Temporal-spatial modeling of precipitation using an approach based on MLR, ANN, HBA-ANN hybrid algorithm

Introduction
Life on Earth is influenced by precipitation. Precipitation is one of the most significant factors that affect the hydrological cycle. Considering that precipitation is non-linear, complex, and can be changed according to spatial and temporal, estimating the amount of this important atmospheric factor in each month or year for each region and watershed is particularly important in managing and optimizing water resources. Various optimization models and algorithms have been proposed for modeling hydrological systems in recent decades. These algorithms have significantly reduced errors and increased accuracy. Still, since hydrological systems rely on random events, none of the methods can be completely and accurately selected as a superior model for modeling and estimating. The honey badger algorithm is an innovative algorithm that requires a few iterations to achieve an optimal solution, and this increases the speed of reaching the desired results. In current study investigates the performance of three models, including multiple linear regression (MLR), artificial neural network (ANN), and hybrid artificial neural network with honey badger optimization algorithm (HBA-ANN) for modeling the temporal and spatial precipitation in East Azarbaijan province. The best-developed model was selected by evaluation criteria such as R, RMSE, NRMSE, MBE, and NSE, the best model is selected.
 
Materials and Methods
The MLR model is one of the methods to analyze and investigate several variables. In this method, the model has one dependent variable and several independent variables, so that a linear equation is generated between the independent variables called X1, X2, ..., Xn and the dependent variable Y. ANN is a black box model of neural networks in the human brain. One of the most used methods is the BP method, which includes two stages. In the first stage, which is entitled feed-forward, the error value is calculated, after comparing output and objective values. In the second stage, which is labeled the back-propagation, the error value calculated in the previous step is corrected. The mentioned two stages continue until the output of the model approaches the desired output. The HBA is a new algorithm that simulates the honey-seeking behavior of a creature called the honey badger. The HBA includes two stages. In the first phase, the locations of this creature are calculated, and in the second phase, the exact distance between the HBA and the prey (dj) is calculated based on the honey intensity (S) and the honey smell intensity (Ij), as well as its new and optimal location for the prey Xnew. In the HBA-ANN model, the HBA algorithm is used to determine the most optimal output value in the ANN and increase performance in modeling. Therefore, the developed hybrid model can have the characteristics of both ANN and HBA methods.
 
Results and Discussion
In this study, in the first stage, the temporal modeling, and in the second stage, the spatial modeling of the monthly precipitation of 18 stations in East Azarbaijan province during the period of 1996-2022 using MLP, ANN, and HBA-ANN models has been paid. For temporal modeling of precipitation, one and two-month precipitation delay steps of the stations were used as input parameters. The first 70% of the dataset was selected for the training phase and the last 30% of the dataset was selected for the testing phase. Based on the results obtained from evaluation criteria and graphic diagrams, it can be concluded that the HBA-ANN model indicated significant accuracy compared to other models in the temporal modeling of precipitation. Also, by comparing the results of the stations in the HBA-ANN model, the Heris station with R =0.94, RMSE=2.25, NSE=0.79, NRMSE=0.04, and MBE=1.06 in the testing stage performed better compared with other stations. For spatial modeling of precipitation, the geographic coordinates of the stations, which include longitude, latitude, and altitude, are used as input parameters, and average monthly precipitation is used as the output parameter. From eighteen stations, 70% of the stations were selected for the training phase and 30% of the stations were selected for the testing phase. Based on the results obtained from R=0.95, RMSE=1.03, NSE =0.92, NRMSE = 0.03, and MBE = -0.81 and graphical diagrams, it can be concluded that the HBA-ANN model revealed significant accuracy compared to other models in spatial modeling of precipitation.
 
Conclusion
Precipitation is one of the most important factors that significantly change the hydrological cycle. Therefore, modeling and estimating this parameter is vital. In this study, the performance of multiple linear regression (MLR), artificial neural network (ANN), and hybrid ANN using honey badger algorithm (HBA-ANN) models were used for the spatial and temporal modeling of precipitation in East Azarbaijan province. For spatial modeling, the time delay steps of one and two months of station precipitation were selected as input parameters. Also, for temporal modeling, the longitude, latitude, and altitude parameters were used. The mentioned models were evaluated by R, RMSE, NSE, NRMSE, and MBE assessment criteria. According to the results of temporal modeling, the HBA-ANN model for all stations, especially Heris station with R equal to 0.94, RMSE equal to 2.25, NSE equal to 0.79, NRMSE equal to 0.04, and MBEequal to 1.06 is selected as the superior model. Also, based on the results obtained from spatial modeling, the HBA-ANN model with R equal to 0.95, RMSE equal to 1.03, NSE equal to 0.92, NRMSE equal to 0.03, and MBEequal to -0.81 was selected as the best model. The MLR and ANN models, respectively, presented a relatively poor performance compared to the developed hybrid model.