An empirical analysis about forecasting Tmall air-conditioning sales using time series model Yan Xia

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1 An empirical analysis abou forecasing Tmall air-condiioning sales using ime series model Yan Xia Deparmen of Mahemaics, Ocean Universiy of China, China Absrac Time series model is a hospo in he research of saisics. On November 11, 2015, Tmall plaform s urnover was more han $91.2 billion which caused he aenion of scholars boh a home and abroad. So his paper aims o forecas sales of Tmall, which is helpful o he enerprises. Research mehods are ARIMA model and VAR model. The firs model is single-variable model and he laer is muli-variable model. In he sudy, ARIMA model makes he sequence smooh by using wo difference operaion. In VAR model, five explanaory variables are ransformed ino one main componen. By conras, VAR model does no give deailed accurae predicion, bu ARIMA model does. Therefore, single-variable ime series model is more suiable for sales forecas han muli-variable model. Keywords ARIMA model, VAR model, Sales forecas. I. INTRODUCTION In recen years, he prosperiy of elecronic commerce makes he enerprises pay more and more aenion o markeing sraegy, especially accurae sales forecas. Time series model is ha using hisorical daa relaed o pas behavior o infer he fuure behavior of ime sequence. So his paper aims o do an empirical analysis abou sales forecas using ime series model. Indusry daa such as sales, price, online acive sores, online sores clinching a deal, online producs and producs clinching a deal need collecing. Use MATLAB and EVIEWS sofware o process he daa. The body of his paper includes four pars. The firs par is inroducion. The second par is model definiion abou ARIMA model and VAR model. The hird par is modeling and forecasing sales. The las par is conclusion. Generally, AR(p) formula is as follow: II. MODEL DEFINITION X X X X (1) p p If he error erm is whie noise sequence, hen he process is described as pure AR(p). On he conrary, process as follow: q q is pure MA(q) (2) ARMA model[1] process has he model srucure as follow: X X X X (3) p p q q Auoregressive inegraed moving average model, ARIMA model for shor, is srucure is as follow: d ( B) x ( B) 2 E( ) 0, Var ( ), E( ) 0, s s Ex 0, s s (4) Is basic idea is ha le non-saionary ime series smooh by difference, so ha i can use ARMA model o esablish saionary ime series model. Vecor auoregressive model [2], VAR model for shor, is srucure is as follow: Page 45

2 X AX AX A X U (5) p p A and i X are m dimension marix. T E( UU ) 0, s s U and are m dimension vecor. Le T uu E( U ) 0, E( U U ), and Above wo models, when he reciprocal of characerisic roos are inside he uni circle, hey are smooh. III. MODELING AND FORECAST Monhly daa are used covering he period from January 2010 o December I was provided by a enerprises. 3.1 ARIMA modeling We need o do pure random inspecion. Because only he non-whie noise sequence modeling makes sense. LB[3] es resul is as follows. TABLE 1 LB TEST Delayed order LB saisic P value Table 1 show ha P value is less han he significance level of Therefore, he alernaive hypohesis may be acceped. This sequence is no whie noise sequence. Then le us do saionariy es. ADF es shows P value approximaes 1, so his sequence is non-saionary sequence. Modeling process is as follows: To eliminae seasonal flucuaions, do firs-order difference operaion by sep 12. To eliminae rend changes, do firs-order difference operaion by sep 1. Differenced sequence is smooh. Considering sample size, le p and q change beween 0 o 3. When AIC funcion reaches minimum value, we ge a row vecor including p and q [4]. TABLE 2 AIC FUNCTION VALUE p q AIC p q AIC Table 2 shows ha he srucure of his model should be confirmed as ARMA(2,3). Nex, le us esimae he parameers and forecas sales. Is formula is as follow: X X X (6) Forecas can be seen from Fig.1 Page 46

3 FIG.1 SALES FORECAST OF 2014 USING ARIMA MODEL The average error rae calculaed is , accurae for sales. The residual sequence is whie noise. 3.2 VAR modeling FIG.2 THE RESIDUALS OF ARMA MODEL AND ITS ACF AND PACF Facors ha affec sales are five variables, so i is suiable o use principal componen analysis or PCA [5] for shor o reduce dimensions before esablishing he VAR model. TABLE 3 RESULTS OF PCA Number Principal Componen Conribuion Rae Cumulaive Conribuion Rae 1 z z z z z The conribuion rae of he firs principal componen is above 90%.Is formula as follow: z x x x x x (7) Page 47

4 The VAR model uses wo variables, namely z1 and sales. Firsly, saionariy es shows ha when p 6, ha is, he reciprocal of characerisic roos are inside he uni circle, model is smooh. 1.5 Inverse Roos of AR Characerisic Polynomial 1.5 Inverse Roos of AR Characerisic Polynomial VAR(6) model VAR(7) model FIG.3 THE RECIPROCAL OF CHARACTERISTIC ROOTS DISTRIBUTION ABOUT VAR(6) AND VAR(7) Similar wih ARIMA model, using MATLAB sofware o calculae AIC funcion. The srucure of his model is evenually deermined as VAR (6). Is parameers calculaed can be seen on equaion (8): sal es sal es( 1) sal es( 2) sal es( 3) sal es( 4) sal es( 5) sal es( 6) z1( 1) z1( 2) z1( 3) z1( 4) z1( 5) z1( 6) (8) FIG.4 VAR MODEL FORECAST SALES OF 2014 VAR model can grasp he change of ime series from he macroscopic aspec. Bu is precision is no high. Page 48

5 IV. CONCLUSION As single variable model, ARIMA model esablishes equaion by calculaing ACF and PACF, he residuals included. According o akaike informaion crierion, he model srucure is idenified as ARMA (2,3). I is very precise and credible. Wih regard o muli-variable ime series model, i did principal componen analysis o reduce dimensions. Five explanaory variables are ransformed o one main componen. VAR model has wo variables. On he conrary, predicion accuracy of VAR model is no high. I is suiable for macro-economic analysis. This paper sudies he issues of sales forecas, which belongs o he caegory of microeconomic. I comes o a conclusion ha single-variable ime series model is more suiable for his problem, especially he sochasic ime series models. Alhough we find an ideal ime series model, sales forecas in November may have oo big error. This is a problem of srucural breaks, which need furher research. REFERENCES [1] Shuyuan He. Random process [M]. Peking Universiy Press, [2] Tiemei Gao. Economeric analysis mehod and modeling: applicaion and insance EViews [M]. Beijing: Tsinghua Universiy press, 2006: [3] Jianhong Wu, Lixing Zhu. Mulivariae ime series GARCH ype model of diagnosic es [J]. Journal of mahemaical physics, 2010, 30 (3) : [4] Sifan Yang. Time series modeling and forecasing [D]. Tsinghua universiy, [5] Shoukui Si. Mahemaical modeling algorihm and applicaion [M]. Naional defense indusry press, 2011 Page 49

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