Prediction of Copper and Manganese Concentrations in Citrus Leaves Based on Easily Measured Soil Characteristics: Comparison of Stepwise Regression Models and Gene Expression Programming

Background and Objectives
The availability of micronutrients, particularly copper (Cu) and manganese (Mn), is critical for sustainable citrus production. Cu serves as a vital component of enzymes involved in photosynthesis and respiration, while Mn is crucial for enzyme activation and chlorophyll synthesis. Deficiencies in these nutrients can severely compromise plant vitality and yield, posing a significant economic threat to agricultural regions. This challenge is particularly acute in arid areas like southern Kerman, Iran, where the calcareous soils (characterized by high pH and low organic matter content) severely limit the bioavailability of Cu and Mn. Consequently, there is an urgent need to develop accurate predictive methods for assessing the nutritional status of citrus trees to guide precision fertilization strategies. Traditionally, researchers have relied on linear statistical methods, such as Stepwise Regression (SWR), to model the relationship between soil properties and nutrient uptake. However, these models are fundamentally limited by their assumption of linearity, whereas soil-plant systems are inherently non-linear and interactive. This limitation has prompted a shift toward more sophisticated machine learning approaches, such as Gene Expression Programming (GEP). GEP is an evolutionary algorithm that offers a unique advantage by producing explicit, transparent mathematical equations, thereby combining high predictive accuracy with interpretability. This study aims to fill a critical research gap by directly comparing the performance of traditional stepwise regression against GEP for predicting citrus leaf Cu and Mn concentrations under the specific soil conditions of southern Kerman. The central hypothesis was that the non-linear GEP model would significantly outperform its linear counterpart.
 
Materials and Method
This study was conducted across 40 commercial Valencia orange orchards in southern Kerman, Iran. From each orchard, composite soil samples were collected from two depths (0–30 cm and 31–60 cm), alongside leaf samples from 4- to 6-month-old spring shoots. In the laboratory, leaf Cu and Mn concentrations were quantified using atomic absorption spectrometry. Soil samples were analyzed for key physicochemical properties using standard protocols, including texture, pH, electrical conductivity (EC), organic carbon (OC), total neutralizing value (TNV), and available phosphorus (P). Two distinct modeling approaches were developed and compared: (1) Stepwise Multiple Linear Regression (SWR) was employed to generate linear predictive models, and (2) GEP was implemented using GeneXproTools software to generate non-linear models. To ensure robust model validation, the full dataset (n=40) was randomly partitioned into a training set (70%, 28 samples) and a testing set (30%, 12 samples). The performance of all developed models was rigorously evaluated using the Coefficient of Determination (R²), Root Mean Square Error (RMSE), and Mean Absolute Error (MAE).
 
Results
Descriptive statistics showed that the soils were predominantly neutral to alkaline (mean pH 7.75). Pearson correlation analysis revealed that citrus leaf Cu concentration was negatively correlated with soil pH (r=-0.41) and positively with OC (r=0.39). Similarly, citrus leaf Mn concentration showed negative correlations with both pH (r=-0.33) and EC (r=-0.29).
The SWR models yielded limited predictive accuracy on the test data. For leaf Cu, the SWR model (predictors: pH, clay, P) resulted in an R² of 0.36 and an RMSE of 1.30 mg/kg. The model for leaf Mn concentration (predictors: EC, pH) exhibited lower performance, with an R² of 0.28 and an RMSE of 5.31 mg/kg. In stark contrast, the GEP models demonstrated vastly superior predictive power. The optimal GEP model for Cu (GEP3; inputs: pH, OC, and clay) achieved an R² of 0.60 and an RMSE of 0.82 mg/kg on the test set. The best model for Mn (GEP2; inputs: pH and EC) yielded an R² of 0.53 and an RMSE of 3.67 mg/kg. These results represent a 67% improvement in Cu concentration and a near-doubling for Mn concentration compared to the SWR models. Furthermore, the prediction error (RMSE) was reduced by 37% for Cu concentration and 31% for Mn concentration.
 
Conclusions
The findings of this study demonstrate that Gene Expression Programming (GEP) is a significantly more powerful and accurate tool than conventional stepwise regression for predicting citrus leaf Cu and Mn concentrations in calcareous soils. The results confirm that the relationships between soil properties and nutrient uptake are fundamentally non-linear, and GEP’s capacity to model these complex interactions is the primary driver of its superior performance. The models consistently identified soil pH as a master variable controlling micronutrient availability. Moreover, the GEP model for Cu highlighted the crucial, non-linear role of organic carbon in protecting Cu from precipitation a nuance that the linear model failed to capture. The practical implication of this research is twofold. First, the explicit mathematical equations generated by the GEP models can be directly embedded into spreadsheet software or mobile applications to serve as a preliminary screening tool. This allows growers to input routine soil test data and receive instant, reliable predictions of potential nutrient deficiencies. Second, on a broader scale, this work provides a methodological framework for other regions, encouraging the adoption of non-linear modeling to develop locally calibrated decision-support systems. Ultimately, this study offers a tangible pathway toward empowering farmers with data-driven tools for enhancing citrus productivity in challenging agricultural environments.

Author Contributions

Heidari and S.A. Ghaffari Nejad conceived and planned the experiments. J. Sarhadi carried out the experiments. S. Heidari and S.A. Ghaffari Nejad analyzed data S. Heidari and S.A. Ghaffari Nejad wrote the first manuscript. All authors contributed to the interpretation of the results. All authors provided critical feedback and helped shape the research, analysis and manuscript.

Data Availability Statement
Data available on request from the authors.

Acknowledgements
The authors would like to thank the research council of the Soil and Water Research Institute, Iran for the financial support of this research.

Ethical considerations
The authors avoided data fabrication, falsification, plagiarism, and misconduct.

Conflict of interest
The author declares no conflict of interest.