3 Autocorrelation. 3.1 Time series plot. 3.2 Lagged scatterplot

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1 3 Auocoelaon Auocoelaon efes o he coelaon of a me sees wh s own pas and fuue values. Auocoelaon s also somemes called lagged coelaon o seal coelaon, whch efes o he coelaon beween membes of a sees of numbes aanged n me. Posve auocoelaon mgh be consdeed a specfc fom of pessence, a endency fo a sysem o eman n he same sae fom one obsevaon o he nex. Fo example, he lelhood of omoow beng any s geae f oday s any han f oday s dy. Geophyscal me sees ae fequenly auocoelaed because of nea o cayove pocesses n he physcal sysem. Fo example, he slowly evolvng and movng low pessue sysems n he amosphee mgh mpa pessence o daly anfall. O he slow danage of goundwae eseves mgh mpa coelaon o successve annual flows of a ve. O soed phoosynhaes mgh mpa coelaon o successve annual values of ee-ng ndces. Auocoelaon complcaes he applcaon of sascal ess by educng he numbe of ndependen obsevaons. Auocoelaon can also complcae he denfcaon of sgnfcan covaance o coelaon beween me sees (e.g., pecpaon wh a ee-ng sees). Auocoelaon can be exploed fo pedcons: an auocoelaed me sees s pedcable, pobablscally, because fuue values depend on cuen and pas values. Thee ools fo assessng he auocoelaon of a me sees ae () he me sees plo, (2) he lagged scaeplo, and (3) he auocoelaon funcon. 3. Tme sees plo Posvely auocoelaed sees ae somemes called pessen because posve depaues fom he mean end o be followed by posve depaues fom he mean, and negave depaues fom he mean end o be followed by negave depaues (Fgue 3.). In conas, negave auocoelaon s chaacezed by a endency fo posve depaues o follow negave depaues, and vce vesa. Posve auocoelaon mgh show up n a me sees plo as unusually long uns, o seches, of seveal consecuve obsevaons above o below he mean. egave auocoelaon mgh show up as an unusually low ncdence of such uns. Because he depaues fo compung auocoelaon ae elave he mean, a hozonal lne ploed a he sample mean s useful n evaluang auocoelaon wh he me sees plo. Vsual assessmen of auocoelaon fom he me sees plo s subjecve and depends consdeably on expeence. Sascal ess based on he obseved numbe of uns above and below he mean ae avalable (e.g., Dape and Smh 98), hough none ae coveed n hs couse. I s a good dea, howeve, o loo a he me sees plo as a fs sep n analyss of pessence. If nohng else, hs nspecon mgh show ha he pessence s much moe pevalen n some pas of he sees han n ohes. 3.2 Lagged scaeplo The smples gaphcal summay of auocoelaon n a me sees s he lagged scaeplo, whch s a scaeplo of he me sees agans self offse n me by one o seveal me seps (Fgue 3.2). Le he me sees of lengh be x,,...,. The lagged scaeplo fo lag s a scaeplo of he las obsevaons agans he fs obsevaons. Fo example, fo lag-, obsevaons x, x, x ae ploed agans obsevaons x 2 3,, x 2,, x. A andom scaeng of pons n he lagged scaeplo ndcaes a lac of auocoelaon. Such a sees s also somemes called andom, meanng ha he value a me s ndependen oes_3, GEOS 585A, Spng 205

2 of he value a ohe mes. Algnmen fom lowe lef o uppe gh n he lagged scaeplo ndcaes posve auocoelaon. Algnmen fom uppe lef o lowe gh ndcaes negave auocoelaon. Fgue 3.. Tme sees plo llusang sgnaues of pessence. Tendency fo hghs o follow hghs o lows o follow lows (ccled segmens) chaaceze sees wh pessence, o posve auocoelaon. Fgue 3.2. Lagged scaeplos of ee-ng sees MEAF. These ae scaeplos of he sees n Fgue 3. wh self offse by, 2, 3 and 4 yeas. Annoaed above a plo s he coelaon coeffcen, he sample sze, and he heshold level of coelaon needed o ejec he null hypohess of zeo populaon coelaon wh 95 pecen sgnfcance (α=0.05). The heshold s exceeded a lags, 2, and 4, bu no a lag 3. A an offse of 3 yeas, he juxaposon of hgh-gowh 999 wh low-gowh 2002 exes hgh nfluence (pon n ed ecangle). oes_3, GEOS 585A, Spng 205 2

3 An abue of he lagged scaeplo s ha can dsplay auocoelaon egadless of he fom of he dependence on pas values. An assumpon of lnea dependence s no necessay. An oganzed cuvaue n he paen of dos mgh sugges nonlnea dependence beween mesepaaed values. Such nonlnea dependence mgh no be effecvely summazed by ohe mehods (e.g., he auocoelaon funcon [acf], whch s descbed lae). Anohe abue s ha he lagged scaeplo can show f he auocoelaon s chaacesc of he bul of he daa o s dven by one o moe oules. The scae plo n Fgue 3.2 fo lag-3 (lowe lef plo), fo example, has a dsnc lowe-lef o uppe-gh slan suppong posve lag-3 auocoelaon, bu an oule (hghlghed) pobably eeps he lag-3 auocoelaon fom eachng sascal sgnfcance. Influence of oules would no be deecable fom he acf alone. Fed lne. A sagh lne can be f o he pons n a lagged scaeplo o faclae evaluaon lneay and sengh of elaonshp of cuen wh pas values. A sees of lagged scaeplos a nceasng lags (e.g.,, 2, 8 ) helps n assessng whehe dependence s esced o one o moe lags. Coelaon coeffcen and 95% sgnfcance level. The coelaon coeffcen fo he scaeplo summazes he sengh of he lnea elaonshp beween pesen and pas values. I s helpful o compae he compued coelaon coeffcen wh ccal level of coelaon equed o ejec he null hypohess ha he sample comes fom a populaon wh zeo coelaon a he ndcaed lag. If a me sees s compleely andom, and he sample sze s lage, he lagged-coelaon coeffcen s appoxmaely nomally dsbued wh mean 0 and vaance / (Chafeld 2004). I follows ha he appoxmae heshold, o ccal, level of coelaon fo 95% sgnfcance ( 0.0 5) s /, whee s he sample sze. Accodngly, he equed level of coelaon fo sgnfcance becomes vey small a lage sample sze (Fgue 3.3). Fgue 3.3. Ccal level of coelaon coeffcen (95 pecen sgnfcance) as a funcon of sample sze. The ccal level dops fom =0.20 fo a sample sze of 00 o =0.02 fo a sample sze of 0, Auocoelaon funcon (coelogam) An mpoan gude o he pessence n a me sees s gven by he sees of quanes called he sample auocoelaon coeffcens, whch measue he coelaon beween obsevaons a dffeen mes. The se of auocoelaon coeffcens aanged as a funcon of sepaaon n me s he sample auocoelaon funcon, o he acf. An analogy can be dawn beween he auocoelaon coeffcen and he poducmomen coelaon coeffcen. Assume pas of obsevaons on wo vaables x and y. The coelaon coeffcen beween x and y s gven by x x y y / 2 / x x y y () oes_3, GEOS 585A, Spng 205 3

4 whee he summaons ae ove he obsevaons. A smla dea can be appled o me sees fo whch successve obsevaons ae coelaed. Insead of wo dffeen me sees, he coelaon s compued beween one me sees and he same sees lagged by one o moe me uns. Fo he fs-ode auocoelaon, he lag s one me un. The fs-ode auocoelaon coeffcen s he smple coelaon coeffcen of he fs obsevaons, x,, 2,..., and he nex obsevaons, x, 2, 3,...,. The coelaon beween x and x s gven by x x ( ) x x ( 2 ) / 2 / x x ( ) x x ( 2 ) 2 whee x s he mean of he fs obsevaons and x s he mean of he las ( ) ( 2 ) obsevaons. As he coelaon coeffcen gven by (2) measues coelaon beween successve obsevaons, s called he auocoelaon coeffcen o seal coelaon coeffcen. Fo easonably lage, he denomnao n equaon (2) can be smplfed by appoxmaon. Fs, he dffeence beween he sub-peod means x and x can be gnoed. Second, he ( ) ( 2 ) dffeence beween summaons ove obsevaons o - and 2 o can be gnoed. Accodngly, can be appoxmaed by (2) x x x x x x 2 (3) whee x x s he oveall mean. Equaon (3) can be genealzed o gve he coelaon beween obsevaons sepaaed by me seps: x x x x x x 2 (4) The quany s called he auocoelaon coeffcen a lag. The plo of he auocoelaon funcon as a funcon of lag s also called he coelogam. Ln beween acf and lagged scaeplo. The coelaon coeffcens fo he lagged scaeplos a lags, 2,...8 ae equvalen o he acf values a lags,,8. oes_3, GEOS 585A, Spng 205 4

5 Ln beween acf and auocovaance funcon (acvf). Recall ha he vaance s he aveage squaed depaue fom he mean. By analogy he auocovaance of a me sees s defned as he aveage poduc of depaues a mes and + (5) c x x x x whee c s he auocovaance coeffcen a lag. The auocovaance a lag zeo, c, s he 0 vaance. By combnng equaons (4) and (5), he auocoelaon a lag can be wen n ems of he auocovaance: c c (6) Alenave equaon fo auocovaance funcon. Equaon (5) s a based (hough asympocally unbased) esmao of he populaon covaance. The acvf s somemes compued wh he alenave equaon c x x x x (7) The acvf by (7) has a lowe bas han he acvf by (5), bu s conjecued o have a hghe mean squae eo (Jenns and Was 968, chape 5). 3.4 Tesng fo andomness wh he coelogam The fs queson ha can be answeed wh he coelogam s whehe he sees s andom o no. Fo a andom sees, lagged values of he sees ae uncoelaed and we expec ha 0. I can be shown ha f x..., x ae ndependen and dencally dsbued andom vaables wh abay mean, he expeced value of s E ( ) (8) he vaance of s V a( ) / (9) and s asympocally nomally dsbued unde he assumpon of wea saonay. The 95% confdence lms fo he coelogam can heefoe be ploed a / 2, and ae ofen fuhe appoxmaed o Thus, fo example, f a sees has lengh 00, he appoxmae 95% confdence band s Any gven has a 5% chance of beng ousde he 95% confdence lms, so ha one value ousde he lms mgh be expeced n a coelogam ploed ou o lag 20 even f he me sees s dawn fom a andom (no auocoelaed) populaon. Facos ha mus be consdeed n judgng whehe a sample auocoelaon ousde he confdence lms ndcaes an auocoelaed pocess o populaon ae () how many lags ae beng examned, (2) he magnude of, and (3) a wha lag he lage coeffcen occus. A vey lage s less lely o occu by chance han a smalle baely ousde he confdence bands. And a lage a a low lag (e.g., ) s moe lely o epesen pessence n mos physcal sysems han an solaed lage a some hghe lag. 3.5 Lage-lag sandad eo Whle he confdence bands descbed above ae hozonal lnes above and below zeo on he coelogam, he confdence bands you see n he assgnmen scp may appea o be naowes a oes_3, GEOS 585A, Spng 205 5

6 lag and o wden slghly a hghe lags. Tha s because he confdence bands poduced by he scp ae he so-called lage-lag sandad eos of (Andeson 976, p. 8). Successve values of can be hghly coelaed, so ha an ndvdual mgh be lage smply because he value a he nex lowe lag,, s lage. Ths nedependence maes dffcul o assess jus a how many lags he coelogam s sgnfcan. The lage-lag sandad eo adjuss fo he nedependence. The vaance of, wh he adjusmen, s gven by V a ( ) 2 K 2 (0) whee K. The squae oo of he vaance quany gven by (0) s called he lage-lag sandad eo of (Andeson 976, p. 8). Compason of (0) wh (9) shows ha he adjusmen s due o he summaon em, and ha he vaance of he auocoelaon coeffcen a any gven lag depends on he sample sze as well as on he esmaed auocoelaon coeffcens a shoe lags. Fo example, he vaance of he lag-3 auocoelaon coeffcen, V a, s geae han / by an amoun ha depends on he auocoelaon coeffcens a 3 lags and 2. Lewse, he vaance of he lag-0 auocoelaon coeffcen, V a 3, depends on he auocoelaon coeffcens a lags -9. Assessmen of he sgnfcance of lag- auocoelaon by he lage-lag sandad eo essenally assumes ha he heoecal auocoelaon has ded ou by lag, bu does no assume ha he lowe-lag heoecal auocoelaons ae zeo (Box and Jenns 976, p. 35). Thus he null hypohess s OT ha he sees s andom, as lowe-lag auocoelaons n he geneang pocess may be non-zeo. An example fo a ee-ng ndex me sees llusaes he slgh dffeence beween he confdence neval compued fom he lage-lag sandad eo and compued by he ough appoxmaon 2, whee s he sample sze (Fgue 3.4). The alenave confdence nevals dffe because he null hypoheses dffe. Thus, he auocoelaon a lag 5, say, s judged sgnfcan unde he null hypohess ha he sees s andom, bu s no judged sgnfcan f he heoecal auocoelaon funcon s consdeed o no have ded ou unl lag 5. Fgue 3.4. Sample auocoelaon wh 95% confdence nevals fo MEAF ee-ng ndex, Doed lne s smple appoxmae confdence neval a ± 2/, whee s he sample sze. Dashed lne s lage-lag sandad eo. oes_3, GEOS 585A, Spng 205 6

7 3.6 Hypohess es on The fs-ode auocoelaon coeffcen s especally mpoan because fo physcal sysems dependence on pas values s lely o be songes fo he mos ecen pas. The fs-ode auocoelaon coeffcen,, can be esed agans he null hypohess ha he coespondng populaon value 0. The ccal value of fo a gven sgnfcance level (e.g., 95%) depends on whehe he es s one-aled o wo-aled. Fo he one-aled hypohess, he alenave hypohess s usually ha he ue fs-ode auocoelaon s geae han zeo: H : 0 () Fo he wo-aled es, he alenave hypohess s ha he ue fs-ode auocoelaon s dffeen fom zeo, wh no specfcaon of whehe s posve o negave: H : 0 (2) Whch alenave hypohess o use depends on he poblem. If hee s some eason o expec posve auocoelaon (e.g., wh ee ngs, fom cayove food soage n ees), he one-sded es s bes. Ohewse, he wo-sded es s bes. Fo he one-sded es, he Wold Meeoologcal Oganzaon ecommends ha he 95% sgnfcance level fo be compued by, whee s he sample sze. Moe geneally, followng Salas e al. (980), who efe o Andesen (94), he pobably lms on he coelogam of an ndependen sees ae (3) (9 5 % ) o n e s d e d (4).9 6 (9 5 % ) w o s d e d whee s he sample sze and s he lag. Equaon (3) comes fom subsuon of = no equaon (4). 3.7 Effecve Sample Sze If a me sees of lengh s auocoelaed, he numbe of ndependen obsevaons s fewe han. Essenally, he sees s no andom n me, and he nfomaon n each obsevaon s no oally sepaae fom he nfomaon n ohe obsevaons. The educon n numbe of ndependen obsevaons has mplcaons fo hypohess esng. Some sandad sascal ess ha depend on he assumpon of andom samples can sll be appled o a me sees despe he auocoelaon n he sees. The way of ccumvenng he poblem of auocoelaon s o adjus he sample sze fo auocoelaon. The numbe of ndependen samples afe adjusmen s fewe han he numbe of obsevaons of he sees. Below s an equaon fo compung so-called effecve sample sze, o sample sze adjused fo auocoelaon. Moe on he adjusmen can be found elsewhee (WMO 966; Dawdy and Maalas 964). The equaon was deved based on he assumpon ha he auocoelaon n he sees epesens fs-ode auocoelaon (dependence on lag- only). In ohe wods, he govenng pocess s fs-ode auoegessve, o Maov. Compuaon of he effecve sample oes_3, GEOS 585A, Spng 205 7

8 sze eques only he sample sze and fs-ode sample auocoelaon coeffcen. The effecve sample sze s gven by: ' oes_3, GEOS 585A, Spng whee s he sample sze, s he effecve samples sze, and s he fs-ode auocoelaon coeffcen. The ao s a scalng faco mulpled by he ognal sample sze o compue he effecve sample sze. Fo example, an annual sees wh a sample sze of 00 yeas and a fs-ode auocoelaon of 0.50 has an adjused sample sze of ( 0.5 ) 0.5 ' ye a s The adjusmen o effecve sample sze becomes less mpoan he lowe he auocoelaon, bu a fs-ode auocoelaon coeffcen as small as =0.0 esuls n a scalng o abou 80 pecen of he ognal sample sze (Fgue 3.5). Fgue 3.5. Scalng faco fo compung effecve sample sze fom ognal sample sze fo auocoelaed me sees. Fo a gven fs-ode auocoelaon, he scalng faco s mulpled by he ognal me sees. Refeences (5) Andeson, R.L., 94, Dsbuon of he seal coelaon coeffcens: Annals of Mah. Sascs, v. 8, no., p. -3. Andeson, O., 976, Tme sees analyss and foecasng: he Box- Jenns appoach: London, Buewohs, p. 82 pp. Box, G.E.P., and Jenns, G.M., 976, Tme sees analyss: foecasng and conol: San Fancsco, Holden Day, p. 575 pp. Chafeld, C., 2004, The analyss of me sees, an noducon, sxh edon: ew Yo, Chapman & Hall/CRC. Dawdy, D.R., and Maalas,.C., 964, Sascal and pobably analyss of hydologc daa, pa III: Analyss of vaance, covaance and me sees, n Ven Te Chow, ed., Handboo of appled hydology, a compendum of wae-esouces echnology: ew Yo, McGaw-Hll Boo Company, p Jenns, G.M., and Was, D.G., 968, Specal analyss and s applcaons: Holden-Day, 525 p. Salas, J.D., Delleu, J.W., Yevjevch, V.M., and Lane, W.L., 980, Appled modelng of hydologc me sees: Lleon, Coloado, Wae Resouces Publcaons, 484 pp. Wold Meeoologcal Oganzaon, 966, Techncal oe o. 79: Clmac Change, WMO-o, 95.TP.00, Geneva, 80 pp.