6/10/2014. Definition. Time series Data. Time series Graph. Components of time series. Time series Seasonal. Time series Trend

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1 6//4 Defiiio Time series Daa A ime series Measures he same pheomeo a equal iervals of ime Time series Graph Compoes of ime series Q 7 Q 7 Q 3 7 Q 4 8 Q 8 Q 8 Q 3 8 Q 4 9 Q 9 Q 9 Q 3 9 Q 4 Q Q Tred: uderlyig log-erm moveme Cycle: medium-erm cyclical movemes abou he red Seasoal (S: facors ha occur oe or more imes per year. Sable i size ad direcio from year o year. Irregular (I: residual afer oher compoes have bee removed. Should exhibi o paer. We combie he red ad he cycle o form redcycle (C, bu refer o his as he red. 3 4 Time series Tred Time series Seasoal Tred 5 5 Seasoal Q 7Q 7Q3 7Q4 8Q 8Q 8Q3 8Q4 9Q 9Q 9Q3 9Q4 Q Q 7Q 7Q 7Q3 7Q4 8Q 8Q 8Q3 8Q4 9Q 9Q 9Q3 9Q4 Q Q 5 6

2 6//4 Time series Irregular Decomposiio models Irregular The compoes are uobserved ad cofouded. They ca oly be separaely ideified by makig assumpios abou heir form. Two models 7Q 7Q 7Q 3 7Q 4 8Q 8Q 8Q 3 8Q 4 9Q 9Q 9Q 3 9Q 4 Q Q Addiive: Y C + S + I Muliplicaive: Y C x S x I 7 8 Time series Addiive Time series Muliplicaive Addiive Muliplicaive Q 7 Q 7 Q 3 7 Q 4 8 Q 8 Q 8 Q 3 8 Q 4 9 Q 9 Q 9 Q 3 9 Q 4 Q Q 7 Q 7 Q 7 Q 3 7 Q 4 8 Q 8 Q 8 Q 3 8 Q 4 9 Q 9 Q 9 Q 3 9 Q 4 Q Q 9 Seasoal adusme Esimaio The aim of seasoal adusme is o esimae he seasoal compoe of he ime series ad remove i. Addiive: SA(Y Y - S C + I Muliplicaive: SA(Y Y / S C x I Seasoal adusme approach The X- approach o esimaio is sequeial, based o Macaulay (93 esimae easies compoe firs remove easies compoe leavig ohers esimae ohers X- is a produc of he US Cesus Bureau 954: X-, X-, (Julius Shiski 967: X- Varia of he Cesus Mehod II Seasoal Adusme Program (Shiski, Youg,

3 6//4 The X- mehod X- uses a sysem of filerig o esimae he differe compoes I rus hrough a cycle of esimaio ad improveme hree imes. The primary filers used movig averages (MA, which are a example of liear filers. If Movig averages { y } + + is a ime series of values M y w y ( + + where M y is a MA of order < N successive erms ad ofe: w + If he M y is said o be "cered" + If w w for all he M y is said o be "symmeric" 3 4 Cerig movig averages Cerig movig averages A MA wih a order which is odd ca be cered o is middle value. A simple 3-erm MA: M y y y y 3 ( Period Values 3-erm MA For a MAwih a eve order, he cere of he MA falls bewee values. For example, a simple 4-erm MA ( y + y + y+ + y + 4 The middle of he MA is bewee he secod ad hird erms. The soluio: ake a -erm MA of he origial 4-erm MA. Period Value 4 erm MA X4 MA Cerig movig averages This is kow as a x4 MA. More geerally, we ca express such "composie" MAs as x m. For he x 4 composie MA: M y y y y y y 4 4 y y y y + y + y + y + + y The weighs are wrie [,,,, ]. 8 ( ( Seasoal adusme MA problem The X- mehod uses movig averages (MAs for seasoal adusme The problem wih MAs is 7 8 3

4 6//4 Example: MAs Seasoal adusme MA problem Ja 7 Apr 7 Jul 7 Oc 8 Ja 8 Apr 8 Jul 8 Oc 9 Ja 9 Apr 9 Jul 9 Oc Uadused 3x MA 3x5 MA The X- mehod uses movig averages (MAs for seasoal adusme The problem wih MAs is ed pois ad ouliers The soluio for X- was asymmeric movig averages However, research i he 97 s proved ARIMA forecasig - o eable symmeric movig averages - is beer (ie lower revisios 9 Seasoal adusme X- Seasoal adusme X- X- is a produc of he US Cesus Bureau 954: X-, X-, (Julius Shiski 967: X- Varia of he Cesus Mehod II Seasoal Adusme Program (Shiski, Youg, 975: Box-Jekis ARIMA models used o develop X--ARIMA (Dagum, Saisics Caada X- is a produc of he US Cesus Bureau 954: X-, X-, (Julius Shiski 967: X- Varia of he Cesus Mehod II Seasoal Adusme Program (Shiski, Youg, 975: Box-Jekis ARIMA models used o develop X--ARIMA (Dagum, Saisics Caada 998: RegARIMA opios used i he developme of X--ARIMA (Fidley, USCB Seasoal adusme X- X- is a produc of he US Cesus Bureau 954: X-, X-, (Julius Shiski 967: X- Varia of he Cesus Mehod II Seasoal Adusme Program (Shiski, Youg, 975: Box-Jekis ARIMA models used o develop X--ARIMA (Dagum, Saisics Caada 998: RegARIMA opios used i he developme of X--ARIMA (Fidley, USCB : Added SEATS decomposiio ad reamed X-3ARIMA-SEATS 3 Aalysig a ime series usig X- ARIMA. Choose a decomposiio. Fi a regarima model o clea ad forecas he series 3. Seasoally adus wih X- mehod 4. Use diagosics o assess your adusme 4 4

5 6//4 The X- algorihm The X- approach coais four cycles (labelled A-D. The B-cycle (assumig mohly daa ad a addiive model. Prelimiary esimaio of C usig a x MA: X The A-cycle cosiss of prior adusme (cleaig he daa. While he B- o D-cycles represe he ieraive par of he algorihm. Each ieraio leads o beer esimaes of he compoes 5 6 Prelimiary esimaio of red usig a x MA The B-cycle (assumig mohly daa ad a addiive model. Prelimiary esimaio of C usig a x MA: X. Prelimiary esimaio of S+I compoe ( ( S + I y C 7 8 Prelimiary esimaio of S+I compoe The B-cycle (assumig mohly daa ad a addiive model. Prelimiary esimaio of C usig a x MA: X. Prelimiary esimaio of S+I compoe ( ( S + I y C 3 Prelimiary esimaio of S usig a 3x3 MA: ( 3X 3 + S M S I S% S M S X 9 3 5

6 6//4 Prelimiary esimaio of seasoal usig a 3x3 MA The B-cycle (assumig mohly daa ad a addiive model Jauary Augus. Prelimiary esimaio of C usig a x MA: X. Prelimiary esimaio of S+I compoe ( ( S + I y C 3 Prelimiary esimaio of S usig a 3x3 MA: ( S M S + I 3X 3 S% S M X S 4. Prelimiary esimaio of SA: SA C + I y S % 3 3 Prelimiary esimaio of SA Seasoal adusme X- algorihm Afer prior adusme ( A cycle Apply Movig Average (MA o B o esimae C Remove C from B o leave S ad I (SoI Apply MA o SoI o esimae S Remove S from B o esimae CoI (seasoally adused Apply Hederso MA o CoI o esimae C Remove C from B o leave S ad I (SoI Apply MA o SoI o esimae S Remove S from B o esimae CoI Repea i C ad D cycles Hederso red filers MA filers work well oly whe he series coais a liear red. For his reaso, he secod esimaes of red is performed usig a Hederso filer. The weighs for a Hederso filer are compued o allow polyomial reds up o degree hree o be exacly reproduced. Hederso red filers Hederso weighs ca be cosruced for ay odd order from 5 upwards. Example weighs: 5 erm : {,84,6} 86 3 erm : { 35, 468,,, 475,36, 43} 6796 The Hederso filers used i X- are: 5-,7-erm for quarerly series 9-,3-,3-erm for mohly series

7 6//4 Selecio of filer leghs Legh of red filer A loger filer produces a smooher red (or seasoal facors bu is less resposive o chages i he red (or seasoaliy. The soluio buil io he X- is o use sigal o oise raios: For example, he raio for he red i a addiive decomposiio is: I C C I I C A large raio idicaes ha a log filer is eeded. 37 Mohly series Quarerly series I/C < 9-erm 5-erm <I/C<3.5 3-erm 5-erm I/C > erm 7-erm 38 Selecio of seasoal filer leghs Legh of filer for he seasoal MA (sma Agai, as wih he red, he choice of legh of filer is based o he oise o sigal raio. Bu here he raio is bewee oise ad seasoaliy. For, a addiive series:,, Ii Ii I i S Si, Si, ( i where i year, moh or quarer I/S<.5.5<I/S< <I/S< <I/S<6.5 I/S<6.5 sma3x3 Remove las year ad recalculae sma3x3 Remove las year ad recalculae sma3x