INEAR UTISCAE AUTOREGRESSIVE ODE FOR FORECASTING SEASONA TIE SERIES DATA Brodjol S.S.U. Suartono and A.J. Endarta Department of Statistics Faculty of atematic and Natural Sciences Sepulu Nopember Institute of Tecnology Indonesia Abstract. Te aim of tis researc is to study furter some latest progress of aelet transform for time series analysis particularly about ultiscale Autoregressie (AR). Tere are tree main issues tat be considered furter in tis researc. Te first is some properties of scale and aelet coefficients from aximal Oerlap Discrete Waelet Transform (ODWT) decomposition particularly at seasonal time series. Te second is about deelopment of model building procedures of AR based on te properties of scale and aelet coefficients. Ten te tird is empirical study about te implementation of procedures ic ae been deeloped and comparison study about te forecast accuracy of AR to oter models. Te results so tat scale coefficients of ODWT at seasonal time series as seasonal pattern and aelet coefficients are stationer. Te results of model building procedure deelopment yield a ne procedure for seasonal time series. In general tis procedure accommodates input lags of scale and aelet coefficients introduced by Renaud et al. () and oter additional seasonal lags. Te result of empirical study sos tat tis procedure orks ell for finding te best AR model for seasonal time series forecasting. Te comparison study of forecast accuracy sos tat te AR model yields better forecast tan ARIA model. Keyords: ultiscale; ODWT; AR; ARIA; Seasonal; Time series. Introduction Waelet transformation becomes an alternatie for time series analysis. Waelet is a function tat cuts te data matematically to different components and learns eac component using te suitable scale resolution (Daubecies 99). Te adantage of using aelet is separating te trend from te data automatically (Renaud Starck and urtag ). Nason and Sacs (999) mentioned te oter adantage i.e. te ability of modeling and estimating te autocorrelated trend data. Zang and Coggins () said tat aelet transformation is multiresolution decomposition tecnique to sole modelling problems yielding local representation signal bot in time and frequency domains. Waelet analysis as been applied by many researcers in statistical problems. Abramoic Bailey and Sapatinas () gae a reie corresponding to aelet analysis application and some researces tat ae been done in statistical problems including nonparametric regression density estimation inerse problem cange point problem and some certain aspects in time series analysis. Renaud Starck and urtag () also proposed ultiscale Autoregressie (AR) approac for time series forecasting using Haar Waelet. Te use of AR model brougt a better result compared to classical metod suc as ARIA (Damayanti 8).Tere are only fe researces about AR model for seasonal time series especially Indonesian tourism. Terefore tis researc is done to find te appropriate AR model for seasonal data. Te aelet transformation ic is used in tis researc is aximal Oerlap Discrete Waelet Transform using Daubecies and Haar aelet.
. Teories Tis section explains te teories and metods tat are used in te researc. Te teories are about Waelet aximal Oerlap Discrete Waelet Transform (ODWT) and AR model.. Waelet and aximal Oerlap Discrete Waelet Transform (ODWT) Waelet is a term of up-don small aes itin certain period. As te comparison tere is big aes suc as sinusoidal function (Percial and Walden ). Waelet families used in tis researc are Haar and Daubecies aelet. Waelet filter coefficients in Haar aelet are as follo. and () Scale filter coefficients in Haar aelet are as follo. g and g () Waelet and scale filter coefficients in Daubecies aelet are calculated as follo. + +. () + + g g g g. () aximal Oerlap Discrete Waelet Transform (ODWT) is a modification of Discrete Waelet Transform. ODWT as oter names suc as undecimated-discrete Waelet Transform (DWT) Sift inariant DWT aelet frames translation DWT non decimated DWT (Percial and Walden ). ODWT aelet filter } { l troug l l and ODWT scale filter } { l g troug l g l g Te aim of ODWT formulation is to sole te sensitiity of DWT especially in te initial point selection. Te sensitiity is about te donsampling of te aelet and scale filter outputs at eac steps of pyramid algoritm. Define A as matrix N N including circular filter g for ODWT and B as matrix N N including filter. For example for first leel define B as matrix N N in Eq. (7) so tat x B. Analogy to A e get x A. atrix A equals to B by replacing l it l g. B.
Terefore ODWT first leel can be ritten as follos. T P is ortonormal matrix. B x A P (5) B x ere P and (6) A Equation (6) for reconstructing data x from ODWT coefficient if first leel decomposition is done is as follos. T P x T - P Because P is an ortonormal matrix P ten and T x P x P - [ ] T A T x B. (7) x B + A. (8) To get an efficient calculation for ODWT scale and aelet coefficients it leel-j e use pyramid algoritm for ODWT. Smooting and detail coefficients in different leels could be found by using pyramid algoritm ic is deeloped by allat (Popoola Amas S. and Amad K. 7).. ultiscale Autoregressie (AR) Assume tat stationarity signal X ( X X... Xt ) ill be predicted. Te main idea is to use coefficients from te decomposition i.e. j for k... A j t ( k ) j and j... J and also J for k... A J t ( k ) j (Renaud dkk. ). Type of te prediction is focused on autoregressie (AR) type. Autoregressie (AR) process it order p or AR(p) can be ritten as follos. Xˆ + p T k k T Xˆ ˆ t φ X. (9) t ( k ) ˆ φ ˆ ˆ. () t+ Xt + φ Xt +... + φp Xt p+ In te use of te decomposition AR prediction ill be ultiscale Autoregressie (AR) ic is gien by Renaud et al. (). J A j A j t + aˆ j k j + aˆ j t ( k ) J + k j k k Xˆ () J J t ( k )
Were j is te number of leel ( j... J ) A j is order of AR model ( k... Aj ) j t is aelet coefficient is scale coefficient and a is AR coefficient. j t For example t6 J and Xˆ t+ j j k A ( k ) AR() is as follos. aˆ + j k j aˆ j t ( k) a j k t + a X X X X X X X 5 X 6 t 5 + a k atrix equation in Eq. () can be ritten sortly as t J + k + a 5 J J t ( k) 8 9 t + a 5 t 8 9 + a t a a a a a a () s Aα () ere s is ( X X... X 6 ) A is aelet and scale coefficients as inputs in AR model and α is parameter. Parameter estimation is done by using east Square principle. Terefore te parameters are estimated by ˆ α ( A' A) A' s. () AR model in Eq. () is a model tat can be used for stationary time series forecasting (Renaud et al. ). Renaud et al. introduce an input process of aelet model isually dran in Fig.. Figure represents common form of aelet model it leel decomposition and order AR model. Figure. Illustration of aelet model it J and A j
. ultiscale Autoregressie (AR) odel Conergence ultiscale Autoregressie (AR) model is proposed to follo AR model for eac scale of multiresolution transformation. Teorem tat sos if AR model follos AR model troug te forecasting procedure conergence to optimal procedure and asymptotically equals to te best forecast is as follos. Teorem (Renaud et al. ) ' Assume { X t } follo AR causal process it order p and parameters φ { φ φ... φ p } and X t φ X t +... + φp X t p + εt ere { ε t } is IID ( σ ). If order A j is cosen at eac scale more j tan or equals to p for j... J ten te multiresolution model is J A j A j t + aˆ j k j + aˆ j t ( k ) J + k j k k ere te increasing of sample size αˆ is asymptotically to: Xˆ. (5) n J J t ( k ) ' ( ˆ α α) N( σ ( R' W Γ W R) ). (6) / α is defined as coefficient of te best linear procedure of X t+ based on te prior obseration α is equialent to Ω φ and Γ B is autocoariance matrix. ΓB [ γ ( t k) ] t k... B ere γ (l) is te autocoariance of lag l series.. Researc etodology Te data ic is used in tis researc is te montly number of foreign tourist isiting Bali troug Ngura Rai airport. Te data is since January 986 until April 8. Te in-sample data is first 6 obserations and te rest is out-sample. Te data as trend and seasonal pattern Te analysis starts it ODWT decomposition to find te aelet and scale filter coefficients. ODWT decomposition is done to te stationary data so differencing is applied to te data at first. Te data is modelled by using AR model it te appropriate procedure. Te analysis is ended by selecting te best AR model based on te out-sample criteria. Figure sos te steps of analysis for te trend seasonal data. B B B
Start Detrending is applied if tere is deterministic trend and differencing is done if tere is stocastic trend in te data. Tis process is done to get te stationary data. ODWT decomposition leel- using Haar and Daubecies aelet family to get scale and aelet coefficients. Determine te lags of scale and aelet coefficients suc as te lags proposed by Renaud et al. () and additional seasonal lags. First and second orders AR modelling using te lags of scale and aelet coefficients. Are all AR model parameters significant? No Stepise metod applied to get te significant parameters of AR model Yes odel ealuation based on te RSE and te fulfilment of te assumption suc as residual ite noise and normality of residual Finis Figure. Te steps of AR model for seasonal trend data
. Empirical Results Te plot of te data is son in Fig.. From te plot of te data it can be knon tat te data as a stocastic trend and seasonal patterns. Te data sould be stationary for te AR model; ence regular differencing is applied. ODWT leel- decomposition is done to te differencing data. Te decomposition use Haar and Daubecies aelet families. Figure sos tat aelet coefficients are stationer and Figure sos tat scale coefficients ae seasonal pattern. Te coefficients are used in te AR model. Te most appropriate procedure for tis type of data is procedure i.e. AR model procedure for seasonal time series it trend. Te complete procedure for AR model can be seen in Prasetiyo (9). Time Series Plot of Zt Time Series Plot of Differencing Data 8 Bali Bomb I Bomb II 5 Bali Bomb I Bomb II 6 5 Tourists (Zt) 8 Xt -5 6-5 6 69 9 5 8 6 8 7-75 6 69 9 5 8 6 8 7 Figure. Time series plot of te number of foreign tourists isiting Bali troug Ngura Rai airport Plot of Xt and D() Waelet Coefficients Plot of Xt and Haar Waelet Coefficients 5 5 5 5-5 -5-5 -75 Variable Xt W W W W -5-75 Variable Xt W W W W 6 69 9 5 8 6 8 7 6 69 9 5 8 6 8 7 Figure. Plot of differencing data D() and Haar aelet coefficients
Plot of Xt D() Scale Coefficients and Haar Scale Coefficients 5 5-5 -5-75 Variable Xt V_D() V_Haar 6 69 9 5 8 6 8 7 Figure. Plot of differencing data D() and Haar scale coefficients First AR models ic are tried in tis researc are Haar-AR() and D()-AR() models. Te predictors are te input lags proposed by Renaud et al. and additional multiplicatie seasonal period lags i.e. multiplicatie lags. ags proposed by Renaud et al. and additional additie seasonal period lags are also tried as te predictors. Te parameters ic are not significant are taken out from te model. Tis process is sortened by using stepise metod. By using stepise metod for all models te significant parameters are knon. Using te significant parameters as te predictors AR model is built. After AR() models bot using leel- Haar and Daubecies or D() ae been built AR() model is built using same aelet families. Same it oter classical time series analysis AR model also needs some assumptions tat sould be fulfilled. Te assumptions are ite noise residual and normality of residual. Te ite noise of residual can be cecked isually by plotting te ACF of te residual. Te normality of te residual can be tested by using normality test suc as Kolmogoro-Smirno test statistics. Te AR models are summarized in Table. Table. Summary of AR odels for Foreign Tourists Visiting Bali odel RSE of in-sample Wite Noise Normality of Residual ultiplicatie Haar-AR() 585.89 - ultiplicatie D()-AR() 9.6 ultiplicatie Haar-AR() 585.89 - ultiplicatie D()-AR() 8.6 - Additie Haar-AR() 9.7 Based on Table models tat satisfy bot assumptions are multiplicatie D()-AR() and additie Haar-AR() models. Altoug tese models don t bring least RSE tese models are selected as te best models because tey satisfy bot assumptions. ultiplicatie D()-AR() model it only significant parameters for te differencing data is D()-AR(J;[6][8][][7][]) model. Additie Haar-AR() model in tis researc is Haar-AR(J;[6][6][6]). Tese bot models ill be compared it anoter type model.
Tabel. odel Comparison Based on RSE odel In-sample RSE Out-sample ARIA()().98 8.87 D()-AR(J; ;[6][8][][7][]) 9.6 757.57 Haar-AR(J;[6][6][6]) 9.7 6.85 Te oter type model in tis researc is ARIA model. Te ARIA model ic is appropriate for te data is ARIA()(). Te comparison beteen te best AR model and ARIA model is son in Table. Based on RSE of in-sample ARIA()() is te best model. Based on RSE of out-sample Haar-AR(J;[6][6][6]) is te best model. Besides RSE te model comparison is also done isually. Figure 5(a) sos te plot of actual data and te forecasts of bot ARIA and AR models. Time Series Plot of Actual ARIA D() Haar Plot of Residual of ARIA D() Haar 5 Variable ARIA D() Haar 5 People (Zt) Residual 5 5 Variable Actual ARIA D() Haar 6 8 6-5 6 8 Index 6 (a) Figure 5. (a) Plot of actual data and forecasts. (b) Plot of residual from ARIA and AR models Based on Figure 5(a) te best model is Haar-AR(J;[6][6][6]) because its forecasts are most releant to te actual data. Figure 5(b) sos tat residual of Haar-AR(J; [6][6][6]) is te nearest to zero. Terefore te best model for forecasting number of foreign tourist isiting Bali troug Ngura Rai airport is Haar-AR(J;[6][6][6]) model. (b)
5. Conclusion Commonly AR model building procedures inole te input lags of scale and aelet coefficient proposed by Renaud et al. () and additional input lags of scale and aelet coefficient especially in seasonal lags. AR model yields better forecast tan ARIA model. It is son by RSE of out-sample from Haar-AR(J;[6][6][6]) model is less tan from ARIA()( ). Te forecasts from AR model are also most releant to te actual data. Te residual of AR model is te nearest to zero compared to residual of ARIA model. References [] Abramoic F. Bailey T.C. and Sapatinas T.. Waelet analysis and its statistical applications. Te Statistician 9 Part pp. -9. [] Damayanti I. 8. etode Waelet untuk Peramalan Time Series yang Non Stasioner. Unpublised aster Tesis. Department of Statistics Sepulu Nopember Institute of Tecnology Indonesia. [] Daubecies I. 99. Ten ectures on Waelets Society for Industrial and Applied atematics. Piladelpia: SIA. [] Nason G.P. and on Sacs R. 999. Waelets in Time Series Analysis Pil. Trans. Royal Soc. ond. A. [5] Percial D.B. and Walden A.T.. Waelet etods for Time Series Analysis. Cambridge: Cambridge Uniersity Press. [6] Popoola A. Amad S. and Amad K. 7 Fuzzy-Waelet etod for Time Series Analysis submitted for te Degree of Doctor of Pilosopy from te Uniersity of Surrey England. [7] Prasetiyo. D. 9. Te Application of Waelet etod to ultiscale Autoregressie odel for Forecasting Te Seasonal Time Series. Unpublised Bacelor Project Department of Statistics Sepulu Nopember Institute of Tecnology Indonesia. [8] Renaud O. Starck J.. and urtag F.. Prediction based on a ultiscale Decomposition Int. Journal of Waelets ultiresolution and Information Processing ol. no. pp. 7-. [9] Zang B. and Coggins R.. ultiresolution Forecasting for Futures Trading Using Waelet Decompositions. Neural Netork ol. no..