Tamilnadu, with a daily milk production of 145.88 lakh litres, is one of the leading states in milk production in India. And the Tamilnadu milk cooperatives play a major role in the development dairy within the state. This study aims at forecasting milk production in Tamilnadu, based on data on milk production during the years from 1978 to 2008. The study considered Autoregressive (AR), Moving Average (MA) and Autoregressive Integrated Moving Average (ARIMA) processes to select the appropriate stochastic model for forecasting milk production in Tamilnadu. Based on ARIMA (p, d, q) and its components ACF, PACF, Normalized BIC, Box-Ljung Q statistics and residuals estimated, ARIMA (1, 1, 0) was selected. Based on the chosen model, it could be predicted that the milk production would increase to 7.15 million tons in 2015 from 5.96 million tons in 2008 in Tamilnadu

Prabakaran, R

Sankar, T Jai

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International Multidisciplinary Research Journal 2012, 2(1):10-15 ISSN: 2231-6302 Available Online: http://irjs.info/    Forecasting milk production in Tamilnadu   T. Jai Sankar1 and R. Prabakaran2  1Assistant Professor, Department of Statistics, Bharathidasan University, Tiruchirappalli – 620 024, Tamilnadu, India. 2Vice-Chancellor, Tamilnadu Veterinary and Animal Sciences University, Chennai – 600 051, Tamilnadu, India.  Abstract  Tamilnadu, with a daily milk production of 145.88 lakh litres, is one of the leading states in milk production in India. And the Tamilnadu milk cooperatives play a major role in the development dairy within the state. This study aims at forecasting milk production in Tamilnadu, based on data on milk production during the years from 1978 to 2008. The study considered Autoregressive (AR), Moving Average (MA) and Autoregressive Integrated Moving Average (ARIMA) processes to select the appropriate stochastic model for forecasting milk production in Tamilnadu. Based on ARIMA (p, d, q) and its components ACF, PACF, Normalized BIC, Box-Ljung Q statistics and residuals estimated, ARIMA (1, 1, 0) was selected. Based on the chosen model, it could be predicted that the milk production would increase to 7.15 million tons in 2015 from 5.96 million tons in 2008 in Tamilnadu.  Keywords: Milk production, Normalized BIC, forecasting, ARIMA.  INTRODUCTION      On global basis India is able to produce milk at very competitive prices by virtue of utilizing crop residues for rearing the animals. India is predominantly an agrarian society where animal husbandry forms the backbone of national economy. Dairying provides millions of small marginal farmers and landless labours means for their subsistence. Milch animals are reared mainly through the utilization of crop residues; thus milk production is essentially a subsidiary activity to agriculture. The planners recognized dairying, because of the potential impact it can make, as an instrument to bring about socio-economic transformations in the rural sector. Cows account was about 44% of the milk produced in India, buffalo 53%, and goat, sheep and camel etc. account for the balance 3%. Thus, dairying in India is a sharp contrast to the pattern of milk production in the advanced countries of the world, where specialized dairy farming is in vogue. Tamilnadu State's share in total milk production at the All India level was 5.38%.       Tamilnadu, with a daily milk production of 145.88 lakh litres, is one of the leading states in milk production in India. And the Tamilnadu milk cooperatives play a major role in the development dairy within the state. The milk cooperatives of Tamilnadu, with the help from the government and National Dairy Development Board (NDDB), have played substantial roles in taking the state to the current position. Besides, NDDB also undertakes methodical approach and appropriate strategy for the upliftment of Tamilnadu milk cooperatives. The dairy development programmes in Tamilnadu have been implemented through a wide network of co-operatives, which follow the three-tier structure where primary milk producers' co-operative societies remain at the base level. In the district level, there is a union of producers' co-operative societies, whereas, Federation of District Co-operative Milk Producers' Union remain at the top (state) level of the structure.      In Tamilnadu, Milk Producers' Cooperative Societies function at the village level, where milk producers get enrolled as members. The members get animal health cover for their animals; breed improvement is also carried out. District unions collect the milk produced at the village societies. In this background, this study was conducted to forecast the future milk production in the State, so as to help the policy planners to formulate needed strategies for achieving and sustaining the targets in the sector.  MATERIAL AND METHODS       As the aim of the study was to forecast milk production, various forecasting techniques were considered for use. ARIMA model, introduced by Box and Jenkins (1970), was frequently used for discovering the pattern and predicting the future values of the time series data. Akaike (1970) discussed the stationary time series by an AR(p), where p is finite and bounded by the same integer. Moving Average (MA) models were used by Slutzky (1973). Hannan and Quinn (1979) suggested obtaining the order of a time series model by minimizing the errors for pure AR models, and Hannan (1980) for ARMA models. A second order determination method could be considered as a variance of Schwarz's Bayesian Criterion (SBC) which gives a consistent estimate of the order of an ARMA model. Hosking (1981) introduced a family of models, called fractionally differenced autoregressive integrated moving average models, by generalizing the ‘d’ fraction in ARIMA (p, d, q) model.      Stochastic time-series ARIMA models were widely used in time series data having the characteristics (Alan Pankratz, 1983) of parsimonious, stationary, invertible, significant estimated coefficients and statistically independent and normally distributed residuals. When a time series is non-stationary, it can often be made stationary by taking first differences of the series i.e., creating a new time series of successive differences (Yt-Yt-1). If first differences do not convert the series to stationary form, then first differences can be created. This is called second-order differencing. A distinction is made between a   Received: Sept 11, 2011; Revised: Dec 18, 2011; Accepted: Dec 26, 2011.  *Corresponding Author  T. Jai Sankar  Assistant Professor, Department of Statistics, Bharathidasan University,  Tiruchirappalli – 620 024, Tamilnadu, India. Tel: +91-9444469629 Email: tjaisankar@gmail.com    International Multidisciplinary Research Journal 2012, 2(1):10-15  11second-order differences (Yt-Yt-2).       While Mendelssohn (1981) used Box-Jenkins models to forecast fishery dynamics, Prajneshu and Venugopalan (1996) discussed various statistical modeling techniques viz., polynomial, ARIMA time series methodology and nonlinear mechanistic growth modeling approach for describing marine, inland as well as total fish production in India during the period 1950-51 to 1994-95. Tsitsika et al. (2007) also used univariate and multivariate ARIMA models to model and forecast the monthly pelagic production of fish species in the Mediterranean Sea during 1990-2005.      The time series when differenced follows both AR and MA models and is known as autoregressive integrated moving averages (ARIMA) model. Hence, ARIMA model was used in this study, which required a sufficiently large data set and involved four steps: identification, estimation, diagnostic checking and forecasting. Model parameters were estimated using the Statistical Package for Social Sciences (SPSS) package and to fit the ARIMA models.  Autoregressive process of order (p) is, tptpttt YYYY εφφφµ +++++= −−− ....2211 ; Moving Average process of order (q) is, tqtqtttY εεθεθεθµ +−−−−= −−− ....2211 ; and the general form of ARIMA model of order (p, d, q) is tqtqttptpttt YYYY εεθεθεθµφφφ +−−−−++++= −−−−−− ........ 22112211         where Yt is milk production, tε’s are independently and normally distributed with zero mean and constant variance2σ for t = 1,2,..., n; d is the fraction differenced while interpreting AR and MA and φs and θs are coefficients to be estimated.  Trend fitting       The Box-Ljung Q statistics was used to transform the non-stationary data in to stationarity data and to check the adequacy for the residuals. For evaluating the adequacy of AR, MA and ARIMA processes, various reliability statistics like R2, Stationary R2, Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE), and Bayesian Information Criterion (BIC) [as suggested by Schwartz, 1978] were used. The reliability statistics viz. RMSE, MAPE, BIC and Q statistics were computed as below:  2/112)(1 −= ∑=∧niii YYnRMSE; ∑=∧−=ni iiiYYYnMAPE1)(1 BIC(p,q) = ln v*(p,q) + (p+q) [ ln (n) / n ]        where p and q are the order of AR and MA processes respectively and n is the number of observations in the time series and v* is the estimate of white noise variance σ2.  )()2(12knrknnQki−+=∑=      where n is the number of residuals and rk is the residuals autocorrelation at lag k.      In this study, the data on milk production in Tamilnadu were collected from the Tamilnadu Cooperative Milk Producers Federation, Government of Tamilnadu for the period from 1978 to 2008 and were used to fit the ARIMA model to predict the future production.   RESULTS AND DISCUSSION Model identification       ARIMA model was designed after assessing that transforming the variable under forecasting was a stationary series. The stationary series was the set of values that varied over time around a constant mean and constant variance. The most common method to check the stationarity is to explain the data through graph and hence is done in Figure 1.       Figure 1 reveals that the data used were non-stationary. Again, non-stationarity in mean was corrected through first differencing of the data. The newly constructed variable Yt could now be examined for stationarity. Since, Yt was stationary in mean, the next step was to identify the values of p and q. For this, the autocorrelation and partial autocorrelation coefficients (ACF and PACF) of various orders of Yt were computed and presented in Table 1 and Figure 2.   Fig 1. Time plot of milk production in Tamilnadu  T. Jai Sankar and R. Prabhakaran   12   Fig 2. ACF and PACF of differenced data  Table 1. ACF and PACF of milk production  Lag Auto Correlation Box-Ljung Statistic Partial Auto Correlation Value Df Sig. Value Df Value Df 1 0.027 0.174 0.025 1 0.876 0.027 0.183 2 0.084 0.171 0.266 2 0.875 0.083 0.183 3 -0.376 0.168 5.302 3 0.151 -0.384 0.183 4 -0.222 0.165 7.127 4 0.129 -0.234 0.183 5 0.004 0.161 7.128 5 0.211 0.096 0.183 6 0.026 0.158 7.155 6 0.307 -0.085 0.183 7 -0.002 0.155 7.155 7 0.413 -0.232 0.183 8 -0.086 0.151 7.476 8 0.486 -0.131 0.183 9 0.027 0.148 7.509 9 0.584 0.067 0.183 10 -0.011 0.144 7.515 10 0.676 -0.108 0.183 11 0.124 0.141 8.293 11 0.687 -0.045 0.183 12 -0.013 0.137 8.302 12 0.761 -0.022 0.183                    The tentative ARIMA models are discussed with values differenced once (d=1) and the model which had the minimum normalized BIC was chosen. The various ARIMA models and the corresponding normalized BIC values are given in Table 2. The value of normalized BIC of the chosen ARIMA was 11.195.  Table 2. BIC values of ARIMA (p, d, q)  ARIMA (p, d, q) BIC values (0, 1, 0) 11.245 (0, 1, 1) 11.197 (0, 1, 2) 11.290 (1, 1, 0) 11.195 (1, 1, 1) 11.243 (1, 1, 2) 11.410 (2, 1, 0) 11.340 (2, 1, 1) 11.487 (2, 1, 2) 11.483    Model estimation        Model parameters were estimated using SPSS package and the results of estimation are presented in Tables 3 and 4. R2 value was 0.96. Hence, the most suitable model for milk production was ARIMA (1, 1, 0), as this model had the lowest normalized BIC value, good R2 and better model fit statics (RMSE and MAPE). Table 3. Estimated ARIMA model of milk production   Estimate SE t Sig. Constant -5980.471 9723.478 -0.615 0.544 AR 1 0.018 0.193 0.091 0.928  International Multidisciplinary Research Journal 2012, 2(1):10-15  13Table 4. Estimated ARIMA model fit statistics  Fit Statistic Mean Stationary R-squared 0.015 R-squared 0.966 RMSE 227.534 MAPE 4.491 Normalized BIC 11.195   Diagnostic checking       The model verification is concerned with checking the residuals of  the model to see if they contained any systematic pattern which still could be removed to improve the chosen ARIMA, which has been done through examining the autocorrelations and partial autocorrelations of the residuals of various orders. For this purpose, various autocorrelations up to 12 lags were computed and the same along with their significance tested by Box-Ljung statistic are provided in Table 5. As the results indicate, none of these autocorrelations was significantly different from zero at any reasonable level. This proved that the selected ARIMA model was an appropriate model for forecasting milk production in Tamilnadu.   Table 5. Residual of ACF and PACF of milk production  Lag ACF PACF Mean SE Mean SE Lag 1 -0.001 0.183 -0.001 0.183 Lag 2 0.084 0.183 0.084 0.183 Lag 3 -0.360 0.184 -0.362 0.183 Lag 4 -0.204 0.206 -0.232 0.183 Lag 5 0.032 0.213 0.108 0.183 Lag 6 0.014 0.213 -0.086 0.183 Lag 7 -0.021 0.213 -0.240 0.183 Lag 8 -0.107 0.213 -0.135 0.183 Lag 9 0.013 0.215 0.047 0.183 Lag 10 -0.018 0.215 -0.133 0.183 Lag 11 0.120 0.215 -0.061 0.183 Lag 12 -0.021 0.217 -0.044 0.183       The ACF and PACF of the residuals are given in Figure 3, which also indicated the ‘good fit’ of the model. Hence, the fitted ARIMA model for the milk production data was: ttt YY ε+−−= −1018.0471.5980                                                                                                                                Fig 3. Residuals of ACF and PACF T. Jai Sankar and R. Prabhakaran   14 Forecasting       Based on the model fitted, forecasted milk production (in million tons) for the year 2009 through 2015 respectively were 6081, 6219, 6400, 6584, 6770, 6959 and 7150 million tons (Table 6). To assess the forecasting ability of the fitted ARIMA model, the measures of the sample period forecasts’ accuracy were also computed. This measure also indicated that the forecasting inaccuracy was low. Figure 4 shows the actual and forecasted value of milk production (with 95% confidence limit) in the State.  Table 6. Forecast of milk production (in million tons) in Tamilnadu Year Actual Predicted LCL UCL 1978 1681 -- -- -- 1979 1727 1785 1264 2307 1980 1860 1817 1318 2316 1981 1738 1957 1481 2433 1982 1886 1809 1342 2277 1983 1788 1925 1459 2391 1984 2562 1904 1440 2367 1985 2846 2786 2323 3250 1986 3118 3280 2818 3743 1987 3295 3227 2765 3690 1988 3109 3364 2902 3827 1989 3238 3203 2741 3666 1990 3410 3262 2799 3724 1991 3375 3597 3134 4059 1992 3357 3522 3059 3984 1993 3468 3352 2890 3815 1994 3524 3563 3101 4026 1995 3694 3709 3247 4172 1996 3791 3818 3355 4280 1997 3977 3925 3462 4387 1998 4061 4128 3666 4591 1999 4273 4220 3758 4683 2000 4574 4411 3948 4873 2001 4909 4797 4334 5259 2002 4988 5171 4709 5634 2003 4622 5154 4692 5617 2004 4753 4568 4105 5030 2005 4784 4725 4263 5188 2006 5474 5052 4590 5515 2007 5705 5780 5317 6242 2008 5961 6052 5589 6514 2009 -- 6081 5618 6543 2010 -- 6219 5477 6961 2011 -- 6400 5316 7484 2012 -- 6584 5243 7926 2013 -- 6770 5213 8327 2014 -- 6959 5213 8705 2015 -- 7150 5234 9067  CONCLUSION The most appropriate ARIMA model for milk production forecasting was found to be ARIMA (1, 1, 0). From the forecast available from the fitted ARIMA model, it can be found that forecasted production would increase to 7.15 million tons in 2015 from 5.96 million tons in 2008. That is, using time series data from 1978 to 2008 on milk production, this study provides evidence on future milk production in the State, which can be considered for future policy making and formulating strategies for augmenting and sustaining milk production in the State.     International Multidisciplinary Research Journal 2012, 2(1):10-15  15 Fig 4. Actual and estimate of milk production  REFERENCES [1] Akaike H. 1970. Statistical Predictor Identification. Annals of Institute of Statistical Mathematics 22: 203-270.  [2] Alan Pankratz. 1983. Forecasting with univariate Box–Jenkins models - concepts and cases. John Wiley, New York, Page 81. [3] Box G E P and Jenkins J M. 1970. Time Series Analysis - Forecasting and Control. Holden-Day Inc., San Francisco. [4] Hannan E J and Quinn B G. 1979. The determination of the order of an autoregression. Journal of Royal Statistical Society B(41): 190-195. [5] Hannan E J. 1980. The estimation of the order of an ARMA process. Annuals of Statistics 8: 1071-1081. [6] Hosking J R M. 1981. Fractional differencing. Biometrika 68(1): 165-176. [7] Mendelssohn R. 1981. Using Box-Jenkins models to forecast fishery dynamics: identification, estimation and checking. Fishery Bulletin 78(4): 887-896. [8] Prajneshu and Venugopalan  R. 1996. Trend analysis in all India marine products export using statistical modeling techniques. Indian Journal of Fisheries 43(2): 107-113. [9] Slutzky E. 1973. The summation of random causes as the source of cyclic processes. Econometrica 5:105-146. [10] Tsitsika E V, Maravelias C D and Haralabous J. 2007. Modeling and forecasting pelagic fish production using univariate and multivariate ARIMA models. Fisheries Science 73: 979-988.  

Forecasting milk production in Tamilnadu

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Forecasting milk production in Tamilnadu

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