Comparison of Singular Spectrum Analysis and ARIMA

Transcription

1 Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 0, Dubln (Sesson CPS009) p.99 Comparson of Sngular Spectrum Analss and ARIMA Models Zokae, Mohammad Shahd Behesht Unverst, Department of Statstcs Evn ehran 989-6, Iran. E-mal: Mahmoudvand, Rahm Shahd Behesht Unverst, Department of Statstcs Evn ehran 989-6, Iran. E-mal: Naar, Nader Shahd Behesht Unverst, Department of Statstcs Evn ehran 989-6, Iran. E-mal: Introducton and Prelmnares Sngular Spectrum Analss (SSA) s a relatvel new powerful non-parametrc technque for tme seres analss ncorporatng the elements of classcal tme seres analss, multvarate statstcs, multvarate geometr, dnamcal sstems and sgnal processng (Golandna et al, 00). SSA s desgned to look for nonlnear, non statonar, and ntermttent or transent behavor n an observed tme seres, and has ganed successful applcaton n the varous scences such as meteorologcal, bomechancal, hdrologcal, phscal scences, economcs and fnance, engneerng and so on (see Khan and Posktt (00) and references theren). In spte of the successful applcaton of SSA for dfferent data n comparson wth other models such as ARIMA (see for example Hassan (007)), lttle of the current lterature analzes the statstcal propertes of SSA from an abstract theoretcal perspectve. So, to obtan a theoretcal vew of the forecastng wth SSA, accurac of forecastng wth SSA s compared wth ARIMA b means of statstcal smulatons. In what follows we gve a bref explanaton of the SSA method (for more detals see for example Golandna et al, 00). Consder the real-valued non-zero tme seres Y =(,, ) of suffcent length. Let K = L +, where L ( L /) s some nteger called the wndow length. he general structure of the SSA can be descrbed n four basc steps:

2 Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 0, Dubln (Sesson CPS009) p.99 Step, Embeddng: Defne the traector matrx X where: LK, =( x ), == X L L+ L+ K () Note that X s a Hankel matrx, whch means that all the elements along the off dagonal are equal. Step, Decomposton: Defne the matrx XX. Let λ λ λ L 0 denote egenvalues of XX and U be the normalzed egenvector correspondng to the egenvalue λ ( =,, L). hen sngular value decomposton (SVD) of the traector matrx X can be wrtten as: X= E E L Where, E = λ UV, V = X U / λ ( =,..., L). Step, Separaton: Partton the set of ndces {,, L} nto m dsont subsets { I I } separaton leads to the followng decomposton: Where, X =, =,..., m. I E I X= X X I I m,, m. hs Step4, Reconstructon: he last step n SSA, transforms each matrx of the grouped decomposton (step) nto a new seres of length. hs wll be dong wth dagonal averagng as follow. If z stands for an element of a matrx Z, then the k-th term of the resultng seres s obtaned b averagng z over all ; such that + = k+. B performng the dagonal averagng of all matrx components of X I n the expanson of X above, we obtan another expanson: X= X X, where I I m X I s the dagonalzed verson of the matrx X. hs s equvalent to the decomposton of the ntal seres Y =(,, ) nto a sum of I m seres; t m =, where ( ) ( ) ( ) Y =(,, ) corresponds to the matrx = ( ) t X I. It must be noted that the general purpose of the SSA analss s decomposton of orgnal seres wth addtve components that are ndependent and dentfable tme seres. Sometmes, however, one can also

3 Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 0, Dubln (Sesson CPS009) p.99 be nterested n partcular tasks, such as extracton of sgnal from nose, extracton of oscllator components and smoothng. Forecastng b SSA he Basc SSA recurrent forecastng algorthm dscussed n Golandna et al (00) should be regarded as the man forecastng algorthm. Although, there exst several natural modfcatons to ths algorthm that can gve better forecasts n specfc stuatons (see, e.g Golandna et al (00)), here we consder recurrent forecastng algorthm and call t b R-SSA. Let ( ) ˆ s denotes the R-SSA forecast at tme for lead tme s or s steps ahead. Accordng to the R-SSA, the followng recursve formula can be used to obtan forecasts:,,,, Where, Such that, π s the last component of the vector components ofu, ( =,..., L). π π π U and U s the vector consstng of the frst L Comparson Crtera Frst of all recall that an ARIMA(p,d,q) model can be explaned wth an equaton such as: p q d αl ( L ) t = θl εt = = Where L s the lag operator, the α 's are the parameters of the autoregressve part of the model, the θ 's are the parameters of the movng average part and the ε 's are error terms. he error terms are generall assumed to be ndependent, dentcall dstrbuted varables sampled from a normal dstrbuton wth zero mean. here are several crtera to measure the accurac of the forecastng model and all of them are based on the behavor of the forecast errors. As there s no one best crtera, we use root mean square errors (RMSE) whch commonl used for the evaluaton of alternatve forecastng models. he MSE of some ARIMA models wth respect to the number of step ahead, are shown n able. t

4 Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 0, Dubln (Sesson CPS009) p.994 able : MSE of several ARIMA models wth respect to the number of steps ahead Model Steps ahead s, where s> AR() AR() MA() σ σ ( + α ) σ ( + α 4 + α ) ( ( s σ + α ) α ) ( ) σ ( σ + α ) σ α ( α α) Depends on s σ σ ( + θ ) σ ( + θ ) σ ( + θ ) MA() σ σ ( + θ ) σ ( + θ + θ ) σ ( + θ + θ ) Smulaton Studes In order to evaluate the performance of the SSA procedure n comparson wth ARIMA models, data were smulated usng a set of statonar autoregressve and movng average models of orders and where the coeffcents were changed stepwse (wth varance of the whte nose process be one). For all data generatng processes, N = 0000 seres of length = 00 were smulated b means of arma.sm functon n statstcal software R. he smulated data were used to forecast, applng the R-SSA algorthm presented above. hen standard smulaton procedure was used to obtan estmates of root mean square errors (RMSE) for forecastng. Results he results are summarzed n Fgures, and ables and. Fgures and show RMSE of the out of sample forecastng b R-SSA and AR() (MA()). he parameters α and θ are vared from 0. to 0.9 n order to generate dfferent AR and MA processes for comparson. Moreover,,, and 4 step ahead outof-sample forecasts are depcted n the fgures (from left to rght) to assess the forecastng power of the consdered technques n the short and long lead. As t ndcated n both of fgures and, ARIMA models gves accurate forecastng ust for the parameters less than around 0.6 n step ahead, but generall, SSA has superort to ARIMA models for, and 4 steps ahead. It means that ARIMA acts well for short perod forecastng, whereas SSA s good for both short and long forecastng. Smlar results presented n ables and for AR() and MA(). Of Course t s worth menton that, accordng to the results of able and, R- SSA s better than ARIMA for all consdered values of the parameters.

5 Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 0, Dubln (Sesson CPS009) p.995 Fg : RMSE of the out of sample forecastng b R-SSA and AR() Fg : RMSE of the out of sample forecastng b R-SSA and MA() able : RMSE of the out of sample forecastng b R-SSA and AR() for several selected AR coeffcents Lead Method AR parameters: ( α, α ) (0.,0.) (0.,0.) (0.,0.4) (0.,0.) (0.,0.) (0.,0.4) (0.4,0.) (0.4,0.) (0.4,0.4) 4 R-SSA AR() R-SSA AR() R-SSA AR() R-SSA AR()

6 Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 0, Dubln (Sesson CPS009) p.996 able : RMSE of the out of sample forecastng b R-SSA and MA() for several selected MA coeffcents Lead Method MA parameters: ( θ, θ ) (0.,0.) (0.,0.) (0.,0.4) (0.,0.) (0.,0.) (0.,0.4) (0.4,0.) (0.4,0.) (0.4,0.4) 4 R-SSA MA() R-SSA MA() R-SSA MA() R-SSA MA() Concluson Despte the successful applcatons of SSA compared to smpler models, lttle of the current lterature analzes the statstcal propertes of SSA. herefore, to obtan a theoretcal vew of the forecastng wth SSA, basc SSA forecastng algorthm s compared wth ARIMA b means of a Monte Carlo stud. he evaluaton crtera used s the accurac of out of sample pont forecasts va RMSE. It has been shown n the smulatons that forecastng b SSA can be advantageous compared to ARIMA models. REFERENCES - Golandna N, Nekrutkn V, Zhglavsk A. Analss of me Seres Structure: SSA and related technques. Chapman & Hall/CRC, Rahman Khan, M, A.; and Posktt, D.S. (00). Descrpton Length Based Sgnal Detecton n Sngular Spectrum Analss. Workng Paper. Department of Econometrcs and Busness Statstcs, Monash Unverst. Australa. - Hassan, H. (007). Sngular Spectrum Analss: Methodolog and Comparson. Journal of Data Scence, 5(), pp ABSRAC Sngular Spectrum Analss (SSA) s a non-parametrc method that can be appled to analze tme seres of complex structure. he man purpose of SSA s a decomposton of the orgnal seres nto a sum of seres, so that each component n ths sum can be dentfed as ether a trend, perodc or quas-perodc component (perhaps, ampltude modulated), or nose. In ths paper, b means of statstcal smulaton, we compare the performance of the SSA and ARIMA models. We show that, n general, the performance of forecastng usng these methods are dfferent and depends on parameters of the models. Kewords: SSA, ARIMA, Forecastng.