A nonlinear mathematical production model for crop yield estimation and resource optimization in Indian agriculture

Introduction

Agriculture is significant in ensuring food security, rural livelihoods, and economic growth. Estimation of crop yield is vital in agricultural planning, resource management, food supply, and risk assessment. Crop yield is influenced by various interacting factors such as land under cultivation, production factors, rainfall, temperature, and crop-management practices [1,2] . Production function methods offer a framework that enables an understanding of the association between agricultural inputs and outputs. Methods such as Cobb-Douglas framework and subsequent statistical production functions have extensively been used in modeling the associations between input and output and assessing the contributions of production factors [1,3] . Nevertheless, agricultural data can be highly variable, skewed, exhibit multicollinearity, and nonlinearities in relationships that can influence the statistical estimation process. The availability of vast data on agriculture and weather has promoted the adoption of machine learning and deep learning methods for crop yield prediction. Such methods can model the interactions between agricultural, climatic, and environmental variables in yield predictions and have shown tremendous promise in yield prediction [4] . In spite of the fact that modern sophisticated prediction methods have been invented, the development of easily understandable, statistically proven, and mathematically formulated models which will help measure the dependency between agricultural production and climatic parameters is required. That is why the purpose of this research is the construction of a logarithmic OLS production model which will include the area of cultivation, production volume, annual precipitation, and average temperature. Moreover, it is necessary to check the presence of multicollinearity, heteroscedasticity, residual behavior, and influential observations.

Literature Survey

The problem of agricultural production has been examined using the statistical, econometric, process-based, remote sensing, and machine learning methods. Previous works in the area of agricultural production have found that the relationship between agricultural output and major factors of production like land, labor, capital, and others is vital. The framework of the Cobb-Douglas production function helped to establish the input-output relationship and production elasticities [5] . Another important factor in agricultural productivity is the climatic factor. Lobell and Field [6] have found that there are considerable relations between climate and global crop yields. They emphasized the role of temperature and precipitation in agricultural productivity. Later researches included climatic and environment variables in their models of agricultural production and yield estimation. Statistical and econometric methods remain very useful since they allow establishing interpretative relations between inputs and outputs of agricultural production. The production function models were extensively used in estimation of influence of land, water, fertilizers, climatic, and other factors on agricultural production [3][4][5] . However, nonlinearities, multicollinearity, heteroscedasticity, and influential observations can be present in agricultural systems. The development of data-driven approaches has greatly widened the scope of yield prediction studies. The application of neural networks and data mining approaches in wheat, sugarcane, and apple showed that nonlinear computational models have great capabilities to model the interrelations between agricultural and environment-related features [7-10]. Then Random Forest, Support Vector Machines, Decision Trees, Gradient Boosting, and ensemble-learning were extensively applied [11-15]. It is possible to use many predictors without the need for a specific functional form of relations. Remote sensing has made a significant contribution to yield estimation by means of satellite images, vegetation indices, weather data, and crop properties [16-18]. Thus, there are opportunities for making a spatially distributed evaluation of crop production and to get information which cannot be collected using standard field data sets. A number of systematic reviews show that machine learning and deep learning have been rapidly developing in agriculture. For example, Van Klompenburg et al. [19] analyzed machine learning and deep learning studies and found that the most popular features are temperature, rainfall, and soil properties. Other recent investigations have made use of CNN, LSTM, DNN, and recurrent models, using remote sensing and environmental data to increase the level of accuracy [20-22]. Jabed and Murad [23] also noted the popularity of Random Forest, ANN, and SVM methods, with CNN, LSTM, and DNN being used in deep-learning models. Furthermore, Yadav et al. [24] have pointed out the rising popularity of machine learning, deep learning, and hybrid models from 2017 through 2024 to enhance the accuracy of crop yield and resource planning. Moreover, comparative analysis and the combination of process-based and data-driven models are being increasingly carried out. Such research indicates that the combination of process-based and data-driven modeling approaches can provide more accurate prediction while retaining the information on crop-growth and environmental processes [24] . In spite of these improvements, however, there is still a need for interpretable mathematically formulated models that estimate the contribution and elasticity of important agricultural and climatic factors. The majority of the previous research has paid more attention to the accuracy of prediction without much concern about multicollinearity, heteroscedasticity, non-normality of residuals, and influential observations. This study aims at filling this gap by proposing a log-transformed OLS production model that includes area, production, rainfall, and temperature factors while assessing multicollinearity, heteroscedasticity, residuals, and influential observations.

Materials and Methods

Data Set

The research will employ the crop-yield data set consisting of 19, 689 observations. This data set encompasses various crops, seasons, states, and years in relation to agricultural production and climatic variables. The significant gap between the mean and median as well as the very high skewness value suggest that Yield is significantly right-skewed. Thus, logarithmic transformation was performed to minimize the effect of outliers.

Yield Summary

As per the descriptive analysis, the mean and median of Yield are 79.95 and 1.03 respectively, which suggests a large gap between average and outlier values. Its standard deviation is 878.31, suggesting very high variability. The values for Yield range from 0 to 21, 105. As seen from high positive skewness (12.79), there is a strong right skewness in its distribution whereas the high kurtosis (175.58) reflects the presence of outliers. Hence, transformation of Yield through ln(Yield+1) will be done prior to regression analysis.

Heatmap Correlation Analysis

From the above heatmap, one can see the strength and the direction of the linear relationship between crop production variables. Positive correlations were found between the Area, Fertilizer, and Pesticide as well as between different temperature variables, indicating the existence of multicollinearity. While there is a moderate positive correlation (r=0.57) between Production and Yield, weak correlations are present with other variables.

OLS Regression Result

The OLS regression model analysis was done with Log_Yield as the dependent variable. The regression model can explain 89.5% of the variability in crop yield (R² = 0.895; adjusted R² = 0.895), and is statistically significant overall (F = 28, 020; p < 0.001). All the six independent variables were statistically significant at the 1% level or even more apart from Pesticide that was statistically significant at the 1% level. Log_Production is highly positively correlated with Log_Yield (β = 0.6904), whereas Log_Area has a large negative coefficient (β = -0.6238). Temperature contributes only slightly to the production, positively. Rainfall, fertilizer, and pesticide have negative coefficients. Thus, the model shows high explanatory power, though the significant test results for the diagnostic statistics (Jarque-Bera p < 0.001 and Durbin-Watson = 1. The regression model has an R squared value of about 89.52%. The extremely large F-statistic and p-value < 0.001 suggest that the whole regression model is statistically significant.

Assessment of Multicollinearity

The Variance Inflation Factor (VIF) test was conducted to detect multicollinearity in the explanatory variables. The values from the VIF indicate that there is very high multicollinearity between Log_Area, Log_Fertilizer, and Log_Pesticide. There is moderate multicollinearity between Log_Production (VIF=5.08), but there is no multicollinearity between rainfall and temperature. Consequently, dimensionality reduction should be done prior to regression. The reduced OLS regression model accounts for 89.5% of the variations in Log_Yield and is significant (F = 41, 970, p < 0.001). All four variables in the regression are significant. The relationship between Log_Production and yield is strongly positive (β = 0.6899), whereas the relationship between Log_Area and yield is strongly negative (β = -0.6984). Elimination of Fertilizer and Pesticide solves the problem of multicollinearity without affecting the explanatory power of the regression model. The two models have near-identical explanatory capabilities (R² ≈ 0.895). However, the Initial Model has higher explanatory power, AIC, and BIC than the Reduced Model, but the latter has fewer predictors. Therefore, the Reduced Model gives a simpler model without significant reduction in explanatory capability. Hence, the Reduced Model is preferred when simplification of the model is important. Residuals are almost randomly distributed around the zero line, however, the distinct curvilinear/funnel shape along with several extreme residuals suggest non-linearity and existence of outliers. Widening of residuals in certain areas also raises questions regarding heteroscedasticity. Thus, the model might be violating assumptions of homoscedasticity and linearity. The Q-Q plot clearly demonstrates a deviation of the residuals from the reference line, especially in the upper tail. It means that the residuals are not normally distributed and are skewed to the right. Thus, the normality assumption regarding the regression residuals is violated, which means that there are outliers in the data set. The LM test and F-test statistics are significant (p < 0.001), and hence the variance of the error term is not constant. The data therefore exhibit heteroscedasticity, and hence it would be prudent to consider the use of robust standard errors. Once corrected for heteroscedasticity through HC3 robust standard errors, all predictors retain their statistical significance (p < 0.001). The model still explains a very large proportion of the data (R² = 0.895), meaning that the key regression relations are robust even in the face of heteroscedasticity. Log_Production is the most strongly correlated predictor with Log_Yield (β = 0.6899), whereas Log_Area is negatively correlated (β = -0.6984). The regression model has a value of R² = 0.9043, which means that 90.43% of variability in the yield is explained by the model. The low values of MAE (0.2044) As seen from the scatterplot between Actual and Predicted, the fit is quite accurate at lower levels of yields but shows large under prediction for extreme and high levels of yields. It implies that the model is good enough to capture the overall level of the relationship between the variables but does not have much scope to reproduce the extreme values of yield. The estimates are fairly accurate as compared to the actuals. Nevertheless, observation 326 has relatively high error (32.74), which means that there is an outlier or challenging point. The relationship from the equation shows that Production has a positive impact on yield, whereas Area and Annual Rainfall have negative coefficients. Average Temperature has a minor positive coefficient. All the coefficients are statistically significant at p < 0.001.

Cook's Distance -Influential Observations

The threshold value for Cook's Distance is 0.000203 and 632 observations have been identified as influential observations. Table 19 shows the top 20 influential observations as per the Cook's Distance. According to the Cook's Distance measure, there are 632 observations which are above the threshold, which means that there could be some influence on the regression. The highest value (0.3130) is much higher than the threshold and represents a very influential observation. The interesting thing about the highest influential observations is that they have Yield = 0 and Log_Yield = 0, meaning that the zero yield observations seem to have a very strong influence on the estimated model.

Results

There are 19, 689 observations in the analysis. Crop yield was highly variable and positively skewed (12.79), with mean = 79.95, median = 1.03, and std dev = 878.31. Production is correlated with the crop yield the most (r=0.5708, " " p < 0.001). Rainfall and temperature variables show small correlations.

There was a severe multicollinearity among Area, Fertilizer and Pesticide. That is why Fertilizer and Pesticide were removed from the model. The final semi-logarithmic OLS model is:

ln (𝑌𝑖𝑒𝑙𝑑 + 1) = 1.0361 -0.6984ln (𝐴𝑟𝑒𝑎) + 0.6899ln (𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛) -0.0348ln (𝐴𝑛𝑛𝑢𝑎𝑙𝑅𝑎𝑖𝑛𝑓𝑎𝑙𝑙) + 0.00494𝐴𝑣𝑔𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 + 𝜀

It is evident that the model was statistically significant (F=41, 965.98, " " p < 0.001), accounting for 89.50% of the variance in the transformed variable Yield (R^2=0.8950, adjusted R^2=0.8950). All four independent variables were statistically significant, where Production and Avg.Temp have positive influence, while Area and Annual Rainfall have negative conditional influence. The residual analysis reveals the presence of significant nonnormality of residuals (Jarque-Bera = 2, 849, 661.96, p < 0.001) and heteroscedasticity (Breusch-Pagan p < 0.001). The use of HC3 Robust Standard Errors is hence recommended, where all four predictors will still be statistically significant.

There are 632 potentially influential observations according to Cook's Distance, including some

Conclusion

The research presented here developed a semi-logarithmic production mathematical model to estimate crop yield using 19, 689 data points on Indian agriculture. The model was able to explain 89.50% of the variance in transformed crop yield (R^2 = 0.8950), with production, area, rainfall, and temperature having statistically significant impacts. Nevertheless, issues like heteroscedasticity, non-normality of error terms, and presence of influential observations call for the use of robust estimation techniques and sound interpretation of the findings.

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