Statistical Modeling for Wheat (Triticum Aestivum) Crop Production

The present investigation was carried out to study the trends in area, production and productivity of wheat crop grown during the period 1950-1951 to 2009-2010 in India. Different non-linear models were employed to study the trends in area, production and productivity. When none of the non-linear models were found suitable to fit the trends nonparametric regression model was employed. None o f the non-linear model was found suitable to fit the trends in area data. The Sinusoidal model was found suitable to fit the trends in production as well as productivity of wheat crop grown in Ind ia. The results indicated that area, production and productivity of wheat crop grown in India, had been shown in the increasing trend. The area of cult ivation had played a major ro le in increasing the trend in production.


Introduction
Wheat is a cereal grain wh ich originated in fro m the Levant region o f the North east but is now cu ltivated globally. In 2007 world production of wheat was 607 Million Tons, making it the third most produced cereal crop after maize and rice. Wheat's particular characteristic i.e. it's adaptability has favoured its growth worldwide. Wheat is adapted to a wide variety of climatic conditions. It is grown where annual temperatures of 4.9 to 27.8℃ p revail [3].
When we look at the major producers Globally, Ch ina emerges as the leading producer of wheat fo llowed by India at second place, and USA ranks third amongst the top producers globally. On an average, China produces 108,712 TMT of wheat annually, making it the worlds largest producer of wheat but it is not enough to feed its 1.2 Billion population thus it is forced to import 4,247 TMT of wheat and it also exports a very little amount to the neighbouring Asian countries about 657 TMT annually. India is second largest producer of wheat in the world, averaging an annual production of 65,856 TMT. Although it is less when compared to china but India imports less comparatively, also India has a population almost comparable with that of China. India imports 990 TMT of wheat, and, for various reasons, exports an average of 767 TMT of wheat annually [2].
The country' s wheat production is estimated to achieve an all-time high of 81.47 million tons and the overall food grain output is projected to rise by over 6% to 232.07 million tons in 2010-2011 crop year. The bu mper harvest is expected to contribute significantly to the economic g rowth. The government had projected 8.6% expansion in the economy, aided by an impressive 5.4% rise in agricu lture and allied activities [3,4]. Thus this shows the significance of wheat in Ind ia and therefore the statistical information on the area, production and productivity of the crop emerges as a subject of great importance. Measurement of growth rate in agricultural productions looks apparently simp le, but it is not free fro m complications. Thus the experimental analysis of different statistical model like the parametric (Linear or Non-linear models) by assuming the linear or exponential functional forms and nonparametric models is mandatory to get the statistical estimation of the productivity of the crop and its future predictions.
A number of research workers [8, 9, 13, 14, 15, 18, 19, 21, 22, 28 and 29] have used parametric models to estimate growth rates. There are the drawbacks of these models i.e. the data may not be following these linear or exponential models or may require fitting of higher degree polynomials or non linear models. Further, these models lack the economic considerations i.e. normality and randomness of residuals [26]. Thus recourse to the nonparametric regression approach is taken which is based on fewer assumptions [11].
The same is done in this paper, non-linear and nonparametric models are applied, to study the trends in production of the wheat crop and developing a best possible statistical model for analy zing the area, production and productivity of the wheat crop data.

Materials and Methods
The time-series data on area, production and productivity for the period 1950-1951 to 2009-2010 were co llected through the web site www.agricoop.nic.in. The data were analyzed using the non-linear as well as the nonparametric regression models and the conclusion is given based on the best model to study the trends and growth rate of wheat crop production in India.

Non-Li near Models
In parametric model, different non-linear models [5, 10, 17, 25 and 27] g iven in Table 1 were emp loyed. Among the non-linear models, the model having highest adjusted R 2 with significant F value was selected, so that it satisfied test for goodness of fit [17]. Normality of residuals was examined by using Shaprio-Wilks test [1]. Further mo re, wh ile dealing with time-series data it may be possible that successive observations may be auto-correlated among themselves [30]. To overcome all these problems, performing residual analysis was being carried out. Rando mness assumption of the residuals required to be tested before taking any final decision about the adequacy of the model developed. To carry out the above analysis "Run test" procedure developed in the literature [25] was used. Further, to test the presence or absence of auto-correlation in the data set Durbin-Watson test procedure [16] was utilized In case of more than one model being the good fit for the data, the best model was selected based on lower values of Root Mean Square Error (RM SE) and Mean Absolute Error (MA E) values.
Levenberg-Marquardt algorith m [25] which is widely used was utilized for fitting Logistic, Go mpertz Relat ion, Sinusoidal and Rational Function. Different sets of init ial parameter values were tried so as to ensure global convergence. The iterative procedure was stopped whenever the successive iterations parameter estimates values were negligibly low. The standard SPSS Ver. 16.0 package was used to fit all the models given in Table 1. In the Y is the area/ production/ productivity and X is the time points; A, B, C and D are the parameters and e is the error term. The parameter A represents carrying capacity; C is the intrinsic gro wth rate ; B represents different functions of the init ial value y(0) and D is the added parameter.

Nonparametric Regression Model[11, 12]
The nonparametric regression model with the addit ive error is of the form , ) ( where Yi is the observation (area, production and productivity) of the i th time point, m is the trend function, which is assumed to be smooth, and i ε is random error with mean zero and finite variance ∞ < 2 σ . The kernel weighted linear regression smoother is used to estimate the trend function. The value of the local linear regression smoother at time x is the solution of a 0 to the following weighted least squares problem: The optimu m bandwidth h can be obtained by the method of cross-validation. The slope m | (x) of m(x) can be considered as the simple linear growth rate at the t ime point x.
The estimate of m | (x) is given by

Results and Discussion
Different non-linear and nonparametric regression models were emp loyed to study the trends in the area, production and productivity data of the wheat crop. The findings are discussed in sequence, as follows.

Trends in Area
The data presented in Table 2. for the area under the wheat crop revealed that among the non-linear models fitted to the area under the wheat crop, the maximu m adjusted R 2 of 97 per cent was observed in case of Morgan-Mercer-Flodin (MMF) model with co mparatively lo wer values of RMSE (0.9667) and MA E (0.8010) in co mparison to that of other non-linear models. The Shapiro-W ilks test (test for normality) was found to be non-significant indicating that the residuals due to this model were found to be normally distributed. However the run test (test for randomness) value was significant indicat ing that the residuals were correlated. Y=(10.72*3898.99+29.54*X^2.52)/(3898.99+X^2.52) (R 2 =97%) Since none of the non-linear models was found suitable to fit the trends in area under the wheat crop, the nonparametric regression model was employed to fit the trends in area data. The optimu m bandwidth was co mputed as 0.50 using the cross-validation method. Nonparamet ric estimates of underlying growth function were co mputed at each time point. Residual analysis showed that the assumptions of independence of errors were not violated at 5%level of significance. The RMSE and MAE values were found to be 0.5069 and 0.4107 respectively. These values were found to be much lo wer than those obtained through the parametric models, indicat ing thereby the superiority of this approach over the parametric approach. Hence the nonparametric regression model was selected to fit the trends in area under the wheat crop. The graph of the fitted trend for area under the wheat crop using the nonparametric regression is depicted in the Fig 1. Similar trends were observed for Gu jarat wheat production [7].
Nonparametric regression model was used [24] to fit the trends in tobacco crop grown in Anand district of midd le Gu jarat in Ind ia for the period 1949-1950 to 2007-2008.  It [23] was also reported that none of the parametric models were found suitable to fit the trends in area, data of castor crop grown in Anand district of middle Gujarat in India for the period 1949-1950 to 2007-2008.

Trends in Producti on
In case of the production of wheat crop, the Sinusoidal model had maximu m ad justed R 2 of 99 % with the comparatively lower values of RMSE (2.6877) and MAE (2.0565) among the parametric models fitted to the production of the wheat crop. Moreover, the run tests as well as the Shapiro-Wilks test values were non-significant indicating that the residuals due to this model were independently normally d istributed. All the estimated parameter values were in the 95 % confidence interval indicating that the parameter values were significant (Tab le  3

Trends in Producti vi ty
The data presented in Table 4 for productivity of wheat crop revealed that among the non-linear models fitted, the ma ximu m ad justed R 2 of 99 per cent was observed in the case of Sinusoidal model with the comparat ively lower values of RMSE (85.5915) and MAE (69.5027). All the estimated values of the parameters in the model were found to be within the 95% confidence interval indicating that the parameters were significant at 5% level of significance. The Shapiro-Wilks test (test for normality) and run test (test for randomness) values were found to be non-significant indicating that the residuals fulfilled model selection criteria. Among the non-linear models, the sinusoidal model was found suitable to fit the trends in productivity of wheat crop. Y = 1756.28+1043.13* Cos(0.06* X+2.92) ( R 2 = 99 * %) The graph of the fitted trend in productivity of wheat crop using the Sinusoidal Model is depicted in the Fig 3.

Discussions in area, production and producti vity of wheat Crop
In the present study exponential model was not found suitable as the best fitted model because of lack of assumptions of residuals, though some earlier reports [6,31] found exponential model as suitable. Hence the nonparametric regression model was selected as the best fitted trend function for the area under the wheat crop. The Sinusoidal model was found suitable to fit the trends in production as well as productivity of wheat crop gro wn in India.

Conclusions
Fro m the above discussion on the analysis of area, production and productivity of wheat crop data based on different non-linear as well as nonparametric regression models, it can be finally concluded that, the parametric models (non-linear) are based on many assumptions and they go on the current condition of the crop and other parameters at a particular instant of time. It is not dynamic enough to be considered as a suitable method for the calcu lation or the estimation of the growth rate of the wheat crop.
Nonparametric model was found to be a suitable measure to estimate the growth rates of the wheat production data grown in India because it is based on fewer assumptions i.e. this model can be used as a most suitable one as it is dynamic and versatile enough to be considered for the statistical interpretation for the growth and trends for the wheat crop for the years to come.
The Sinusoidal model was found suitable to fit the trends in production as well as productivity of wheat crop grown in India. The area, production and productivity of wheat crop grown in India, had been shown in the increasing trend. The area of cult ivation had played a major ro le in increasing the trend in production.