.31 60.46 66.41 39.73 35.81 38.89 34.24 35.27 42.28 
Change mean (%) −0.06 −0.06 −0.06 −0.02 −0.02 −0.02 −0.03 −0.03 −0.03 

The MARS algorithm's observed and predicted rainfall box plot for three GCMs under three scenarios in different stations is shown in Figure 4 and Supplementary material, Figure S3. Comparing the observed data and predicted results in different stations revealed that rainfall decreased in all stations except for Takab station. This is due to the assumptions of GCM scenarios. For example, in each scenario, the increase in greenhouse gas emissions is distinct from other ones. According to box plots, the median value in all scenarios and stations was increased in the future. According to the predicted values at different stations, the amount of outlier data at Tabriz, Sahand, Urmia, and Takab stations in C5 was lower than in the other GCMs. Other stations in the M2 had fewer outliers. Furthermore, the data predicted by C5 and M2 had less dispersion and less uncertainty.
Figure 4

Box plot diagram observed and predicted rainfall by the MARS algorithm.

Figure 4

Box plot diagram observed and predicted rainfall by the MARS algorithm.

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Table 8 and Supplementary material, Table S4 show the results of rainfall prediction in investigated scenarios and stations by the LSSVM-RUN. According to these tables, in Tabriz and Sahand stations, the highest rainfall changes (−0.17 and −0.20%) were related to A1 and SCS. Also, the lowest rate of rainfall changes was predicted to be −0.08% by C5 and SCS in Tabriz and −0.6% by M2 and its two scenarios (SCM and SCS). At Urmia station, the highest rainfall change was predicted (−0.05%) by M2 and its three scenarios, and the lowest rate (−0.03%) was related to A1 and its three scenarios. At Maragheh station, the most changes (−0.20%) in rainfall were in the A1 and its three scenarios. Also, the lowest rate (−0.05%) was related to C5 and its three scenarios. At Mianeh station, the highest rate of rainfall changes was predicted to be −0.09% to the C5 and SCF, and the lowest rate (−0.05%) was related to A1 and SCM. At the Mahabad station, the highest changes (−0.11%) were in the A1 model and the SCM. Also, the lowest rainfall changes were predicted to be −0.06% by C5 (SCF) and M2 (SCM and SCS). In Saqez stations, the highest rate of rainfall changes (−0.15%) was predicted by M2 and SCM, and, the lowest rate was predicted to be −0.04% by A1 (SCS) and C5 in its three scenarios. Takab station is the only station where rainfall will increase in the future. The highest rate of rainfall changes was predicted to be 0.03% by the C5 and SCF. Also, the lowest rate (−0.03%) was related to A1 (SCF) and M2 (SCF). At Zanjan station, the highest rate of rainfall changes (−0.05%) was predicted by the M2 and its three scenarios, and, the lowest rate (−0.01%) was related to C5 and its three scenarios. Similar to the prediction with the MARS algorithm, the highest increase of rainfall prediction with MARS and LSSVM-RUN algorithms in the Takab station was predicted at 0.06 and 0.03%, respectively. Also, the lowest rate of rainfall changes was related to the Zanjan station by −0.02 and −0.01%, respectively. In most scenarios, the rate of rainfall changes will decrease by −0.04% in most stations.

Table 8

Results of prediction rainfall at the different stations by LSSVM-RUN algorithm

A1 SCFA1 SCMA1 SCSC5 SCFC5 SCMC5 SCSM2 SCFM2 SCMM2 SCS
 Tabriz 
Mean-Obs 31.88 31.88 31.88 31.88 31.88 31.88 31.88 31.88 31.88 
Mean-Pred 28.43 28.22 26.52 28.91 28.84 28.88 28.78 28.89 28.81 
Min-Obs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
Min-Pred 5.11 7.48 15.04 15.25 16.64 16.61 4.50 0.12 2.73 
Max-Obs 214.90 214.90 214.90 214.90 214.90 214.90 214.90 214.90 214.90 
Max-Pred 73.51 63.55 42.19 43.46 46.41 44.84 65.84 65.20 67.79 
Change mean (%) −0.11 −0.11 −0.17 −0.09 −0.10 −0.08 −0.10 −0.09 −0.10 
 Urmia 
Mean-Obs 26.02 26.02 26.02 26.02 26.02 26.02 26.02 26.02 26.02 
Mean-Pred 25.16 25.20 25.21 24.86 24.87 24.88 24.74 24.66 24.65 
Min-Obs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
Min-Pred 6.37 7.36 5.22 5.68 2.76 3.91 0.00 3.12 0.00 
Max-Obs 165.72 165.72 165.72 165.72 165.72 165.72 165.72 165.72 165.72 
Max-Pred 43.02 42.47 41.60 51.76 45.43 44.80 61.01 74.24 54.84 
Change mean (%) −0.03 −0.03 −0.03 −0.04 −0.04 −0.04 −0.05 −0.05 −0.05 
 Zanjan 
Mean-Obs 20.52 20.52 20.52 20.52 20.52 20.52 20.52 20.52 20.52 
Mean-Pred 19.74 19.71 19.77 20.26 20.23 20.23 19.53 19.55 19.54 
Min-Obs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
Min-Pred 5.63 4.14 5.74 4.03 4.21 6.65 4.02 3.22 1.63 
Max-Obs 114.84 114.84 114.84 114.84 114.84 114.84 114.84 114.84 114.84 
Max-Pred 50.32 68.18 72.84 38.97 36.42 38.22 34.97 35.58 41.99 
Change mean (%) −0.04 −0.04 −0.04 −0.01 −0.01 −0.01 −0.05 −0.05 −0.05 
A1 SCFA1 SCMA1 SCSC5 SCFC5 SCMC5 SCSM2 SCFM2 SCMM2 SCS
 Tabriz 
Mean-Obs 31.88 31.88 31.88 31.88 31.88 31.88 31.88 31.88 31.88 
Mean-Pred 28.43 28.22 26.52 28.91 28.84 28.88 28.78 28.89 28.81 
Min-Obs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
Min-Pred 5.11 7.48 15.04 15.25 16.64 16.61 4.50 0.12 2.73 
Max-Obs 214.90 214.90 214.90 214.90 214.90 214.90 214.90 214.90 214.90 
Max-Pred 73.51 63.55 42.19 43.46 46.41 44.84 65.84 65.20 67.79 
Change mean (%) −0.11 −0.11 −0.17 −0.09 −0.10 −0.08 −0.10 −0.09 −0.10 
 Urmia 
Mean-Obs 26.02 26.02 26.02 26.02 26.02 26.02 26.02 26.02 26.02 
Mean-Pred 25.16 25.20 25.21 24.86 24.87 24.88 24.74 24.66 24.65 
Min-Obs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
Min-Pred 6.37 7.36 5.22 5.68 2.76 3.91 0.00 3.12 0.00 
Max-Obs 165.72 165.72 165.72 165.72 165.72 165.72 165.72 165.72 165.72 
Max-Pred 43.02 42.47 41.60 51.76 45.43 44.80 61.01 74.24 54.84 
Change mean (%) −0.03 −0.03 −0.03 −0.04 −0.04 −0.04 −0.05 −0.05 −0.05 
 Zanjan 
Mean-Obs 20.52 20.52 20.52 20.52 20.52 20.52 20.52 20.52 20.52 
Mean-Pred 19.74 19.71 19.77 20.26 20.23 20.23 19.53 19.55 19.54 
Min-Obs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
Min-Pred 5.63 4.14 5.74 4.03 4.21 6.65 4.02 3.22 1.63 
Max-Obs 114.84 114.84 114.84 114.84 114.84 114.84 114.84 114.84 114.84 
Max-Pred 50.32 68.18 72.84 38.97 36.42 38.22 34.97 35.58 41.99 
Change mean (%) −0.04 −0.04 −0.04 −0.01 −0.01 −0.01 −0.05 −0.05 −0.05 

Figure 5 and Supplementary material, Figure S4 compare the data observed and predicted by the LSSVM-RUN at different stations. According to the box plots at different stations, the predictions made by the LSSVM-RUN, such as the MARS, showed a decrease in rainfall in all scenarios and stations in the future period. As in these figures, the median value was increased in all future scenarios and stations. Also, the amount of outlier's data and dispersion was less in the C5 and M2 models, which indicates the accuracy and clarity of these models.
Figure 5

Box plot diagram observed and predicted rainfall by the LSSVM-RUN algorithm.

Figure 5

Box plot diagram observed and predicted rainfall by the LSSVM-RUN algorithm.

Close modal

The results of rainfall prediction in 2021–2050 by meta-algorithms showed that rainfall would decrease in all scenarios and stations (except Takab station). Decreased rainfall can have several causes: (1) increasing temperature and global warming due to population growth and excessive energy consumption, (2) increased air pollution and dust that prevents the accumulation of fine vapor particles and the formation of raindrops, (3) reduce soil moisture and vegetation and reduce vertical air movement, (4) increase in greenhouse gases. Decreasing rainfall can cause major water scarcity and drought, reduced soil moisture, increased air pollution, and plant destruction.

Uncertainty analysis

Figure 6 shows uncertainty analysis based on the BM for MARS and LSSVM-RUN as meta-algorithms. The BM calculates the 95% confidence interval of the uncertainty prediction. The prediction scenarios are less uncertain when the R-factor coefficient is lower. In the 2021–2050 horizon, according to the results in Figure 6, the mean R-factor coefficient in the LSSVM-RUN was 55% lower than the MARS, indicating less uncertainty in the results of the LSSVM-RUN. This means that the predicted rainfall results by the LSSVM-RUN are more probable than the MARS. In comparison to two algorithms and all stations, the top three results are as follows:
  • R-factor = 0.63; in Takab station, A1 and SCF from LSSVM-RUN had less uncertainty (for this model and scenario the amount of rainfall decreased by −0.03%, which had more certainty than other models and scenarios).

  • R-factor = 0.66; in Mianeh station, M2 and SCS from MARS had less uncertainty (for this model and scenario the amount of rainfall decreased by −0.09%, which had more certainty than other models and scenarios).

  • R-factor = 0.67; At Zanjan station, M2 and SCF from LSSVM-RUN had less uncertainty (for this model and scenario the amount of rainfall decreased by −0.01%, which had more certainty than other models and scenarios).

Figure 6

Obtained R-factor by BM at the different stations.

Figure 6

Obtained R-factor by BM at the different stations.

Close modal
Figure 7 and Supplementary material, Figure S5 show the models' and scenarios' uncertainty analysis results with the least uncertainty by the LSSVM-RUN. The shaded area (95PPU) shows the results of the predicted rainfall values in the upper and lower bands. The models and prediction scenarios are less uncertain if there are fewer differences between the upper and lower bands. The results revealed a high degree of uncertainty in GCMs and emission scenarios of CMIP6. The least uncertainty was related to Takab station. Also, the M2 model and the SCF and SCM scenarios had the least uncertainty. Among the meta-algorithms, the LSSVM-RUN algorithm reported less uncertainty than MARS, therefore it was used to predict rainfall in the 2021–2050 periods with higher accuracy.
Figure 7

The uncertainty bound for GCMs and scenarios with less uncertainty by the LSSVM-RUN algorithm.

Figure 7

The uncertainty bound for GCMs and scenarios with less uncertainty by the LSSVM-RUN algorithm.

Close modal

Predicting rainfall in climate change conditions is crucial for water resource planning and management. However, rainfall modeling and prediction under the influence of climate change are often associated with bias and uncertainty. One of the most important approaches to investigate rainfall changes from the mentioned conditions is the use of stacking ensemble ML algorithms. Hence, in the present study, the novel modeling technique (using SEML) for downscaling the rainfall in Lake Urmia and Sefidrood basins was introduced. This downscaling method used MLR, MNLR, MARS, M5, RF, LSBoost, LSSVM-GS, and the novel hybrid algorithm (LSSVM-RUN) for modeling and predicting rainfall (as base and meta-algorithms); also, the TOPSIS method was used for selecting meta-algorithms. The results were examined for the date (1985–2014) and future (2021–2050). The main results of this research are as follows:

  1. The accuracy of the base algorithms at level 1 of the SEML was low. The TOPSIS method's results revealed that MARS and LSSVM-RUN (score 0.99) were the best algorithms and were chosen as meta-algorithms.

  2. In the second level of the SEML, the accuracy of rainfall modeling became very good. The highest modeling accuracy was obtained by MARS in the Mianeh (MAE = 4.53, RMSE = 6.87, RRMSE = 0.31, R = 0.96), and Takab (MAE = 3.16, RMSE = 6.37, RRMSE = 0.36, R = 0.93) stations in model C5. Also, in modeling using the LSSVM-RUN of the Takab station (MAE = 3.46, RMSE = 6.20, RRMSE = 0.37, R = 0.93) in the C5 model, it had the highest accuracy among the stations. The reason for this was the combination of the results of the base algorithms in level 2. In addition, at this level, thanks to the combination of the advantages of all the algorithms, the accuracy of the meta-algorithms became very high, which showed the power of SEML.

  3. According to the prediction results, the mean rainfall in all stations (except Takab station in C5 model scenarios) decreased in all models and scenarios. The Maragheh station (0.20%) had the highest decrease, while the Takab station (0.06%) had the highest increase.

  4. GCMs and CMIP6 scenarios were highly uncertain. However, the results obtained thanks to the new hybrid algorithm were less uncertain. Among the GCMs and scenarios considered, the A1 model and the SCF scenario in the Takab stations (R-factor = 0.63) had the least uncertainty.

The employed approach for reasonable performance needs to have sufficient lengths of data. Also predicting future meteorological and hydrological components requires large-scale data in the future.

The findings of this study (values of rainfall in the future) can assist water resource management decision-makers in managing the investigated basin in the future. For example, the obtained rainfall in future periods can be used to reduce the damages of climate change and drought, planning to define cropping patterns in the near future horizon, and optimal operation of reservoirs.

Furthermore, the technique that was introduced can be extended to other new GCMs and emission scenarios from CMIP6. Also, this technique can be used to predict other meteorological parameters in other basins to encourage locals about their climate change and severe weather concerns, as warned in Bruine de Bruin & Dugan (2022) article.

S.F. conceptualized the whole article; M.V.A., M.K. and A.M.-B. rendered support in data curation; M.V.A., M.K., A.M.-B. and S.F. rendered support in formal analysis; M.V.A., M.K. and S.F. investigated the data; M.V.A., M.K., A.M.-B. and S.F. developed the methodology; S.F. administered the project; M.V.A., M.K. and A.M.-B. brought the resources; M.V.A., M.K. and A.M.-B. worked on preparing the software; S.F. supervised the work; M.V.A., M.K. and A.M.-B. validated the data; M.V.A., M.K. and A.M.-B. visualized the project; M.K. and A.M.-B. wrote the original draft; S.F. and M.V.A. wrote the review and edited the article. All authors have seen and approved the final manuscript.

The authors have agreed to publish the study in Environment, Development and Sustainability.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.

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Supplementary data