Comparative evaluation of growth models for pulse crop performance in Telangana state

Introduction

Pulses occupy a central position in the agricultural and nutritional systems of the Indian subcontinent, complementing cereals as a primary food source. Commonly consumed in the form of dal, pulses are rich in high-quality protein, essential amino acids, dietary fibre, vitamins, and minerals, making them indispensable for ensuring nutritional security, particularly among vegetarian populations. Owing to their affordability and nutritional value, pulses are often referred to as the "protein source of the poor" (Guntukula and Goyari, 2022) [5] .

Beyond their dietary importance, pulse crops contribute significantly to the sustainability of agricultural systems. As leguminous crops, they enhance soil fertility through biological nitrogen fixation, improve soil structure, and reduce dependence on synthetic fertilizers. Pulses are frequently incorporated into crop rotations, intercropping, and mixed cropping systems, thereby supporting resource conservation and long-term farm productivity. Recognizing their agronomic and economic significance, pulse cultivation in India has received considerable policy support over the years (Pasala and Rudra, 2017). Consequently, the area under pulses expanded from 19.09 million hectares in 1950-51 to 31.1 million hectares in 2021-22, accompanied by an increase in production from 8.41 to 27.3 million tonnes and a notable improvement in productivity (Guntukula, 2018) [4] . Among the various pulse crops grown in India, Bengal gram (Cicer arietinum L.) holds a dominant position, accounting for nearly half of the total pulse output. India is the leading producer of Bengal gram globally, contributing more than 60 percent of world production.

In Telangana, the crop is predominantly cultivated during the rabi season. Red gram (Cajanus cajan), another major pulse crop, ranks second in importance and is mainly grown during the kharif season (Kalamkar et al, 2002) [6] . Apart from its use as dal, red gram also provides fodder and green vegetables, often being cultivated as an intercrop. Given the economic and nutritional relevance of these crops, analysing their longterm performance using appropriate statistical models is essential for informed planning and policy formulation.

Material and Methods

Data Source

The present research investigation comprising of secondary time-series data on area, production, and productivity of Bengal gram and red gram for a period of 30 years (1993-2022). District-wise data for Adilabad, Mahbubnagar, and Nizamabad districts of Telangana state were collected from the Statistical Abstracts of Telangana, the Directorate of Economics and Statistics (DES), and the Indiastat database. These data were utilized to examine long-term growth behaviour and to identify suitable models for trend analysis and forecasting

Description of Growth Models

To analyse temporal variations, both linear and nonlinear growth models were fitted to the data using five different methods.

A. Linear Growth Model

The linear model assumes a constant rate of change over time and is expressed as: Where, 𝑌𝑌 𝑡𝑡 denotes area, production, or productivity at time 𝑡𝑡; 𝛽𝛽 0 is the intercept; 𝛽𝛽 1 represents the linear growth rate; and 𝜀𝜀 𝑡𝑡 is the random error term (Gautam and Sisodia, 2018) [3] .

B. Quadratic Growth Model

The quadratic model accounts for curvature in the trend and is defined as:

Where, 𝛽𝛽 2 captures acceleration or deceleration in growth over time period (Satish et al, 2022) [14] .

C. Exponential Growth Model

The exponential model is suitable for describing proportional growth and is specified as:

Where, 𝛼𝛼is the scale parameter, 𝛽𝛽is the exponential growth rate, and 𝑒𝑒 is the base of the natural logarithm (Shah et al, 2014) [12] .

D. Monomolecular Growth Model

The monomolecular model describes growth approaching an asymptote and is given by:

Where, 𝛼𝛼 Represents the asymptotic maximum level, and 𝛽𝛽 and 𝛾𝛾 are model parameters (Kumar et al, 2012) [8] .

E. Generalized Additive Model (GAM)

To capture complex nonlinear patterns, a Generalized Additive Model was employed. The GAM is expressed as:

Where, 𝛼𝛼 is the intercept and 𝑓𝑓(𝑡𝑡) is a smooth nonparametric function of time estimated using spline-based techniques (Khan et al, 2013) [7] . Model estimation was carried out using a backfitting algorithm that iteratively estimates the smooth and parametric components.

Model Estimation and Diagnostic Criteria

Nonlinear models were estimated using nonlinear least squares via the nlsLM function from the minpack.lm package in R. Model performance was evaluated using the following diagnostic measures:

Coefficient of Determination

Mean Absolute Percentage Error (MAPE)

𝑛𝑛

Where, Y t = Observed value, Y t � = Predicted value and 𝑛𝑛 = Number of observations. The model exhibiting the highest 𝑅𝑅 2 and the lowest MAPE and RMSE values was considered the most appropriate (Malhi and Kiran, 2015) [9] .

Residual Diagnostics

Residual independence was assessed using the Runs test, while normality of residuals was examined through the Shapiro-Wilk test. All statistical analyses were performed using R statistical software.

Results and Discussion

The performance of different growth models was evaluated for analysing long-term changes in area, production, and productivity of Bengal gram and red gram in Adilabad, Mahbubnagar, and Nizamabad districts of Telangana during 1993-2022. Model adequacy was assessed using the coefficient of determination (R²), Mean Absolute Percentage Error (MAPE), and Root Mean Squared Error (RMSE). Across crops, districts, and variables, the Generalized Additive Model (GAM) consistently provided superior fit compared to linear and other nonlinear models.

Bengal Gram

Bengal gram exhibited an overall increasing trend in production and productivity across all districts, though interannual variability was observed in cultivated area (Devi et al, 2017) [1] . In Adilabad district, the average area under Bengal gram was 13.35 thousand hectares, with mean production of 14.97 thousand tonnes and productivity of 1132.75 kg ha⁻¹. In Mahbubnagar, the corresponding averages were 13.79 thousand hectares, 17.14 thousand tonnes, and 1105.66 kg ha⁻¹, while in Nizamabad district, mean production and productivity were relatively higher despite moderate area coverage. GAM recorded the highest R² values and lowest RMSE and MAPE across all variables, indicating its ability to capture nonlinear growth behaviour.

Red Gram

Red gram showed mixed trends in cultivated area, while production and productivity generally improved over time (Eswara et al, 1993) [2] . In Adilabad district, the average area, production, and productivity were 40.62 thousand hectares, 23.09 thousand tonnes, and 551.09 kg ha⁻¹, respectively. Mahbubnagar district recorded the largest average area under red gram (84.65 thousand hectares), though productivity remained comparatively lower. In Nizamabad district, red gram was cultivated on a smaller scale, but productivity levels were relatively higher. GAM again outperformed other models across all districts, reflecting its robustness in modelling highly variable datasets, similar to the findings of Sharma et al, 2013 [13] . practices. These results emphasize the suitability of GAM for reliable trend analysis and forecasting of pulse crops, thereby supporting evidence-based agricultural planning and policy formulation.

Summary

The present study examined long-term trends in area, production, and productivity of major pulse crops, namely Bengal gram and Red gram, in three districts of Telangana state Adilabad, Mahbubnagar, and Nizamabad using 30 years of district-level data from 1993-2022. To capture temporal dynamics, several growth models, including linear, quadratic, exponential, monomolecular, and Generalized Additive Models (GAM), were fitted and compared using standard diagnostic criteria such as the coefficient of determination (R²), Mean Absolute Percentage Error (MAPE), and Root Mean Squared Error (RMSE).

The results indicated that Bengal gram showed an overall increasing trend in production and productivity across all three districts, although area under cultivation exhibited district-specific variations. Red gram also displayed improvements in production and productivity, despite declining or fluctuating cultivated area in certain districts. Across crops, districts, and variables, the Generalized Additive Model (GAM) consistently outperformed other models by providing higher explanatory power and lower prediction errors. The fitted GAM curves closely followed observed values, confirming their ability to capture nonlinear patterns and structural changes in pulse crop performance. Overall, the findings or results demonstrate the suitability of GAM for modelling agricultural time-series data and provide reliable insights for understanding crop growth behaviour. The study established the importance of advanced statistical approaches for effective analysis, forecasting, and planning of pulse crop production in selected districts of Telangana state.

Conclusion

The present study applied linear and non-linear growth models to analyse the long-term behaviour of area, production, and productivity of Bengal gram and Red gram in selected districts of Telangana state using district-level data from 1993 to 2022. Model comparison based on diagnostic measures such as R², RMSE, and MAPE revealed that the Generalized Additive Model (GAM) consistently outperformed conventional parametric models across crops, districts, response variables and consideration. The findings showed that Bengal gram exhibited a general increase in production and productivity in all three districts, although changes in cultivated area varied across locations. Red gram also demonstrated improvements in productivity and output despite fluctuations or declines in area in certain districts, indicating the influence of technological progress and improved crop management practices. The fitted GAM curves closely matched the observed data, confirming their ability to capture non-linear trends and temporal variability inherent in agricultural time-series data. Overall, the results highlight the effectiveness of GAM as a robust and flexible modelling framework for analysing crop performance over the period. The study provides valuable insights for researchers and policymakers by offering a reliable approach for understanding growth dynamics and supporting informed decision-making in pulse crop planning and development in Telangana state. Future studies may improve the present existing analysis by including weather-related variables such as rainfall and temperature to better explain variations in pulse crop performance. The modelling framework can be extended using spatial and time-based approaches to capture regional differences across districts. Advanced methods, including machine learning techniques, may also be used and applied to enhance prediction accuracy. Further validation of forecasts using future or unseen data would strengthen the reliability of the results. In addition, the approach can be used to assess the impact of improved technologies and policy interventions on the growth and sustainability of pulse crop production.

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