Complex exponenial Smoohing Ivan Sveunkov Nikolaos Kourenzes 3 June 24
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Inroducion Exponenial Smoohing mehods performed ver well in man compeiions: M-Compeiions in 982 and 2, Compeiion on elecommunicaion daa in 998 and 28, Tourism forecasing compeiion in 2. In pracice forecasers usuall use: SES for he level ime series, Hol s mehod for rend ime series, Hol-Winers mehod for a rend-seasonal daa.
Inroducion Hol s mehod is no performing consisenl. Examples: M-Compeiions; Talor, 28; Gardner & Diaz-Saiz, 28; Acar & Gardner, 22. Hol s mehod is sill ver popular in publicaions: Gelper e. al, 2; Maia & de Carvalho, 2.
Inroducion Several modificaions for differen pes of rends were proposed over he ears: Muliplicaive rend (Pegels, 969); Damped rend (Gardner & McKenzie, 985); Damped muliplicaive rend (Talor, 23); Prior daa ransformaion using cross-validaion (Bermudez e. al., 29). Model selecion procedure based on IC is usuall used. Is he seleced model alwas appropriae?
Objecives Propose a model overcoming limiaions of Hol's mehod and SES; Sud properies of he model; Carr ou a compeiion on differen daa ses.
Theoreical framework Simple exponenial smoohing: Principle of CES: smooh level and combine i wih correcion, using complex variables (Sveunkov, 22). Basic form of CES: x i i i i i ix 2 i s s,...,, 2, x x f
Theoreical framework Complex variables -> ssem of real variables: Final forecas of CES consiss of wo pars: level, correcion. + + α + α + x α + α = x x α + α α + α = x i i i i i ix
Theoreical framework ARMA(N,N): The order depends on he complex smoohing parameer value: if hen oherwise N j j j N j j j N j j j N j j j b a = x B b B a α +i α +i N N
Theoreical framework Weighs disribuion in ime.6 Im.6.4.4.2.2 weighs -.2 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 24 25 26 27 28 29 3 3 -.6 -.4 -.2.2.4.6 -.2 Re -.4 -.4 -.6 lag -.6 α iα.2 i.5
Theoreical framework Weighs disribuion in ime.2 Im.2.8.8 weighs.6.4.6.4.2 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 24 25 26 27 28 29 3 3 lag.2 Re -.6 -.4 -.2.2.4.6 α iα. 3 i
Theoreical framework Forecasing rajecories 2 α +i α =.9+i 3 α +i α =.2+i. 25 8 2 6 5 4 2 5 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 24 25 26 27 28 29 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 24 25 26 27 28 29 2 α +i α =+i.99 2 5 α +i α =.99+i. 8 6 5 4 2 95 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 24 25 26 27 28 29 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 24 25 26 27 28 29
Empirical resuls: seup M3-Compeiion daa. 33 ime series. Rolling origin. Auomaed ETS was used o spli daa ino caegories: level non-seasonal, level seasonal, rend non-seasonal, rend seasonal.
Empirical resuls: seup M3-Compeiion daa. 33 ime series. Rolling origin. Auomaed ETS was used o spli daa ino caegories. Series pe Number of series Level series Trend series Overall Forecasing horizon Rolling origin horizon ear 255 39 645 6 2 quar 36 45 756 8 6 monh 686 742 428 8 24 oher 6 3 74 8 6 Overall 38 695 33
Empirical resuls: compeiors. Naive (Naive), 2. Simple exponenial smoohing (SES), 3. Hol s addiive rend (AAN), 4. Pegels muliplicaive rend (MMN), 5. Sae-space ETS wih AICc model selecion (ZZN), 6. Gardner s Damped rend (AAdN), 7. Talor s Damped muliplicaive rend (MMdN), 8. Thea using Hndman & Billah, 23 (Thea), 9. Hndman & Khandakar 28 Auo ARIMA (ARIMA),.Complex exponenial smoohing (CES).
Empirical resuls MASE was calculaed for each of he horizons from each of he origins, Nemeni es was conduced o compare mehods for each of he series pe. General resuls for CES: a leas as good as SES on level series, ouperforms MMN and AAN on level series, a leas as good as MMN and AAN on rend series, ouperforms all he mehods on monhl rend series.
Empirical resuls. Nemeni es
Conclusions CES is flexible, is able o idenif rends and levels, does i beer han Hol and Pegels, has an underling varing-order ARMA(N,N), ouperforms all he oher mehods on monhl daa, is a leas as good as SES.
Fuure works Sud he influence of he number of observaions on CES accurac; Derive sae-space form of CES; Derive variance and likelihood funcion; Implemen seasonal ime series forecasing using CES; Implemen exogenous variables in CES; Use oher forms of correcion parameer.
Thank ou! Ivan Sveunkov, Lancaser Universi Managemen School Cenre for Forecasing - Lancaser, LA 4YX email: I.sveunkov@lancaser.ac.uk
Example. Trended series Series N2692 from M3 Series N2692 62 66 7 74 78
Example. Trended series ETS(M,A,N) Series N2692 62 66 7 74 78 82 Forecass from ETS(M,A,N)
Example. Trended series CES Series N2692 62 66 7 74 78 α iα.999993 +.3635i
Example. Trended series CES Series N2692 62 66 7 74 78
Example. Saionar series Series N66 in M3 Series N66 3 5 7
Example. Saionar series ETS(M,N,N) Forecass from ETS(M,N,N) 3 5 7
Example. Saionar series CES Series N66 2 4 6 8 99 99 992 993 994 995 α iα.9673464 +.997947i
Example. Saionar series CES Series N66 2 4 6 8 99 99 992 993 994 995