Implementation of ARIMA Model for Ghee Production in Tamilnadu

Transcription

1 Inter national Journal of Pure and Applied Mathematics Volume 113 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Implementation of ARIMA Model for Ghee Production in Tamilnadu T. Jai Sankar 1 and C. Vijayalakshmi 2 1 Department of Statistics, Bharathidasan University Tiruchirappalli , Tamilnadu, India 2 SAS, Mathematics Division, VIT University, Chennai, India 1 tjaisankar@gmail.com and 2 vijusesha2002@yahoo.co.in Abstract This research is a study model of forecasting ghee production of Tamilnadu for the years, Forecasting plays a vital role in many fields such as agricultural production, animal husbandry and dairy economics, stock prices prediction, etc. The present study was carried out the design of Autoregressive (AR), Moving Average (MA) and Autoregressive Integrated Moving Average (ARIMA) Process to select the appropriate model in the data related to ghee production in Tamilnadu. This paper covers the time series analysis with ARIMA (p, d, q) and its components ACF, PACF, Normalized BIC, Box-Ljung Q Statistics and Residual analysis. Based on the chosen model, it could be predicted that the ghee production would increase to tons in 2015 from 6547 tons in 2008 in Tamilnadu. The results are presented by numerically and graphically. AMS Subject Classification: 37A50 Keywords: Ghee Production, Time Series Analysis, ARIMA Model and Residual Analysis. 1 Introduction Ghee is characterized by a pleasant, nutty flavour and a creamy white to light yellow colour. It should be free from foreign colouring matter, sediment and should have a granular texture. It should be free from any objectionable odours. In this background, this study was conducted to forecast the future ghee production in the State, so as to help the policy planners to formulate needed strategies for achieving and sustaining the targets in the sector. ijpam.eu

2 2 Material and Methods As the aim of the study was to forecast ghee production, various forecasting techniques were considered for use. Hosking introduced a family of models, called fractionally differenced autoregressive integrated moving average models, by generalizing the d fraction in ARIMA (p, d, q) model [5]. Stochastic time-series ARIMA models were widely used in time series data having the characteristics ([2]) 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 (Y t Y t 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 second-order differences (Y t Y t 2 ). While Mendelssohn [7] used Box-Jenkins models to forecast fishery dynamics, Prajneshu and Venugopalan in [8], 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 to Garcia et al. [4] and Abedullah and Sabir [1] were some of the notable studies that look at various aspects of dairy sector in systematic fashion. Burki et al. [3] provided a potential in making impact on the dairy economy and forecasted fresh milk growth by using ARIMA model. Lohano and Soomro [6] used historical time series data employed Random Walk Model with drift trend-stationary autoregressive model forecasted the annual milk production. Tsitsika et al. [10] also used univariate and multivariate ARIMA models to model and forecast the monthly pelagic production of fish species in the Mediterranean Sea during Autoregressive process of order (p) is, Y t = µ + φ 1 Y t 1 + φ 2 Y t φ p Y t p + ɛ t ; Moving Average process of order (q) is, Y t = µ θ 1 ɛ t 1 θ 2 ɛ t 2... θ q ɛ t q + ɛ t ; and the general form of ARIMA model of order (p, d, q) is Y t = µ 1 Y t 1 +φ 2 Y t φ p Y t p +µ θ 1 ɛ t 1 θ 2 ɛ t 2... θ q ɛ t q +ɛ t ijpam.eu

3 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 R 2, Stationary R 2, 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: RMSE = [ 1 n n ] 1/2 (Y i Ŷi )2 and MAP E = 1 n i=1 n (Y i Ŷi ) Y i i=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 σ 2. Q = n(n + 2) k i=1 rk2 n k where n is the number of residuals and rk is the residuals autocorrelation at lag k. In this study, the data on ghee production in Tamilnadu were collected from the Tamilnadu Co-operative Milk Producers Federation Limited, Government of Tamilnadu for the period from 1977 to 2008 and were used to fit the ARIMA model to predict the future production. 3 Results and Discussion Model Identification: ARIMA model was designed after assessing that transforming the variable under forecasting was a stationary series. 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 Y t could now be examined for stationarity. Since, Y t 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 Y t were computed and presented in Table 1 and Figure 2. 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 ijpam.eu

4 Figure 1: Time plot of Ghee production in Tamilnadu BIC values are given in Table 2. The value of normalized BIC of the chosen ARIMA was Model Estimation: R 2 value was 0.75 (Table 3 and 4). Hence, the most suitable model for ghee production was ARIMA (1, 1, 1), as this model had the lowest normalized BIC value, good R2 and better model fit statistics (RMSE and MAPE). Table 1: ACF and PACF of Ghee production Auto Partial Auto Box-Ljung Statistic Lag Correlation Correlation Value Df Sig. Value Df Value Df Table 2: BIC values of ARIMA (p, d, q) ARIMA (p, d, q) BIC values (0, 1, 0) (0, 1, 1) (0, 1, 2) (1, 1, 0) (1, 1, 1) (1, 1, 2) (2, 1, 0) (2, 1, 1) (2, 1, 2) Diagnostic Checking: Autocorrelations and partial autocorrelations ijpam.eu

5 Figure 2: ACF and PACF of differenced data Table 3: Estimated ARIMA model of Ghee production Estimate SE T Sig. Constant AR AR of the residuals of various orders were obtained. 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 ghee production in Tamilnadu. Table 4: Estimated ARIMA model fit statistics Fit Statistic Mean Stationary R-squared R-squared RMSE MAPE Normalized BIC The ACF and PACF of the residuals are given in Figure 3(a), which also indicated the good fit of the model. Hence, the fitted ARIMA model for the ghee production data was: Y t = Y t ɛ t 1 + ɛ t Forecasting: Based on the model fitted, forecasted ghee production (in tons) for the year 2009 through 2015 respectively were 7178, 7830, 8507, 9211, 9941, and tons (Table 6). To assess the forecasting ijpam.eu

6 Table 5: Residual of ACF and PACF of Ghee production Lag ACF PACF Mean SE Mean SE Lag Lag Lag Lag Lag Lag Lag Lag Lag Lag Lag Lag 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 3(b) shows the actual and forecasted value of ghee production (with 95% confidence limit) in the State. Table 6: Forecast of Ghee production (in tons) in Tamilnadu Year Actual Predicted LCL UCL N Residual ijpam.eu

7 (a) (b) Figure 3: (a) Residuals of ACF and PACF and (b) Actual and estimate of Ghee production 4 Conclusion The most appropriate ARIMA model for ghee production forecasting was found to be ARIMA (1, 1, 1). From the forecast available from the fitted ARIMA model, it can be found that forecasted production would increase from 6547 tons in 2008 to tons in That is, using time series data from 1977 to 2008 on ghee production, this study provides an evidence on future ghee production in the State, which can be considered for future policy making and formulating strategies for augmenting and sustaining ghee production in the State. References [1] Z.M. Abedullah, and H. Sabir, Competitive efficiency of milk production in the Central Punjab, European Journal of Scientific Research, 7(1), (2005). [2] A. Pankratz, Forecasting with univariate Box-Jenkins modelsconcepts and cases. John Wiley, New York, Page 81, (1983). [3] A.A Burki, M. Khan and F. Bari, The state of Pakistans dairy: An assessment, CMER Working Paper, 05-34, LUMS: Lahore, (2005). ijpam.eu

8 [4] O. Garcia, K. Mahmood and T. Hemme, A review of milk production in Pakistan with particular emphasis on small-scale producers, PPLPI Working Paper No. 3, IFCN, Rome, (2003). [5] J.R.M. Hosking, Fractional differencing, Biometrika 68(1)(1981), [6] H.D. Lohano and F. M. Soomro, Unit Root Test and Forecast of Milk Production in Pakistan, International Journal of Dairy Science, 1( 2006.), [7] R. Mendelssohn, Using Box-Jenkins models to forecast fishery dynamics: identification, estimation and checking, Fishery Bulletin 78(4)(1981), [8] Prajneshu and R. Venugopalan, Trend analysis in all India marine products export using statistical modeling techniques, Indian Journal of Fisheries, 43(2)(1996), [9] E. Slutzky, The summation of random causes as the source of cyclic processes, Econometrica 5(1973), [10] E.V. Tsitsika, C.D. Maravelias and J. Haralabous, Modeling and forecasting pelagic fish production using univariate and multivariate ARIMA models, Fisheries Science 73(2007), ijpam.eu

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