Time Series Descriptive Methods

Transcription

1 Time Series Descripive Mehods Summar... 1 Daa Ipu... 3 Aalsis Summar... 4 Daa Table... 4 Aalsis Opios... 5 Horizoal Time Sequece Plo... 6 Verical Time Sequece Plo... 7 Auocorrelaios... 8 Auocorrelaio Fucio Parial Auocorrelaios Parial Auocorrelaio Fucio Periodogram Periodogram Plo Iegraed Periodogram Tess For Radomess Crosscorrelaios Crosscorrelaio Plo Save Resuls Calculaios Summar The Descripive Mehods procedure creaes various ables ad plos for ime series daa. A ime series cosiss of a se of sequeial umeric daa ake a equall spaced iervals, usuall over a period of ime or space. The procedure plos he daa ad displas he auocorrelaios, parial auocorrelaios, ad sample periodogram. Tess are coduced o deermie wheher or o he observaios could be samples from a radom or whie oise process. If a secod ime series is supplied, crosscorrelaios bewee he wo series are also calculaed ad displaed. Sample SaFolio: sdescribe.sgp 2013 b SaPoi Techologies, Ic. Time Series Descripive Mehods - 1

2 Sample Daa: The file golde gae.sgd coais mohl raffic volumes o he Golde Gae Bridge i Sa Fracisco for a period of = 168 mohs from Jauar, 1968 hrough December, The able below shows a parial lis of he daa from ha file: Moh Traffic 1/ / / / / / / / / / / / / The daa were obaied from a publicaio of he Golde Gae Bridge b SaPoi Techologies, Ic. Time Series Descripive Mehods - 2

3 Daa Ipu The daa ipu dialog box requess he ame of he colum coaiig he ime series daa: Daa: umeric colum coaiig equall spaced umeric observaios. Time idices: ime, dae or oher idex associaed wih each observaio. Each value i his colum mus be uique ad arraged i ascedig order. Samplig Ierval: If ime idices are o provided, his defies he ierval bewee successive observaios. For example, he daa from he Golde Gae Bridge were colleced oce ever moh, begiig i 01/68. Seasoali: he legh of seasoali s, if a. The daa is seasoal if here is a paer ha repeas a a fixed period. For example, mohl daa such as raffic o he Golde Gae Bridge have a seasoali of s = 12. Hourl daa ha repea ever da have a seasoali of s = 24. If o er is made, he daa is assumed o be oseasoal (s = 1) b SaPoi Techologies, Ic. Time Series Descripive Mehods - 3

4 Tradig Das Adjusme: a umeric variable wih observaios used o ormalize he origial observaios, such as he umber of workig das i a moh. The observaios i he Daa colum will be divided b hese values before beig ploed or aalzed. Selec: subse selecio. Aalsis Summar The Aalsis Summar displas he umber of observaios i he ime series ad he legh of seasoali: Descripive Mehods - Traffic Daa variable: Traffic (Golde Gae Bridge Traffic Volume) Number of observaios = 168 Sar idex = 1/68 Samplig ierval = 1.0 moh(s) Legh of seasoali = 12 Noe: a limied amou of missig daa is permied, providig here are o oo ma missig values close ogeher. Missig values are replaced b ierpolaed values accordig o he mehod oulied i he Calculaios secio. Daa Table The Daa Table displas he ipu daa: Daa Table for Traffic Period Daa Adjused 1/ / / / / / / / / / / / Period: he sample idex. Daa: he observaio. Adjused: he adjused daa, if a adjusme has bee specified usig Aalsis Opios b SaPoi Techologies, Ic. Time Series Descripive Mehods - 4

5 Aalsis Opios Aalsis Opios permis he daa o be rasformed before beig ploed or aalzed: Mah: rasforms he daa b performig he idicaed mahemaical operaio. Wih he excepio of he Box-Cox rasformaio, he selecios are self-explaaor. The Box-Cox rasformaio is used whe ecessar o make he daa more Gaussia. For a deailed discussio, see he documeaio for he Box-Cox Trasformaios procedure. Seasoal: seasoall adjus he daa usig he idicaed mehod. Seasoal adjusmes are desiged o remove a seasoal compoe from he daa. The mehods used are discussed i he documeaio for he Seasoal Decomposiio procedure. Tred: remove a red b fiig ad subracig he idicaed pe of red. The rasformed daa are he residuals from he red lie. The equaios for each pe of red discussed i he documeaio for he Forecasig procedure. Differecig: rasform he daa b akig oseasoal differeces of order d ad/or seasoal differeces of order D. Differecig is someimes used o sabilize a osaioar ime series ha does o have a cosa mea. A oseasoal differece of order d is creaed b subracig cosecuive observaios d imes. For example, a firs differece (d = 1) is give b (1) b SaPoi Techologies, Ic. Time Series Descripive Mehods - 5

6 while a secod differece (d = 2) is give b (2) A seasoal differece of order 1 (D = 1) is give b 3) s A seasoal differece of order 2 (D = 2) is give b (4) s s 2s Iflaio: adjuss he daa for iflaio usig he specified iflaio rae If applied a he begiig of he period, he adjusme is ( 0 1) 1 (5) where 0 is he idex of he firs observaio. If applied a he middle of he period, he adjusme is (1 0.5) ) ( 0 (6) If more ha oe rasformaio is requesed, he are applied i he followig order: 1. radig da adjusme 2. iflaio adjusme 3. mah adjusme 4. seasoal adjusme 5. red adjusme 6. differecig Horizoal Time Sequece Plo The Horizoal Time Sequece Plo displas he ime series daa i sequeial order: 2013 b SaPoi Techologies, Ic. Time Series Descripive Mehods - 6

7 Traffic STATGRAPHICS Rev. 9/16/2013 Time Series Plo for Traffic /68 1/71 1/74 1/77 1/80 1/83 The raffic daa coais several ver ieresig feaures: 1. A geeral upward red. 2. A regular earl ccle aroud ha red, wih peak raffic occurrig i he summer mohs. 3. Dramaic chages o he red lie occurrig lae i 1973, whe he Arab oil embargo made gasolie difficul o ge. A similar hough less dramaic chage occurred durig Pae Opios Pois: plo poi smbols a he locaio of each observaio. Lies: coec he observaios wih a lie. Verical Time Sequece Plo The Verical Time Sequece Plo displas he ime series b drawig verical lies from a baselie o each observaio: 2013 b SaPoi Techologies, Ic. Time Series Descripive Mehods - 7

8 Traffic STATGRAPHICS Rev. 9/16/2013 Time Series Plo for Traffic /68 1/71 1/74 1/77 1/80 1/83 Pae Opios Baselie: posiio from which verical lies are draw. Auocorrelaios A impora ool i modelig ime series daa is he auocorrelaio fucio. The auocorrelaio a lag k measures he sregh of he correlaio bewee observaios k ime periods apar. The sample lag k auocorrelaio is calculaed from r k k 1 1 k 2 (7) The Auocorrelaios pae displas he sample auocorrelaios ogeher wih large lag sadard errors ad probabili limis: 2013 b SaPoi Techologies, Ic. Time Series Descripive Mehods - 8

9 Esimaed Auocorrelaios for Traffic Lower 95.0% Upper 95.0% Lag Auocorrelaio Sd. Error Prob. Limi Prob. Limi The sadard error for r k is calculaed o he assumpio ha he auocorrelaios have died ou b lag k ad are equal o 0 a all lags greaer or equal o k. I is calculaed from: k se [ rk ] 1 2 r k (8) i1 This sadard error is he used o calculae 100(1-)% probabili limis aroud zero, usig a criical value of he sadard ormal disribuio: 0 z / 2 k se[ r ] (9) If = 0.05, a sample auocorrelaios ha fall ouside hese limis are saisicall sigifical differe from 0 a he 5% sigificace level. The SaAdvisor highlighs a such auocorrelaios i red. For he raffic daa, oe ha here are sigifica values for he firs 3 lags ad also i he vicii of s = 12 ad 2s = 24. The sigifica values a low lags occur because observaios close ogeher i ime are correlaed. The sigifica lags aroud 12 ad 24 are caused b he srog seasoal paer b SaPoi Techologies, Ic. Time Series Descripive Mehods - 9

10 Auocorrelaios STATGRAPHICS Rev. 9/16/2013 Pae Opios Number of lags: maximum lag k a which o calculae he auocorrelaio. Cofidece level: value of 100(1-)% used o calculae he probabili limis. Auocorrelaio Fucio The Auocorrelaio Fucio plo displas he sample auocorrelaios ad probabili limis: 1 Esimaed Auocorrelaios for Traffic lag Bars exedig beod he upper or lower limi correspod o saisicall sigifica auocorrelaios. Parial Auocorrelaios Aoher impora ool i modelig ime series daa is he parial auocorrelaio fucio. The parial auocorrelaios are used o help ideif he proper order of auoregressive model o use o describe a observed ime series. The sample lag k parial auocorrelaio ˆ kk is calculaed from he sample auocorrelaios as described i he Calculaios secio. The Parial Auocorrelaios pae displas he sample parial auocorrelaios ogeher wih large lag sadard errors ad probabili limis: 2013 b SaPoi Techologies, Ic. Time Series Descripive Mehods - 10

11 Esimaed Parial Auocorrelaios for Traffic Parial Lower 95.0% Upper 95.0% Lag Auocorrelaio Sd. Error Prob. Limi Prob. Limi The sadard error for ˆ kk is calculaed from: 1 se[ ˆ kk ] (10) This sadard error is he used o calculae 100(1-)% probabili limis aroud zero, usig a criical value of he sadard ormal disribuio: z se[ ˆ ] (11) 0 / 2 kk If = 0.05, a sample parial auocorrelaios ha fall ouside hese limis are saisicall sigifical differe from 0 a he 5% sigificace level. The SaAdvisor highlighs a such parial auocorrelaios i red. For he raffic daa, oe ha here are sigifica values scaered hroughou he firs 13 lags. This implies ha i would ake a raher complicaed auoregressive model o describe he observed daa, which is o surprisig give is osaioar (redig) aure b SaPoi Techologies, Ic. Time Series Descripive Mehods - 11

12 Parial Auocorrelaios STATGRAPHICS Rev. 9/16/2013 Pae Opios Number of lags: maximum lag k a which o calculae he parial auocorrelaio. Cofidece level: value of 100(1-)% used o calculae he probabili limis. Parial Auocorrelaio Fucio The Parial Auocorrelaio Fucio plos he sample parial auocorrelaios ad probabili limis: Esimaed Parial Auocorrelaios for Traffic lag Bars exedig beod he upper or lower limi correspod o saisicall sigifica parial auocorrelaios. Periodogram The auocorrelaios ad parial auocorrelaios describe he behavior of he daa i he ime domai, i.e., b esimaig saisics based o he amou of ime bewee observaios. I is also useful o examie he daa i he frequec domai, b cosiderig how much variabili exiss a differe frequecies. I has bee show ha a discree ime series ca be represeed as he sum of a se of sies ad cosies a a se of frequecies called he Fourier frequecies. A pical compoe has he form 2013 b SaPoi Techologies, Ic. Time Series Descripive Mehods - 12

13 a i 2f b si2f i i i STATGRAPHICS Rev. 9/16/2013 cos (12) where f i is he i-h Fourier frequec. The i-h Fourier frequec is i f i (13) for i = 0, 1,, /2 if is eve ad i = 0, 1,, (-1)/2 if is odd. The periodogram calculaes he power i he daa a each Fourier frequec b calculaig: I 2 2 f a b i i i (14) 2 which is scaled so ha he sum of he periodogram ordiaes across all of he Fourier frequecies excep for i = 0 ields he sum of squared deviaios of he ime series abou is mea, i.e., i b frequec. i1 2. I effec, he periodogram geeraes a aalsis of variace The Periodogram pae displas he followig able: Periodogram for Traffic Cumulaive Iegraed i Frequec Period Ordiae Sum Periodogram E E E The able icludes: Frequec: he i-h Fourier frequec f i = i/ b SaPoi Techologies, Ic. Time Series Descripive Mehods - 13

14 Period: he period associaed wih he Fourier frequec, give b 1/ f i. This is he umber of observaios i a complee ccle a ha frequec. Ordiae: he periodogram ordiae I(f i ). Cumulaive Sum: he sum of he periodogram ordiaes a all frequecies up o ad icludig he i-h. Iegraed Periodogram: he cumulaive sum divided b he sum of he periodogram ordiaes a all of he Fourier frequecies. This colum represes he proporio of he power i he ime series a or below he i-h frequec. For example, he 14-h Fourier frequec correspods o a oscillaio wih a period of 12 mohs. There is a ver large ordiae a ha frequec, sice he daa ed o rise ad fall o a earl basis. If oe were o fi a regressio model a ha frequec, i would ake he form Y 2 2 c a cos bsi e (15) where c is a cosa ad e is he error erm. Fiig his model usig he Muliple Regressio procedure ields: ˆ Y cos si 2 12 (16) A scaerplo of his model is show below: Muliple X-Y Plo Variables Regressio Traffic /68 1/71 1/74 1/77 1/80 1/83 Moh Noice how much of he variabili has bee explaied b ha sigle compoe b SaPoi Techologies, Ic. Time Series Descripive Mehods - 14

15 Ordiae STATGRAPHICS Rev. 9/16/2013 Pae Opios Remove mea: check o subrac he mea from he ime series before calculaig he periodogram. If he mea is o removed, he ordiae a i = 0 is likel o be ver large. Taper: perce of he daa a each ed of he ime series o which a daa aper will be applied before he periodogram is calculaed. Followig Bloomfield (2000), STATGRAPHICS uses a cosie aper ha dowweighs observaios close o i = 1 ad i =. This is useful for correcig bias if he periodogram ordiaes are o be smoohed i order o creae a esimae of he uderlig specral desi fucio. Periodogram Plo The Periodogram Plo displas he periodogram ordiaes: (X 1000) 5 Periodogram for Traffic frequec Noe he huge spike a a frequec of 1/12 mohs. Two small bumps ca be observed a he firs ad secod harmoics (2/12 ad 3/12), sice he seasoal oscillaio is o purel siusoidal. There is also some power a ver small frequecies, caused b he reds ad sudde chages i he raffic ime series b SaPoi Techologies, Ic. Time Series Descripive Mehods - 15

16 Ordiae STATGRAPHICS Rev. 9/16/2013 Pae Opios Remove mea: check o subrac he mea from he ime series before calculaig he periodogram. Pois: if checked, poi smbols will be displaed. Lies: if checked, he ordiaes will be coeced b a lie. Taper: perce of he daa a each ed of he ime series o which a daa aper will be applied before he periodogram is calculaed. Iegraed Periodogram The Iegraed Periodogram displas he cumulaive sums of he periodogram ordiaes, divided b he sum of he ordiaes over all of he Fourier frequecies: Periodogram for Traffic frequec A diagoal lie is icluded o he plo, ogeher wih 95% ad 99% Kolmogorov- Smirov bouds. If he ime series is purel radom, he iegraed periodogram should fall wihi hose bouds 95% ad 99% of he ime. For he raffic daa, i is safe o coclude ha he daa does o form a radom ime series b SaPoi Techologies, Ic. Time Series Descripive Mehods - 16

17 Tess For Radomess The Tess for Radomess pae displas he resuls of addiioal ess ru o deermie wheher or o he ime series is purel radom: Tess for Radomess of Traffic (1) Rus above ad below media Media = Number of rus above ad below media = 32 Expeced umber of rus = 85.0 Large sample es saisic z = P-value = E-16 (2) Rus up ad dow Number of rus up ad dow = 47 Expeced umber of rus = Large sample es saisic z = P-value = 0.0 (3) Box-Pierce Tes Tes based o firs 24 auocorrelaios Large sample es saisic = P-value = 0.0 Three ess are performed: 1. Rus above ad below media: cous he umber of imes he series goes above or below is media. This umber is compared o he expeced value for a radom ime series. A redig series, such as he raffic daa, is likel o show sigifical less rus ha expeced. Small P-values (less ha 0.05 if operaig a he 5% sigificace level) idicae ha he ime series is o purel radom. 2. Rus up ad dow: cous he umber of imes he series goes up or dow. This umber is compared o he expeced value for a radom ime series. A series wih a srog oscillaio, such as he raffic daa, is likel o show sigifical less rus ha expeced. Small P-values idicae ha he ime series is o purel radom. 3. Box-Pierce Tes: cosrucs a es saisic based o he firs k sample auocorrelaios b calculaig: Q k 2 r i i1 (17) This saisic is compared o a chi-squared disribuio wih k degrees of freedom. As wih he oher wo ess, small P-values idicae ha he ime series is o purel radom. Agai, here is o quesio ha he raffic series coais sigifica o-radom srucure b SaPoi Techologies, Ic. Time Series Descripive Mehods - 17

18 Pae Opios Number of Lags: umber of lags k o iclude i he Box-Pierce es. Crosscorrelaios The Crosscorrelaios pae displas crosscorrelaios bewee he mai ime series ad a secod series, specified usig Pae Opios. The crosscorrelaio bewee oe ime series Y a ime ad a secod ime series X a ime -k is deoed b c x (k). A pical use of crosscorrelaios is i ideifig leadigs idicaors or a ipu-oupu relaioship. For example, Box, Jekis ad Reisel (1994) prese daa from he ipu ad oupu of a gas furace a 9 secod iervals, coaied i he file furace.sf6. The daa cosis of: 1. Oupu series Y: % Co2 i oule gas 2. Ipu series X: ipu gas rae i cubic fee per miue The crosscorrelaio able is show below: Esimaed Crosscorrelaios for Oupu wih Ipu Lag Crosscorrelaio Some large egaive correlaios a oiceable, peakig a k = 5. This suggess ha icreases i he ipu gas rae cause decreases i he % of Co2 i he oule gas rae, peakig abou 45 secods laer b SaPoi Techologies, Ic. Time Series Descripive Mehods - 18

19 Crosscorrelaios STATGRAPHICS Rev. 9/16/2013 Pae Opios Secod Time Series: he observaios for he X ime series. Number of Lags: maximum lag k (boh posiive ad egaive) a which o calculae he crosscorrelaios Cofidece Level: value of 100(1-)% used o calculae he probabili limis. Appl adjusmes o secod ime series: wheher he adjusmes applied o he firs ime series should also be applied o he secod ime series. Crosscorrelaio Plo The Crosscorrelaio Plo displas he esimaed crosscorrelaios: Esimaed Crosscorrelaios for Oupu wih Ipu lag Noe he large egaive correlaio a posiive lags b SaPoi Techologies, Ic. Time Series Descripive Mehods - 19

20 Save Resuls The followig resuls ca be saved o he daashee: 1. Daa he origial observaios, ogeher wih a ierpolaed replacemes for missig values. 2. Adjused daa ime series daa afer a adjusmes have bee made. 3. Period labels ideificaio of ime for each observaio. 4. Auocorrelaios sample auocorrelaios. 5. Parial auocorrelaios sample parial auocorrelaios. 6. Crosscorrelaios sample crosscorrelaios. 7. Periodogram ordiaes calculaed periodogram ordiaes. 8. Fourier frequecies Fourier frequecies correspodig o he periodogram ordiaes b SaPoi Techologies, Ic. Time Series Descripive Mehods - 20

21 Calculaios Missig Daa A limied umber of missig values are permied, as log as here are o oo ma missig values close ogeher. Before he daa is aalzed, missig values are replaced b ierpolaed values, deermied usig he followig rule: 1. If, he observaio a ime, is missig, fid he wo observaios i he same seaso ha precede ime ( -s ad -2s ) ad he wo observaios i he same seaso ha come afer ime ( +s ad +2s ). 2. If oe of he four observaiod are missig, he he replaceme value for is: 32s 12s 12s 32s (18) If +2s is missig bu he oher hree are o, he he replaceme value for is: 2s 3s s (19) 3 4. If +s is missig bu he oher hree are o, he he replaceme value for is: 32s 8s s (20) 6 5. If -s is missig bu he oher hree are o, he he replaceme value for is: 2s 8s 32s (21) 6 6. If -2s is missig bu he oher hree are o, he he replaceme value for is: s 3s s (22) 3 7. If +s ad +2s are missig bu he oher wo are o, he he replaceme value for is: 2 2 s s (23) 2013 b SaPoi Techologies, Ic. Time Series Descripive Mehods - 21

22 8. If -s ad +2s are missig bu he oher wo are o, he he replaceme value for is: 2s 2s (24) 3 9. If -s ad +s are missig bu he oher wo are o, he he replaceme value for is: 2s 2s (25) If -2s ad +2s are missig bu he oher wo are o, he he replaceme value for is: s s (26) If -2s ad +s are missig bu he oher wo are o, he he replaceme value for is: 2s 2s (27) If -2s ad -s are missig bu he oher wo are o, he he replaceme value for is: 2s 2s (28) If more ha 2 of he four observaios are missig, a error message will be displaed ad he aalsis will o be performed. The ierpolaed values are desiged o perfecl reproduce a quadraic red (if ol oe observaio is missig) or a liear red (is wo observaios are missig), provided o oise is prese. Parial Auocorrelaios r k 1 1 k1 ˆ ˆ rk k1, jrk j kk j1 for k 1 k1 1 ˆ k1, jrj j1 (29) 2013 b SaPoi Techologies, Ic. Time Series Descripive Mehods - 22

23 where ˆ ˆ ˆ ˆ for j = 1, 2,, k-1 (30) kj k 1, j kkk 1, k j Rus Tess Refer o he documeaio for he Rus Chars procedure. Crosscorrelaios r x cx ( k) ( k) (31) s s x where c c x x 1 ( k) x x k for k = 0, 1, 2, (32) k 1 1 ( k) x k x for k = 0, -1, -2, (33) k 1 x x 1 (34) 1 (35) sx c xx (0) (36) s c (0) (37) 2013 b SaPoi Techologies, Ic. Time Series Descripive Mehods - 23