STAD57 Time Series Analysis. Lecture 5

Transcription

1 STAD57 Time Series Analysis Lecure 5 1

2 Exploraory Daa Analysis Check if given TS is saionary: µ is consan σ 2 is consan γ(s,) is funcion of h= s If no, ry o make i saionary using some of he mehods below: De-rending Differencing Transformaions 2

3 De-rending Plo TS and/or is MA: if µ seems changing, ry o esimae µ & subrac i from series Y esimae rend ˆ Since E[ ], we have How do we esimae µ? work wih Yˆ ˆ E[ Yˆ ] E[ ˆ ] 0 For deerminisic rend (i.e. funcion of ), can use regression 3

4 Esimaing Trend Assume deerminisic rend µ =f() For linear rend, use simple linear regression Y 0 1 For nonlinear rend, can use polynomial regression Y or some oher nonlinear regression mehod Regression s explanaory variable is ime (), or some funcion hereof Response could be MA of (o reduce variabiliy) 4

5 Example Umbrella sales w/ fied linear rend De-rended TS ˆ ˆ

6 Example J&J sales w/ fied quadraic rend De-rended TS ˆ b b b

7 Regression in R Fi linear model Yi 0 11, i... p p, i i wih command: lm( Y ~ n ) Can save oupu o R objec: ou=lm(y~1+ ) Use summary(ou) o see model summary Coefficien & error sd esimaes, significance ess Feed oupu o oher funcions o ge: ANOVA able: aov(ou) Fied values: fied(ou) Akaike s crierion: AIC(ou), ec 7

8 Differencing Alernaive o fiing linear rend is differencing: Y ( ) ( Y Y ) If {Y } saionary {Y Y 1 } also saionary Differencing more appropriae when {Y } is random walk { } is random walk w/ drif Use when { } seems o hover around linear rend In his case, de-rending is no enough o make series saionary Disadvanage of differencing is ha we canno model dynamics of {Y }, only of {Y Y 1 } 8

9 Differencing Define backshif operaor B as B k exend i o powers so ha B 1 k Define 1 s order difference operaor as (1 B) Exend i o higher (d h ) order differences d d (1 B) by algebraically expanding operaor (1 B) d E.g. (1 B) (1 2 B B ) B B 1 and

10 Differencing in R For 1 s order differences, i.e. use diff() 1 For d h order differences, i.e. use diff(, differences=d) d d (1 B) For simple (1 s order) differences a lag k, i.e. k (1 B ) k, use diff(, lag=k) 10

11 gemp Example Global Temperaure TS w/ fied linear rend gempd gempdif de-rended series differenced series ˆ

12 ACF of ˆ Example Global Temperaure ACF ACF of Differencing considerably decreases auo-correlaions random walk w/ drif is more plausible model for series ACF ACF ACF of

13 Seasonaliy Differencing can also be used for seasonaliy If series has seasonal paern wih period d, hen d-lag differences remove paern Define Z (1 B d ) d If d, E[ Z ] E[ d ] 0 paern of period d ( ) d 13

14 ˆ Example Umbrella sales (quarerly daa, annual paern d=4) y ˆ differencing also removed linear rend

15 Time Series Regression Can also use regression model o esimae relaionship of one TS o ohers E.g. Y e, where {,Y } are joinly saionary TS Usual leas squares esimaion is no opimal when dealing wih non-i.i.d. TS daa However, resuls are sill consisen (reliable for large n) if series are saionary Bu never use leas squares for non-saionary TS (e.g. random walks); can ge very bad resuls 15

16 Example M Cardiovascular Moraliy Temperaure T Pariculaes P Scaer-plo marix of TS relaionships Moraliy Temperaure Pariculaes fi model: M 0 1 2T 3T 4P 16

17 Transformaions If TS has non-consan variance, a nonlinear ransformaion can someimes help For posiive series in paricular, he Box-Cox family of power ransforms can be useful: Y ( 1) /, 0 log( ), 0 Try differen values of λ, and check which one seems o give bes resuls Noe: nonlinear ransforms can also change µ 17

18 Example Series w/ rend De-rended series J&J sales Original daa: quadraic rend Transformed daa: Y log( ) linear rend

19 Saionariy Saionariy of γ(s,) is mos difficul o check In pracice, canno verify saionariy since here is no way o esimae γ(s,), only γ(h) Essenially, we assume saionariy in order o carry on wih any TS analysis There are, however, 2 hings we can check: Sample ACF does no change a lo when calculaed from separae sub-sequences of daa Auocorrelaions drop o zero relaively fas (i.e. exponenially) for large lags 19

20 Example Saionary ACF (AR model) Non-saionary ACF (Random walk)