Estimating Yield and Production of Paddy and Maize Crops of Dantewada district of Chhattisgarh using regression analysis

Introduction

Regression analysis is a method for assessing the linear relationship among variables. Multiple regression involves one dependent variable influenced by several independent variables. Studies cited include Wei and Molin (2020) [37] , who estimated soybean yield using linear regression, Nazir et al. (2021) [21] , who forecasted rice yield with a phenology-based algorithm and regression model, and Sellam and Poovammal (2016) [27] as well as Shastry (2017) [28] , both of whom focused on crop yield predictions through regression analysis. In time series analysis, residuals may show serial correlation or heteroscedasticity, contravening classical regression assumptions. A solution is the Generalized Least Squares (GLS) method, which maximizes likelihood estimation in regression models with ARIMA errors, as explained by Otto et al. (1987). Studies by Kadanali et al. (2019) focused on ARIMA model forecasting for wheat production, while Dritsakis et al. (2019) provided insights into time series analysis with ARIMA. Additionally, Kwasi et al. compared the forecasting abilities of multivariate VAR and univariate ARIMA models. In regression analysis, multicollinearity among dependent variables can harm model accuracy, leading to the elimination of high multi-collinear variables via correlation matrices. The identification of multicollinearity is detailed by Shrestha (2020) [30] , while research conducted by Haitovsky (1969) [17] and Daoud (2017) [10] further explores its implications. Kim (2019) [18] elucidates the relationship between multicollinearity and misleading outcomes, and Schroeder et al. examine methods for diagnosing and treating this issue. Regression and ARIMA models can be integrated to enhance forecasting accuracy by leveraging their strengths, especially when data exhibits both linear correlations and temporal dependencies. Shumway et al. (2017) [31] discussed time series analysis and ARIMA models, while Mondal et al. (2014) investigated ARIMA's efficacy in predicting stock prices. Benvenuto et al. (2020) [9] analyzed the ARIMA model's application using COVID-19 pandemic data.

by Bastar and Narayanpur to the north, Bijapur to the west and Sukma to the south and east with Kondagaon.

Data Description

Secondary data forms the basis of this investigation, which focuses on the region of Chhattisgarh, created from the bifurcation of Madhya Pradesh in 2000. Since then, new districts have emerged from the original districts multiple times. The study considers various variables, including area, gross irrigation area, NPK consumption, temperature ranges, humidity, and rainfall, along with yield and production. Outliers were removed to ensure model accuracy. The study employs the ARIMA (p, d, q) model to assess serial correlation in time series data for paddy, maize, and jowar crops in Dantewada, selecting significant predictor variables through a correlation matrix, and utilizing linear and generalized linear models as needed.

Linear Regression

Linear regression is a statistical method that models the relationship between a dependent variable y and one or more independent variables (X1, X2, Xn) assuming a linear relationship where changes in the independent variable(s) correspond to constant changes in the dependent variable.

Where Y is the dependent variable, , ,..., are the independent variables, is the intercept, , ,..., are the coefficients (slopes) of the independent variables, is the error term, which represents the difference between the observed and predicted values of Y.

Generalised least square model

Generalized least squares (GLS) is a method used to estimate unknown parameters in linear regression models when the standard assumptions regarding residuals-such as zero mean, normality, or equal variances-are violated. It addresses these issues by incorporating a weighting matrix that assigns varying weights to data points according to their estimated variances.

• Estimate the error covariance matrix: This matrix captures how the errors are related to each other. It might show higher variances for certain groups of data points. • Apply the weights: Each data point in the regression equation is multiplied by a weight based on the error covariance matrix. This gives more influence to data points with lower error variance and less influence to those with higher variance. • Minimize the weighted squared residuals: Similar to OLS, GLS minimizes the sum of squared residuals, but these residuals are now weighted based on the error covariance matrix.

Multi-collinearity

Multicollinearity occurs in regression analysis when two or more independent predictor variables are strongly correlated. This correlation complicates the evaluation of each variable's individual effect on the dependent variable, making it challenging to identify their true influence. As a result, the model's predictions may become less accurate due to difficulties in assessing the underlying impacts of these correlated variables.

Auto Regressive Integrated Moving Average (ARIMA)

ARIMA, which stands for Auto Regressive Integrated Moving Average, is a statistical method prevalent in time series forecasting and analysis. It effectively captures various temporal patterns, making it applicable in fields such as economics, finance, epidemiology, and weather forecasting.

The general ARIMA (p, d, q) model can be expressed with the following equation:

Where, is the actual value at time t, is the constant mean (optional, may not be present in all models), (phi) are the autoregressive parameters ( to ). These represent the coefficients of the past p values of , to are the lagged values of , influencing the current value ( ), is the white noise error term at time t (represents unpredictable random shocks), (theta) are the moving average parameters ( to ). These represent the coefficients of the past q forecast errors ( to ).

Model evaluation

After putting all of the estimate methods through diagnostic plots and goodness of fit metrics like R 2 , Adj R 2 , and their Pvalues, the process is concluded. From the time series data of the yield of the paddy crop in Dantewada, auto-regressive integrated moving average, ARIMA (p, d, q) are determined to confirm the presence or absence of auto-correlation.

Results and Discussions Paddy crop of Dantewada

Using acf() and pacf() functions of R, p=1 and q=1 as the correlations are beyond the significant bounds after differencing twice to ensure stationarity. Hence ARIMA (1, 2, 1) was obtained. Further, the yield, predicted yield, standard error and confidence interval of the finalized model are given in Table 1. Moreover, goodness of fit plot between predicted yield and yield is further done and depicted in Figure 3. After fitting and finalizing the model, a forecast for the yield and production has been made upto 2025.

Maize crop of Dantewada

From the time series data of the yield of the maize crop in Dantewada, auto-regressive integrated moving average, ARIMA (p, d, q) are determined to confirm the presence or absence of auto-correlation. Using acf() and pacf() functions of R, p=1 and q=1 as the correlations are beyond the significant bounds after differencing twice to ensure stationarity. Hence ARIMA (1, 2, 1) was obtained. The values of p=1 and q=1 from Step-1 indicating the presence of serial correlation. As a result, generalised least square is used instead of linear model. Further, to know the order of predictor variables for entering into the generalised least square model, correlation matrix was used. From the correlation matrix, the variables having highest correlation with dependent variable were selected in descending order. The orderof the variables is as follows: 'area.h', 'X20.per.gia.prop'. The generalized linear model for the maize crop is given below: Since the diagnostic plots were seemed to fit well, the generalised least square model was finalized. Further, the yield, predicted yield, standard error and confidence interval of the finalized model are given in Table 4. Moreover, goodness of fit plot between predicted yield and yield is further done and depicted in Fig. 6. It is very clear that the data points are close to the fitted line indicating a good fit with R 2 = 0.7348 (P-value: 0.003141) and Adj R 2 = 0.6969

Conclusion

In this study, predicted yield is very much close to the actual yield indicating that the finalized model is a good fit. Through forecasting of yield of paddy and maize upto 2025, there is showing an increasing yield in future which is a good index.

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