Complex exponential Smoothing
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1 Complex exponenial Smoohing Ivan Sveunkov Nikolaos Kourenzes 3 June 24
2 This maerial has been creaed and coprighed b Lancaser Cenre for Forecasing, Lancaser Universi Managemen School, all righs reserved. You ma use his maerial for our privae educaional purposes, so long as he are clearl idenified as being creaed and coprighed b Lancaser Cenre for Forecasing, Lancaser Universi Managemen School. You are no permied o aler, change, or enhance hem. You ma no use an of he conen, ex and images, in full or in par, in a uorial, raining, educaion, wrien papers, videos or oher recordings. You are no permied o disribue or make available direcl or indirecl, wihin or ouside our compan, nor exploi hem wihou explici prior wrien permission from Lancaser Cenre for Forecasing, Lancaser Universi Managemen School, raining@forecasingcenre.com
3 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.
4 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.
5 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?
6 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.
7 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
8 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
9 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
10 Theoreical framework Weighs disribuion in ime.6 Im weighs Re lag -.6 α iα.2 i.5
11 Theoreical framework Weighs disribuion in ime.2 Im weighs lag.2 Re α iα. 3 i
12 Theoreical framework Forecasing rajecories 2 α +i α =.9+i 3 α +i α =.2+i α +i α =+i α +i α =.99+i
13 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.
14 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 quar monh oher Overall
15 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).
16 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.
17 Empirical resuls. Nemeni es
18 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.
19 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.
20 Thank ou! Ivan Sveunkov, Lancaser Universi Managemen School Cenre for Forecasing - Lancaser, LA 4YX I.sveunkov@lancaser.ac.uk
21 Example. Trended series Series N2692 from M3 Series N
22 Example. Trended series ETS(M,A,N) Series N Forecass from ETS(M,A,N)
23 Example. Trended series CES Series N α iα i
24 Example. Trended series CES Series N
25 Example. Saionar series Series N66 in M3 Series N
26 Example. Saionar series ETS(M,N,N) Forecass from ETS(M,N,N) 3 5 7
27 Example. Saionar series CES Series N α iα i
28 Example. Saionar series CES Series N
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