Statistical Models for Forecasting milk production - Statistics Project Example

Statistical Models for Forecasting milk production Introduction Dairy farming is a major economic activity in many countries contributing much to the national gross domestic product (GDP), income, employment and food to many small-scale farmers. The growth of the dairy sub sector has been largely driven by high domestic demand for milk and dairy products due to a growing population (Omondi and Meinderts, 2010). The country’s dairy herd size is the biggest in sub- Saharan Africa (Mugambi et al, 2010). Although dairy farming sector has a significant contribution to the national economy, household incomes and food security, the industry faces a number of technical, economic and institutional problems in milk production, processing and marketing (Karanja, 2003). These constraints affect the ability of the sector to participate and compete in the domestic and regional markets. The climate is very favorable for this farming and the breeds plus potential famers who can help in production and expand the industry. Forecasting of milk production enables policy makers and planners to estimate the supply of milk requirement in the future and formulate the appropriate strategies to meet the growing demand. In this project an attempt has been made to forecast milk production using statistical time series modeling techniques – double exponential smoothing and Autoregressive integrated moving average models. Statistical modeling techniques First the trend analysis of the data is observed. The modeling of the time series involves the steps of model specification, model estimation, diagnostic checking. The statistical time series modeling techniques used in this study are the Exponential Smoothing Technique and Auto-Regressive Integrated Moving Average Technique. In the selection of the appropriate technique for time series analysis, the smoothing of the data is done after ensuring the presence of trend. For smoothing, the common techniques discussed by Gardner (1985) are discussed below. In the selection of ARIMA model, the stationary of the series is first checked and if the data is non-stationary, stationary is achieved by differencing technique. (Satya et al, 2007). The next processes are model identification, estimation and diagnostic checking. Simple Exponential Smoothing (SES) This technique is appropriate for time series data which have no trend or seasonality. Its only smoothing parameter is level. The SES is most similar to an ARIMA (0, 1, 1) model with no constant. For the time series Y1, Y2, ……,Yt forecast for the next value Yt+1 say Ft+1 , is based on the weights α and (1-α) to the most recent observation Yt and the recent forecast Ft respectively, where the α is a smoothing constant. The model is: Ft+1 = Ft + (Yt - Ft) The optimum value of α which is considered on the forecast depending on the choice is identified by taking the one exhibiting the minimum Root Mean Squared Error (RMSE). Double Exponential Smoothing (Holt’s) This technique is appropriate for time series data which have a linear trend and no seasonality. Its smoothing parameters are level and trend, which are constrained by each other’s values. Holt’s exponential smoothing is most similar to ARIMA (0, 2, 2) model. The form of the model is Lt = αYt + (1-α) (Lt-1 + bt-1) bt = β(Lt – Lt-1) + (1-β)bt-1 Ft+m = Lt + bt m Where Lt is the level of the series at time t bt is the slope of the series at time t α and β are the smoothing trend parameters. The initial values are obtained as L1 = Y1 and b1 = Y2 – Y1. The optimal pair of parameters α and β which gives the minimum MAPE are taken. Auto-Regressive Moving Average (ARMA) Model The model is denoted by ARMA (p, q) and it indicates p Autoregressive terms and q Moving Average terms. This model contains the AR (p) and MA (q) models given as Where c is a constant φ1, ……..,φp and θ1, …….., θp are the parameters of the model εt ~ WN (0, σ2) Auto-Regressive Integrated Moving Average (ARIMA) Model In time series analysis, an autoregressive integrated moving average (ARIMA) model is a generalization of an autoregressive moving average (ARMA) model. In theory, the most general class of models for forecasting a time series are stationary and can be made stationary by transformations such as differencing and logging. ARIMA models form an important part of the Box-Jenkins approach to time-series modeling. A non-seasonal ARIMA model is classified as an ARIMA (p, d, q) model, where: p is the number of autoregressive terms, d is the number of non-seasonal differences and q is the number of moving average terms. Estimation At the identification stage one or more models are tentatively chosen that seem to provide statistically adequate representations of the available data. The parameters are estimated by modified least squares or the maximum likelihood techniques appropriate to time series data. Diagnostic For adequacy of the model, the residuals are examined from the fitted model and alternative models are considered. Different models can be obtained for various combinations of AR and MA individually and collectively. The satisfactory model is considered which adequately fits the data. Method selection The best model is obtained on the basis of minimum value of Akaike Information Criteria (AIC) which is given by: AIC = -2 log L + 2m Where m = p + q L is the likelihood function p& q are orders of Auto-Regressive and Moving Average models respectively - number of parameters, Akaike (1974) Model performance The performances of different fits are evaluated on basis of Mean Absolute Percentage Error (MAPE) and Root Mean Square Error (RMSE) which are given by; Where n is the number of years in the forecasting period Yt is the actual observed value at time t Ft is the forecasted value at time t DATA The milk production data Table 1. Data on milk production for the period 2000 to 2010. MILK PRODUCTION Year Cow milk Goat milk Camel milk Total 2000 2,224,000 116,000 25,200 2,365,200 2001 2,444,150 97,100 25,200 2,566,450 2002 2,811,950 10,100 25,200 2,847,250 2003 2,819,500 107,230 25,200 2,951,930 2004 2,829,900 118,500 36,000 2,984,400 2005 2,650,000 129,000 25,200 2,804,200 2006 3,500,000 105,000 32,200 3,637,200 2007 4,230,000 108,000 32,500 4,370,500 2008 3,990,000 110,000 27,000 4,127,000 2009 4,276,000 112,000 31,000 4,419,000 2010 4,641,600 120,000 33,600 4,795,200 ANALYSIS The results were calculated from the two statistical models: The Exponential Smoothing Model and the Auto-Regressive Integrated Moving Average model which gave the following results; Exponential smoothing model Time series plot (Figure 1 below) of the milk production data revealed that there is an increasing trend in the data. Holt’s double smoothing was found the most appropriate in smoothing the data. Figure 1: Time plot of the milk production data Using statistical computer package R-i386 3.0.2 an optimal fit was obtained which gave α = 0.099 and β = 0.000028 which gave the least Mean Absolute Percentage Error (5.60935). The fitted model is given by; Lt = 0.099Yt + 0.901(Lt-1 + bt-1) bt = 0.000028(Lt – Lt-1) + 0.999972bt-1 Ft+m = Lt + bt m Where m = 1, 2, 3 &4 and the initial values for the level Lt = 2365200 and bt = 201250. ARIMA Model The stationary check of the time series of milk production data in Kenya for the period 2000 to 2010 revealed that it was non-stationary. The series was made stationary by using the first differencing technique and thus the value of d was 1. The graphs of sample PACFs and ACFs were plotted (Figure2 and 3). On matching plots with the theoretical ones of various ARIMA processes, the PACF as spikes cut off after lag 1. Hence the order of AR component p was taken as 1. Also, in order that the proposed model adequately represents the data and at the same time have lesser number of parameters an MA component of order 1 was also added to the model. In addition, using R-program package for different values of p and q (0, 1 or 2), various ARIMA models were fitted and the appropriate model was chosen corresponding to minimum value of the selection i.e. Akaike Information Criteria (AIC). In this way, ARIMA (1, 1, 1) model was found to be the best model with the following parameters (Table 2). The fitted model is given by Figure 2: sample PACFs graph Figure 3: sample ACFs graph Call: arima(x = milk, order = c(1, 1, 1)) Coefficients: ar1 ma1 0.0136 -1.0000 s.e. 0.3652 0.2728 sigma^2 estimated as 1.872e+12: log likelihood = -156.67, aic = 319. Table 3: Forecasts of milk production using double exponential smoothing (Holt’s) and ARIMA (1, 1, 1) models Year Actual data Forecast of milk production Holt’s ARIMA(1,1,1) 1 2007 4370500 3874948 4061632 2 2008 4127000 4119396 4337532 3 2009 4419000 4363843 4578632 4 2010 4795200 4608291 4811579 MAPE 4.1672 4.0306 RMSE 266273.77 203392.36 Conclusion For the two models fitted, double exponential smoothing and ARIMA models, ARIMA model gave the least MAPE of 2.236 against that of Holt’s 5.815. Model performance evaluation was computed for the forecasted milk production for the years 2007 to 2010 (Table 3) by use of the performance evaluation measures MAPE and RMSE. Comparison of the results revealed that, for the models fitted, ARIMA (1, 1, 1) model performed better than the Holt model and double exponential model. Reference Gardner, E.S., Jr. (1985). Exponential smoothing: The state of the art, Journal of Forecasting, 4, 1-28. Karanja (2003). The Dairy Industry in Kenya: The Post-Liberalization Agenda. Paper presented at a Dairy Industry Stakeholders Workshop held in Nairobi, Kenya (27th August 2002). Mugambi, D. K, Wambugu, S. K, Gitunu, A. M. M and Maina, M (2010) Evaluation of cow milk production efficiency in eastern central highlands of Kenya. Proceedings of the 3rd international e-commerce on Agricultural BioSciences 2010. Pg. 72-73 Omondi S.P.W. and Meinderts, J 2010 the status of good dairy farming practices on small-scale farms in central highlands of Kenya. Satya Pal, Ramasubramanian, V and Mehta, S.C., 2007, Statistical models for Forecasting Milk Production in India, J.Ind.Soc.Agril.Statist, 61(2), 2007: 80-83. Read More