Statistical modeling on production of papaya in Maharashtra state

Introduction

India is known as the basket of fruits and vegetables in the world. Its diverse climate ensures availability of all varieties of fresh fruits & vegetables round the year. India produced 99069.2 thousand MT of fruits and 191769.11 thousand MT of vegetables (source: NHB database 2019-20). Among the fruits, papaya occupies a special place. Like banana it is available throughout the year and it is easy to cultivate (Biswas, 2010) [3] . Papaya is considered as one of the most important fruit crops in terms of production, productivity and export potential in India. In India, around 24 states are growing papaya, the major producing states in India are Andhra Pradesh, Maharashtra, Chhattisgarh, West Bengal, Assam, Tamil Nadu, Jharkhand. Hadolikar et al. (2022) [4] studied status of papaya cultivation in India and showed that the area, production and yield of Papaya at all India level shown significant positive growth. The instability in production of Papaya was more compared to area and yield. Sagar et al. (2012) [6] studied the costs, returns and constraints in production and marketing of papaya (Carica papaya L.) in Middle Gujarat. Results of the study indicated that papaya was highly capital intensive crop and average cost of cultivation (Cost C2) per hectare was Rs. 176660. On an average cost A (paid out cost) formed 61.64%, while cost B accounted for 87.38% of total cost. The main objective of the study is to identify the statistical model for production of papaya, to compare the efficiency of different statistical models and to forecast the papaya production of Maharashtra state by using best fitted model.

Materials and Methods

The secondary data on area, production and productivity of papaya of Maharashtra state (1987-88 to 2022-23) were used for descriptive statistics and production time series data is considered for finding best fit statistical model. The Agricultural production may vary from year to year due to nature of cultivation, environment changes and government policies.

Sometimes production of crop can increase or decrease with a certain rate. To investigate such trend, we consider eight following different statistical models in the present research study i.e.

Linear

𝑦 𝑖 = 𝑎 + 𝑏𝑡 + 𝜀 𝑖

Polynomial 2 nd degree

Polynomial 3 rd degree

4.

Where, 𝑦 𝑖 is the production of papaya, t is the time and 𝜀 𝑖 is the random error. Taking logarithm on both sides of eq. ( 3), we get

Exponential

Logarithmic

𝑦 𝑖 = 𝑎 + 𝑏𝐿𝑛(𝑡) + 𝜀 𝑖 (6)

Power

𝑦 𝑖 = 𝑎𝑡 𝑏 + 𝜀 𝑖 (7) Durbin Watson test

Where 𝑒 𝑡 = (𝑦 𝑡 -𝑦 𝑡 ̂ ), 𝑦 𝑡 and 𝑦 𝑡 ̂ are observed and estimated values of response variable for individual t. DW(d) becomes smaller as the serial correlations increase.

To test theautocorrelation at significance level α, the test statistic d is compared to lower and upper critical values (dL,α and dU,α):  If d<dL,α, there is statistical evidence that the error terms are positively autocorrelated.  If d>dU,α, there is no statistical evidence that the error terms are positively autocorrelated.  If dL,α<d<dU,α, the test is inconclusive.

ARIMA (Autoregressive Integrated Moving Average) Model

Autoregressive Integrated Moving Average (ARIMA) also known as Box-Jenkins methodology was used for the time series modeling. It includes AR and MA terms with their respective order p and q. Differencing eliminates trends and seasonality and stabilizes mean of the time series. The differencing was used in order to make the series stationary. The non seasonal ARIMA (p, d, q) model for papaya production (y) in period t based on data up to period t-1 was used as follows

Where, 𝜇 is the constant, 𝜑 𝑝 is the autoregressive (AR) coefficient at lag p, 𝜃 𝑞 is the moving average (MA) coefficient at lag q and 𝑒 𝑡-𝑞 = 𝑦 𝑡-𝑝 -𝑦 ̂𝑡-𝑝 is the forecast error that was made at period t-p. The main steps of ARIMA modeling are model identification, parameter estimation, diagnostic checking and forecasting.

Results and Discussion

The descriptive statistics of area, production and productivity of papaya in Maharashtra state is presented in Table 1. The results revealed that, mean area, production and productivity of papaya were 4.19 thousand hectares, 130.15 thousand tons and 36.72 MT/ha respectively. The median for area, production and productivity was 1.90 thousand ha., 60.10 thousand MT and 36.30 MT/ha respectively. The maximum area under cultivation was 16.18 thousand hectares; production was 552.50 thousand MT and productivity was 85.00 MT/ha. Whereas, minimum area under cultivation was 0.58 thousand ha with production 7.80 thousand MT and minimum productivity was 6.20 MT/ha. The results of skewness and kurtosis of area, production and productivity of papaya data positively skewed for area and production while negatively skewed for productivity and was not normally distributed. The results of compound growth rate of papaya production of Maharashtra state is presented in Table 2. The production has increased significantly at a compound growth rate of 2.71% per annum with standard error 0.47. The similar results was reported by Bhagat et al (2024) [2] concluded that the export performance of Indian milk and milk products quantity and its value exhibited positive annual growth rate of 17.20 and 24.46% during study period.

For analyzing the pattern of papaya production various linear and non-linear models were fitted. [8] compared the performance of five nonlinear growth models, namely, Monomolecular, Logistics, Gompertz, Richards, and Weibull and studied the area and production of apple crop in Kashmir. The best model was chosen based on various model adequacy test criteria including MAE, RMSE, MSE, and R2. Therefore, Richards and Weibull models were found to be the best fit for the area and production of apple, respectively. The area and production of the apple crop were forecasted for a few years using the best fit models. The estimated parameters along with their Durbin Watson Statistic (DW), R 2 , different errors and AIC values are presented in Table 3. ) model as compared to all other models. Fig. 3 shows the graphical representation of papaya production using best fitted ARIMA model. Sharma et al (2023) [7] stated that most appropriate ARIMA model was selected to capture the forecasting trends in the area and production of papaya for the period from 2021 to 2030 and projected the values for area under cultivation and production of papaya and reached 1538 hectares and 6,384,220 metric tons, respectively, by 2030. This study found that significance in defining the potential demand gap between the area and production for papaya in the future. ). It was observed that there will be increase in papaya production of Maharashtra in next year and it will be 504.99 thousand metric tonnes and it will lies between LCL 345.32 to 664.60 UCL thousand metric tones. The similar results were reported by Murthy et al (2020) [5] and Balaga and Dash (2022) [1] .

References

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