Forecasting Method Selection Using ANOVA and Duncan Multiple Range Tests on Time Series Dataset

Transcription

1 Foecasing Mehod Selecion Using ANOVA and Duncan Muliple Range Tess on Time Seies Daase Adhisya Ena Pemanasai, Dayang Rohaya Awang Rambli, P. Dhanapal Duai Dominic Compue and Infomaion Science Dep., Univesii Teknonologi PETRONAS Banda Sei Iskanda, Tonoh Peak, Malaysia {ohaam, Absac Selecion of a suiable foecasing echnique is of pime impoance in ode o obain a bee pedicion esul. This pape demonsaed he use of wo saisical appoaches namely, Analysis of Vaiance (ANOVA) and Duncan muliple ange ess fo deemining he pefomance of diffeen foecasing mehods. Thee foecasing mehods wee chosen and compaed: egession, decomposiion, and ARIMA. Daa fom monhly incidence of Salmonellosis in US fom 1993 o 26 was colleced and used fo echnical analysis. ANOVA was iniially used o idenify significan diffeence beween he acual daa and hee foecasing mehods. Based on he esuls fom ANOVA, selecion of appopiae mehod was conduced using Duncan muliple ange ess. The esuls showed ha boh egession and ARIMA could be used in he Salmonellosis daa. On he conay, decomposiion mehod yielded he leas pefomance and is no suiable fo being applied on he available daase. Keywods - foecasing; egession; decomposiion; ARIMA; ANOVA; Duncan Muliple Range Tes I. INTRODUCTION Diffeen foecasing mehods have been employed o pedic fuue numbe of diseases occuences. The esuls can be useful in assising he making of policy o educe disease incidence. Vaious echniques ha have been applied o pedic disease incidence in human, include Mulivaiae Makov chain model o pojec he numbe of ubeculosis (TB) incidence in he Unied Saes fom 198 o 21 [1], exponenial smoohing o foecas he numbe of human incidence of Schisosoma haemaobium a Mali [2], ARIMA model o foecas he SARS epidemic in China [3], a Bayesian dynamic model o monio he influenza suveillance as one faco of SARS epidemic [4], seasonal auoegessive models o analyze Cuaneous leishmaniasis (CL) incidence in Cosa Rica fom 1991 o 21 [5], he applicaion of decomposiion mehod [6] and seasonal ARIMA o pedic numbe of Salmonellosis human incidence [7]. Whils some have employed seveal foecasing models, ealie appoaches o foecasing is ofen based on a single foecasing echnique. Due o numeous numbes of available foecasing echniques wih diffeence pefomance, selecion of he mos appopiae pedicion model o yield bee pedicion esuls becomes ciical. Besides, a single echnique may no yield he same pedicion accuacy fo diffeen ype of daases. Thus, his fuhe highlighs he impoance of selecing an appopiae foecasing model fo each specific daase. Analysis of Vaiance (ANOVA) is one of he commonly used saisical mehods o compae among seveal goups. In his pape, ANOVA was used fo selecing foecasing mehod based on compaison of means beween foecasing esuls; ha is fo esing a significan diffeen beween goup means. ANOVA is ofen used when a use needs o compae pefomance which involves moe han wo paamees. The advanage of ANOVA ove ohe ess such as simple -ess is ha ANOVA can deec effec of ineacion beween vaiables. I could also be used o es moe complex hypoheses in exising poblem [8]. When diffeences beween goups exis, a pos hoc es can hen be conduced o idenify which goup ha diffes fom he ohes. In his pape, Duncan muliple ange es was used. This pape aims o povide he empiical esuls fo evaluaing and finding he appopiae mehod in esimaing fuue numbe of Salmonellosis disease incidence. Diffeen foecasing mehods wee used o addess he goal, including egession, decomposiion, and ARIMA. The esuls wee evaluaed using ANOVA and Duncan muliple es. To accommodae he pupose, monhly numbe of Salmonellosis incidence was seleced. Salmonellosis daase in Unied Saes was colleced fo he 168 monh peiod fom Januay 1993 o Decembe 26. The daa was obained fom he summay of noifiable diseases in Unied Saes fom he Mobidiy and Moaliy Weekly Repo (MMWR) ha published by Cenes fo Disease Conol and Pevenion (CDC). The seasonal vaiaion of he oiginal daa is pesened as a cha /1/$26. (c) 21 IEEE

2 pesened in Fig. 1. The plo shows a peak season of incidence in Augus while he minimum numbe of incidence occuences in Januay. Since ime seies plo of he hisoical daa exhibied he seasonal vaiaions which pesen simila end evey yea, SARIMA was chosen as he appopiae appoach o develop a model pedicion. No of Incidence (monh) Figue 1. Monhly numbe of Tubeculosis incidence in US ( ). The emainde of he pape is sucued as follows. Secion 2 pesens model famewok. Secion 3 epos foecasing esuls. Secion 4 pesens ANOVA esuls. Secion 5 evaluaes Duncan muliple ange es. Finally, Secion 6 pesen conclusion. II. MODEL FRAMEWORK esuls of each mehod ae compaed wih acual daa using ANOVA blocked design. Finally, Duncan muliple es was conduced o idenify which mehod poduced he foecas close o acual daa. A Duncan es esul was also used o find he appopiae mehod among hem. III. FORECASTING RESULTS A. Regession Regession model ime seies is used when he independen vaiable is ime and he model focus on pedicing he fuue values. Twelve seasonal componens wee deemined fom he monhly daa. Unfounaely, he welfh monh could no povide any infomaion like he fis 11. Hence, he welfh monh is used as a baseline fo compaison. In his siuaion, a monhly end vaiable is applied. The final egession fomula was: y TR + SN + ε β + β + β s5 s1 s5, x 1 s1, x x x s1 s6 s3, s6, s9, s1, + β s11 s6, s11, s2 s7 s2, 51.22x x s7, x x x s7, s1, s4, s3 s8 s1, s3, s8, x s4 s x s8, x s4, s11, s9, x s5, s2, (1) whee y is he obseved value in ime peiod, TR is he seasonal faco in ime peiod, SN is he seasonal faco in ime peiod, and ε is he eo em in ime peiod. Figue 2. Flow of model developmen. Fuue numbe of Salmonellosis incidence was foecased using hee foecasing mehod, namely egession, decomposiion, and ARIMA. The flow of model developmen is pesened in Fig. 2. As illusaed in Fig.2, he fis sep was collecion Salmonellosis daa. This is followed by daa pocessing using each mehod (egession, decomposiion, ARIMA). Nex, he B. Decomposiion Decomposiion mehod is one of he seasonal smoohing mehods. By using his mehod, a seies will be boken ino some componens pa: end, seasonaliy, cyclical, and iegula (eo). Decomposiion mehod can be divided ino 2 caegoies of model: muliplicaive decomposiion and addiive decomposiion. When a ime seies exhibis an inceasing o a deceasing seasonal vaiaion hen he muliplicaive decomposiion model should be seleced. Addiive decomposiion model is used only when he ime seies display a consan seasonal vaiaion. Based on he cuen ime seies (Figue 1), he ime seies exhibis consan seasonal vaiaion so addiive decomposiion is used [9]. The addiive decomposiion model is:

3 y TR + SN + CL + IR (2) Whee y is he obseved value of he ime seies in ime peiod, TR is he end componen (o faco) in ime peiod, SN is seasonal componen (o faco) in ime peiod, CL is cyclical componen (o faco) in ime peiod, and IR is iegula componen (o faco) in ime peiod. In he model, componen of CL was emoved in eq. (2) because no cyclic componen was idenified in he ime seies. In ode o calculae SN, esimaion of i was used (sn ). Calculaing aveage, sn, fo each monh. Wih L 12 (numbe of peiod a yea), hen he seasonal faco is: L sn sn sn / L sn (3) 1 Finally, he esimaion of fo he end TR could be obained by fiing a egession equaion o he deseasonalized daa. The esulan funcion is b + b (4) 1 C. ARIMA Seasonal ARIMA (SARIMA) is used when he ime seies exhibis a seasonal vaiaion. A seasonal auoegessive noaion (P) and a seasonal moving aveage noaion (Q) foms he muliplicaive pocess of SARIMA as (p,d,q)(p,d,q) s. The subsciped lee s shows he lengh of seasonal peiod. Fo example, in an houly daa ime seies hen s 7, in a quaely daa s 4, and in a monhly daa s 12. Thus, he geneal SARIMA (p,p,q,q) model is φ + (5) s s p( B ) φp( B ) z δ θq( B) θq( B ) a Whee B is non-seasonal backshif opeao, B s is seasonal backshif opeao, φ is he auoegessive opeao of ode p, and θ is he moving aveage opeao. In ode o develop ARIMA model, Box-Jenkins (BJ) mehodology was used. BJ consiss of fou ieaive seps: idenificaion, esimaion, diagnosic checking, and foecasing. Diffeen ARIMA models wee applied o find he bes fiing model. The mos appopiae model was hen seleced by using he Bayesian Infomaion Cieion (BIC) and Akaike Infomaion Cieion (AIC) values. The model AR(9), SAR(12), MA(14), SMA(24) can be wien as SARIMA(9,,14)(12,1,24) 12. Fom his model he paamee esul wee AR(9).154, SAR(12) -.513, MA(14).255, SMA(24) The final model is expessed as eq. (6): 9 (1.154B +.513B ( B B +.78B ) z.219b 38 ) a (6) IV. ANALYSIS OF VARIANCE (ANOVA) ANOVA is a echnique ha fisly developed by R.A. Fishe in 192s. I is used o compae goup mean [1]. ANOVA uses wo hypohesis o deemine he esul, namely null hypohesis and alenaive hypohesis. ANOVA is called as analysis of an analysis of vaiance because ANOVA es compaes wo vaiance esimaions: vaiance wihin goup (he unsysemaic vaiaion/eo in he daa) and vaiance beween goups (effecs due o he expeimen). In his pape, acual daa and foecas esul beween linea egession, decomposiion, and ARIMA wee compaed by ANOVA. The expeimen has a vaiabiliy values fom a sysemaically conolled souces. Time (monhs) is he common souce of vaiabiliy ha can be conolled by blocking. Thus, a blocked design of ANOVA was applied wih monhs as he blocking paamee. Thee diffeen echniques wee compaed, namely egession, decomposiion, and ARIMA. The esimaed values of fom hese echniques wee compaed wih he acual daa using ANOVA. The hypohesis was: H H : μ 1 μ 2 μ 3 μ 4 : μ i, j 1, 2, 3, 4, i 1 i μ j Whee μ 1, μ 2, μ 3, μ 4 wee he aveage esimaion obained fom acual, egession, decomposiion, and ARIMA, especively. The hypoheses was esed by using level of significance (α) assumed as.5. The ANOVA esuls fo his poblem ae shown in Table I. If F value of he mehod measuemen is geae han he coesponding value of F ci, hen i can be concluded ha he esul beween mehods is having a significan diffeen (null hypohesis is ejeced). Ohewise, i can be concluded ha hee ae no significan diffeen beween mehods (null hypohesis is acceped). Fom he ANOVA esul shown in Table I, Sum Squae (SS) a α.5 was , while Mean Squae (MS) was I can be concluded ha in α.5, he null hypohesis was ejeced because: F ci and F , whee F > F ci Beside he P value of beween goup also gave an evidence o ejec he null hypohesis, whee P.953 <.5. The blocked ANOVA esed whehe any of he populaion means diffe fom each ohe. While, a muliple compaison es is conduced o check which populaion means diffe fom he ohes. The following secion is epoed he applicaion of Duncan's Muliple Range Tes. j

4 TABLE I. ANOVA RESULT Goups Coun Sum Aveage Vaiance Acual Regession Decomposiion ARIMA Souce of Vaiaion SS df MS F P F ci (α.5) Blocks (monh) 8.64E E Beween Goup (Mehod) Wihin Goups (Eo) Toal 9.3E V. DUNCAN MULTIPLE RANGE TEST When null hypohesis is ejeced, hen a pos hoc es can be conduced o idenify which goups having diffeen mean. In his sudy, Duncan muliple ange es was chosen. Duncan muliple ange es can mainain a low oveall ype I eo [11] and also can o be applied in goups applicaion ha exhibi no significanly diffeen [12]. Duncan es uses a sudenized ange saisic wihin a muliple sage es, efeed o as a muliple ange es. A leas significance eo is calculaed and associaed wih an inceasing numbe of sample subse means. The populaion mean is significanly diffeen if he ange of subse geae han he leas significan ange (LSR). Fo he confidence level α.5, LSR was found fom he Duncan s Table. LSR wee compued as shown below: Mean sum of squae eo, MS , n 135 Then, sandad eo of each aveage, MS S (7) n Since he oal numbe of seleced goups was 4, he oal numbe of anges was equal o 3. Fom he able of significan anges Mongomey fo 42 degees of feedom (84 is he numbe of degees of feedom fo wihin goups fom ANOVA able) and α.5, he hee anges wee calculaed as given below: 2 α.5(2,42) 3 α.5(3,42) 4 α.5(3,42) LSR can be calculaed fom equaion: R S (8) p p By applying (8), LSR esuls wee R , R , and R The mean of foecasing esuls is pesened in Table II. TABLE II. MEAN OF EACH GROUPS Mehods Mean Symbol Mean Value Acual M Regession M Decomposiion M ARIMA M The means of mehods in Table II need o be soed fis as M1, M2, M4, and M3. Then, each goup was compaed as he following: Acual diffeence M1 M < (R 2 ) Acual diffeence M1 M > (R 3 ) Acual diffeence M1 M < (R 4 ) Acual diffeence M2 M > (R 2 ) Acual diffeence M2 M < (R 3 ) Acual diffeence M3 M > (R 2 ) The decomposiion mehod pefoms pooly compaed o ohe echniques. The decomposiion mehod (coesponding o he mean value: M ) yield moe diffeences when compaed wih ohe mehods; he acual diffeen was geae han is leas significan ange (19.47>76.486). I is shown ha he mean diffeence of decomposiion wih ohe goups was geae fom he leas significan ange (M2 M3 and M3 M4). This indicaes ha decomposiion mehod is no an appopiae echnique o be applied o he cuen daase. Conaily, he mean compaison beween he acual daa and ohe mehod

5 (egession and ARIMA) eveal small diffeences. The diffeence of egession and ARIMA wee less han he leas significan ange (1.626< fo egession and fo ARIMA). The non significan diffeence beween acual daa and he esul fom egession and ARIMA suggess ha boh could be used fo pedicing fuue numbe of human incidence using he same hisoical daa. VI. CONCLUSION This pape focuses on he selecion of foecasing mehods using ANOVA and Duncan muliple ange es. Thee diffeen foecasing mehods wee seleced, including egession, decomposiion, and ARIMA. The fis wo mehods wee adiional appoaches, while ARIMA was he moe sophisicaed han he wo. Due o is seasonal vaiaion ime seies daa ype, Salmonellosis disease was seleced as he case sudy. Fou eamen goups wee idenified as acual daa; egession foecasing esul, decomposiion foecasing esul, and ARIMA foecasing esul. ANOVA was applied ino hese goups o deemine whehe hee was significan diffeen beween goup mean o no. The ANOVA esuls yielded significan diffeen. Duncan muliple ange ess wee fuhe conduced o find which goup had he diffeences. The empiical esuls showed ha decomposiion was no suiable o use in he ime seies daase used. Convesely, boh egession and ARIMA could be used in he exising daa. The implicaion fom his sudy is ha a complex and sophisicaed echnique does necessaily yield bee pedicion. Even hough egession is a simple mehod howeve i could pefom bee han he decomposiion mehod. REFERENCES [1] S. M. Debanne, R. A. Bielefeld, G. M. Cauhen, T. M. Daniel, and D. Y. Rowland, "Mulivaiae Makovian Modeling of Tubeculosis: Foecas fo he Unied Saes," Emeging Infecious Diseases, vol. 6, pp , 2. [2] D. C. Medina, S. E. Findley, and S. Doumbia, "Sae Space Foecasing of Schisosoma haemaobium Time-Seies in Niono, Mali," PLoS Negleced Topical Diseases, vol. 2, pp. 1-12, 28. [3] D. Lai, "Monioing he SARS Epidemic in China: A Time Seies Analysis," Jounal of Daa Science, vol. 3, pp , 25. [4] P. Sebasiani, K. D. Mandl, P. Szolovis, I. S. Kohane, and M. F. Ramoni, "A Bayesian Dynamic Model fo Influenza Suveillance," Saisics in Medicine, vol. 25, pp , 26. [5] L. F. Chaves and M. Pascual, "Climae Cycles and Foecass of Cuaneous Leishmaniasis, a Nonsaionay Veco-Bone Disease," PLoS Medicine, vol. 3, pp , 26. [6] A. E. Pemanasai, D. R. Awang Rambli, and P. D. D. Dominic, "Foecasing of Zoonosis Incidence in Human Using Decomposiion Mehod of Seasonal Time Seies," in Poc NPC 29, Tonoh, Malaysia, 29, pp [7] A. E. Pemanasai, D. R. Awang Rambli, and P. D. D. Dominic, "Pedicion of Zoonosis Incidence in Human using Seasonal Auo Regessive Inegaed Moving Aveage (SARIMA)," Inenaional Jounal of Compue Science and Infomaion Secuiy (IJCSIS), vol. 5, pp , 29. [8] SaSof. (28). ANOVA/MANOVA [Online]. Available: hp:// [9] B. L. Boweman and R. T. O'Connell, Foecasing and Time Seies An Applied Appoach, 3d ed: Duxbuy Thomson Leaning, [1] A. Agesi. (27). Chape 12 Compaing Goups: Analysis of Vaiance (ANOVA) Mehods [Online]. Available: [11] V. Bewick, L. Cheek, and J. Ball, "Saisics eview 9: One-way analysis of vaiance," Ciical Cae, vol. 8, pp , 24. [12] J. P. Geaghan. (29). [Online]. Available: hp:// 19a%2Noes%2Fall29.pdf