Lesson 2, page 1. Outline of lesson 2

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1 Lesson 2, page Ouline of lesson 2 Inroduce he Auocorrelaion Coefficien Undersand and define saionariy Discuss ransformaion Discuss rend and rend removal C:\Kyrre\sudier\drgrad\Kurs\series\lecure doc, KL, , page of

2 Lesson 2, page 2 Inroducing he Auocorrelaion Coefficien Year -2 Year - Year Year We have a ime series { } =, n? We wan o invesigae he relaionship beween and is immediae hisory. We have = a 0 + a - + noise? = a where a 0 is some inercep and a is he coefficien of he relaionship beween - and (We have removed he rend, if presen) (We shall reurn o "noise" laer) How do we measure his coefficien (a )? The obvious choice is as a correlaion. The correlaion coefficien beween and y: r = ( i )( y iy) 2 2 [ ( i ) ( y iy) ] We can ake all observaions one ime sep apar (N-) and find he correlaion: C:\Kyrre\sudier\drgrad\Kurs\series\lecure doc, KL, , page 2 of 2

3 Lesson 2, page 3 N N r N N / ) ( ) ( ) )( ( 2 = = = Here, we have assumed ha he mean of he observaions wihou - and wihou is equal (see Chafield 999, p. 9). By dropping he facor N/(N-) for small samples we ge: = = = N N r 2 ) ( ) )( ( for one ime sep ("lag") apar. By generalising we have: = = = N k N k k r 2 ) ( ) )( ( for ime seps ("lags") k apar. This is called he (empirical) Auocorrelaion Coefficien a lag k. (Normally he covariances are calculaed: = + = k N k k N ) )( ( c, and hen we compue r k = c k /c o ). C:\Kyrre\sudier\drgrad\Kurs\series\lecure doc, KL, , page 3 of 3

4 Lesson 2, page 4 (In splus: acf: Auocovariance funcion) mp <- read.able("c:\\kyrre\\sudier\\drgrad\\kurs\\series\\kilpis.") kilpis <- s(mp, sar=953, frequency=2) s.plo(kilpis, main="clehrionomys rufocanus a Kilpisjarvi", ylab="trap inde", lab="year") s.poins(kilpis) acf(kilpis, ype="correlaion", plo=t) Clehrionomys rufocanus a Kilpisjarvi Series : kilpis Trap inde Year Significance boundaries Quesion: - How can an plo be very wrong? C:\Kyrre\sudier\drgrad\Kurs\series\lecure doc, KL, , page 4 of 4

5 Lesson 2, page 5 # Using series from las ime s.plo(s(simulda[,3], sar=95)) ile("firs series") acf(simulda[,3]) s.plo(s(simulda[,4], sar=95), ly=) ile("second series") acf(simulda[,4]) Firs series Series : simulda[, 3] Significance boundaries Second series Series : simulda[, 4] > Are hese auocorrelaion coefficiens rue? (and why no?) C:\Kyrre\sudier\drgrad\Kurs\series\lecure doc, KL, , page 5 of 5

6 Lesson 2, page 6 We remove he rend... s.plo(s(simul.derend, sar=95)) ile("firs series") acf(simul.derend) s.plo(s(simul.derend, sar=95), ly=) ile("second series") acf(simul.derend) Firs series Series : simul.derend Second series Series : simul.derend > In order for our conclusions abou shor-ime dependencies o be rue, we need o remove such long-erm variabiliy. (In pracise: he should damp down "fas") -> We need saionary. C:\Kyrre\sudier\drgrad\Kurs\series\lecure doc, KL, , page 6 of 6

7 Lesson 2, page 7 Saionariy The ime series {X } is sricly saionary ("serk sasjonær") if (X, X n ) T d = (X +h, X n+h ) T for all h, n d ( = means: Two random vecors have he same join disribuions). The ime series {X } is weakly, or second-order, saionary ("svak sasjonær") if i) E [ X ()] = µ, i.e., µ ( ) = µ ( ) is independen of ime, ii) Cov[ X ( ), X ( + h) ] = γ X ( + h, ) = γ ( h) for all h, i.e. ha he covariance in ime (auocovariance) is only dependen on he lag (disance beween values) and no he locaion. Commens: - Normal disribuion (of he ime series {X }): Compleely deermined by he firs (mean) and second (variance) order momens. - Wha happens if we have weak saionariy and normaliy (gaussian)? - Is a sric saionary series also weak? And vice versa? How o es for (weak) saionariy? Mos imporan: Plo he series. No good ess eiss (in my eperience) Brockwell & Davies (Brockwell, P. J., and R. A. Davies. 99. series: heory and mehods. Springer-Verlag Inc., New York. Brockwell, P. J., and R. A. Davis Inroducion o ime series and forecasing. Springer-Verlag Inc., New York. ) describe 6(5) ess. NB: Wha we are alking abou here is he lack of saionariy due o longerm variabiliy. The process is no saionary if here is a shor-erm dependency, which is wha we are o model C:\Kyrre\sudier\drgrad\Kurs\series\lecure doc, KL, , page 7 of 7

8 Lesson 2, page 8 Transformaion -> Same principles as in ordinary regression (linear models) - To sabilize variance - To make seasonal effec(s) addiive - To make daa normal Chafield (999), p.. In Sen, A., and M. Srivasava. (990). (Regression analysis: Theory, mehods, and applicaions. Springer, New York. ) you can find more abou ransformaions. Typical ransformaions: - Naural logarihm - square roo mp <- read.able("e:\\programfiler\\itsm96\\chocs_b.da") s.plo(s(mp, frequency=2, sar=c(957,7)), main="chocolae producion (Choc_b.da)", ylab="(onnes)") > lengh((mp)) [] 404 > var(mp[202:404,])/var(mp[:202,]) [] # -> The variance in he laer par of he series 2.3 higher han in he firs par of he series mp <- log(mp) s.plo(s(mp, frequency=2, sar=c(957,7)), main="ln(chocolae producion)", ylab="(onnes)") > var(mp[202:404,])/var(mp[:202,]) [] # -> The variance in he laer and firs par of he series are abou he same Chocolae producion (Choc_b.da) Ln(Chocolae producion) (onnes) (onnes) May 58 Sep 66 Jan 75 May 83 Sep 9 May 58 Sep 66 Jan 75 May 83 Sep 9 C:\Kyrre\sudier\drgrad\Kurs\series\lecure doc, KL, , page 8 of 8

9 Lesson 2, page 9 Trend and removal of rend Wha is a rend? -> Very imprecise erm. In ime series analysis: Ofen any long-erm paern in ime. Some definiions: - A monoonic series of values (i.e., +. Also means ha he firs derivaive does no change sign.). E.g. linear rend - Long-erm paern in ime In ime series analysis normally he rend is removed before analysis. There are numerous echniques o esimae he rend. In Chafield (999) several are menioned: C:\Kyrre\sudier\drgrad\Kurs\series\lecure doc, KL, , page 9 of 9

10 Lesson 2, page 0. Curve fiing (eample: Boom coverage) <- 950:998 y <- meanyear[9:67, 2] pfi <- lm(y ~ poly(, 3)) # y = a 0 + a + a a 3 3 plo(, y, ylim=range(y), main="", lab= Years, ylab="mean Boom Coverage, ~ 20 saions") lines(, pfi$fied) acf(y) cover.polyde <- y - pfi$fied plo(cover.polyde) acf(cover.polyde) me(ouer=t, "Mean Boom Coverage, ~ 20 saions", side=3, ce=.5, line=-) Mean Boom Coverage, ~ 20 saions Mean Boom Coverage, ~ 20 saions Series : y Years Series : cover.polyde cover.polyde Inde Filering (see book. Can use he command filer) C:\Kyrre\sudier\drgrad\Kurs\series\lecure doc, KL, , page 0 of 0

11 3. Differencing -> Very common echnique in ime series analysis. Widely used. {y } = - lengh(y) # [] 49 mp <- mari(0, 48,) dim(mp) # [] 48 for (i in :48) {mp[i] <- y[i+] - y[i]} cover.diffde <- s(mp, sar=95) s.plo(cover.diffde) acf(cover.diffde) Lesson 2, page Series : cover.diffde My quesion is: -> Wha have we done? -> Are he ime series he same? -> Have we induced srucure in he daa? -> Do we know wha we are doing? C:\Kyrre\sudier\drgrad\Kurs\series\lecure doc, KL, , page of

12 Lesson 2, page 2 lengh(simulda$socvar) # [] 0 mp <- mari(0, 49,) dim(mp) # [] 49 for (i in :49) {mp[i] <- simulda[i+,2] - simulda[i,2]} diffser <- s(mp, sar=952) s.plo(diffser) acf(diffser) Series : diffser > We had a "rue" process which was whie noise (i.e., no ime srucure), by differencing we have induced an negaive auocorrelaion of order in he series. Wha does a polynomial rend of hird order for boom coverage really mean? Is here any biological eplanaion for his? I am very scepical o derending and differencing A linear rend is ok. Mean a general increase/decrease of level. Easily undersood, and easy o model. C:\Kyrre\sudier\drgrad\Kurs\series\lecure doc, KL, , page 2 of 2

13 Lesson 2, page 3 # Smoohing by moving average # NB: No he same as a MA-process # e.g.: y = /3 ( ) Can be weighed in many ways lengh(simulda$socvar) # [] 0 mp <- mari(0, 49,) dim(mp) # [] 49 s.plo(s(simulda[,2], sar=95)) s.poins(s(simulda[,2], sar=95)) mp[] <- (simulda[,2]+ simulda[2,2])/2 for (i in 2:48) { mp[i] <- (simulda[i-,2]+simulda[i,2]+simulda[i+,2])/3 } mp[49] <- (simulda[48,2]+ simulda[49,2])/2 s.lines(s(mp, sar=95), lwd=4) acf(simulda[,2]) mp.res <- s((simulda[,2]-mp[]), sar=95) s.plo(mp.res) s.poins(mp.res) acf(mp.res) Series : simulda[, 2] Series : mp.res C:\Kyrre\sudier\drgrad\Kurs\series\lecure doc, KL, , page 3 of 3

14 Lesson 2, page 4 # A real biological eample par(mfrow=c(,)) s.plo(s(meanyear[9:67,2], sar=950), lab="year", ylab="mean Boom Coverage, ~ 20 saions") ile("boom Coverage along he Skagerrak Coas, (caegoric 0-5)") s.poins(s(meanyear[9:67,2], sar=950), pch=6) Boom Coverage along he Skagerrak Coas, (caegoric 0-5) Mean Boom Coverage, ~ 20 saions Wha can we do wih his series? -> We know: disease in seagrass (ålegrass Zoesra marina) in he 930s. -> We know: Temperaure decrease in he 960s Year C:\Kyrre\sudier\drgrad\Kurs\series\lecure doc, KL, , page 4 of 4

15 Lesson 2, page 5 # a) # Find a linear rend y <- :lengh(meanyear[9:35,2]) simul.rend.lin <- s(lm(meanyear[9:35,2] ~ y)$fied.values, sar=950) s.lines(simul.rend.lin, ly=6, lwd=4, col=8) # b) # Find a linear rend y <- :lengh(meanyear[36:67,2]) simul.rend.lin2 <- s(lm(meanyear[36:67,2] ~ y)$fied.values, sar=967) s.lines(simul.rend.lin2, ly=6, lwd=4, col=8) Boom Coverage along he Skagerrak Coas, (caegoric 0-5) Mean Boom Coverage, ~ 20 saions Year acf(meanyear[9:35,2]) acf(meanyear[9:35,2]-simul.rend.lin) acf(meanyear[36:67,2]) acf(meanyear[36:67,2]-simul.rend.lin2) C:\Kyrre\sudier\drgrad\Kurs\series\lecure doc, KL, , page 5 of 5

16 Lesson 2, page 6 Series : meanyear[9:35, 2] Series : meanyear[9:35, 2] - simul.rend.l Series : meanyear[36:67, 2] Series : meanyear[36:67, 2] - simul.rend.l Wha if here is an obvious long-erm paern in ime? -> Try o undersand he paern and use a model based reason o remove i (e.g., regress he values on some covaiae, e.g. emperaure, and remove his rend). C:\Kyrre\sudier\drgrad\Kurs\series\lecure doc, KL, , page 6 of 6