5 Auoregressive-Moving-Average Modeling 5. Purpose. Auoregressive-moving-average (ARMA) models are mahemaical models of he persisence, or auocorrelaion, in a ime series. ARMA models are widely used in hydrology, dendrochronology, economerics, and oher fields. There are several possible reasons for fiing ARMA models o daa. Modeling can conribue o undersanding he physical sysem by revealing somehing abou he physical process ha builds persisence ino he series. For example, a simple physical waer-balance model wih precipiaion as inpu and including erms for evaporaion, infilraion, and groundwaer sorage can be shown o yield a sreamflow series as oupu ha follows a paricular form of ARMA model. ARMA models can also be used o predic behavior of a ime series from pas values alone. Such a predicion can be used as a baseline o evaluae he possible imporance of oher variables o he sysem. ARMA models are widely used for predicion of economic and indusrial ime series. Anoher use of ARMA models is simulaion, in which synheic series wih he same persisence srucure as an observed series can be generaed. Simulaions can be especially useful for esablished confidence inervals for saisics and esimaed ime series quaniies. ARMA models can also be used o remove persisence. In dendrochronology, for example, ARMA modeling is applied rouinely o generae residual chronologies ime series of ring-widh index wih no dependence on pas values. This operaion, called prewhiening, is mean o remove biologically-relaed persisence from he series so ha he residual may be more suiable for sudying he influence of climae and oher ouside environmenal facors on ree growh. 5. Mahemaical Model ARMA models can be described by a series of equaions. The equaions are somewha simpler if he ime series is firs reduced o zero-mean by subracing he sample mean. Therefore, we will wor wih he mean-adjused series y Y Y,, N () where Y is he original ime series, Y is is sample mean, and y is he mean-adjused series. One subse of ARMA models are he so-called auoregressive, or AR models. An AR model expresses a ime series as a linear funcion of is pas values. The order of he AR model ells how many lagged pas values are included. The simples AR model is he firs-order auoregressive, or AR(), model y a y e () where y is he mean-adjused series in year, y is he series in he previous year, a is he lag- auoregressive coefficien, and e is he noise. The noise also goes by various oher names: he error, he random-shoc, and he residual. The residuals e are assumed o be random in ime (no auocorrelaed), and normally disribued. Be rewriing he equaion for he AR() model as y a y e (3) Some auhors (e.g., Chafield, 004) wrie he equaion for an AR() process in he form y a y e, which implies a posiive coefficien a for posiive firs-order auocorrelaion. Bu as wrien in (), posiive auocorrelaion goes wih a negaive coefficien a. There is no confusion as long as he equaion being used for describing he process is presened along wih values of parameers. The convenion used in his chaper follows Ljung (995) and Malab s Sysem Idenificaion Toolbox. Noes_5, GEOS 585A, Spring 05
we see ha he AR() model has he form of a regression model in which y is regressed on is previous value. In his form, a is analogous o he negaive of he regression coefficien, and e o he regression residuals. The name auoregressive refers o he regression on self (auo). Higher-order auoregressive models include more lagged y erms as predicors. For example, he second-order auoregressive model, AR(), is given by y a y a y e (4) where a, a are he auoregressive coefficiens on lags and. The model, AR(p) includes lagged erms on years o p. h p order auoregressive The moving average (MA) model is a form of ARMA model in which he ime series is regarded as a moving average (unevenly weighed) of a random shoc series e. The firs-order moving average, or MA(), model is given by (5) y e c e where e, e are he residuals a imes and -, and c is he firs-order moving average coefficien. MA models of higher order han one include more lagged erms. For example, he second order moving average model, MA(), is y e c e c e (6) The leer q is used for he order of he moving average model. The second-order moving average model is MA(q) wih q. We have seen ha he auoregressive model includes lagged erms on he ime series iself, and ha he moving average model includes lagged erms on he noise or residuals. By including boh ypes of lagged erms, we arrive a wha are called auoregressive-moving-average, or ARMA, models. The order of he ARMA model is included in parenheses as ARMA(p,q), where p is he auoregressive order and q he moving-average order. The simples ARMA model is firs-order auoregressive and firs-order moving average, or ARMA(,): y a y e c e (7) 5.3 Seps in modeling ARMA modeling proceeds by a series of well-defined seps. The firs sep is o idenify he model. Idenificaion consiss of specifying he appropriae srucure (AR, MA or ARMA) and order of model. Idenificaion is someimes done by looing a plos of he acf and parial auocorrelaion funcion (pacf). Someimes idenificaion is done by an auomaed ieraive procedure -- fiing many differen possible model srucures and orders and using a goodness-offi saisic o selec he bes model. The second sep is o esimae he coefficiens of he model. Coefficiens of AR models can be esimaed by leas-squares regression. Esimaion of parameers of MA and ARMA models usually requires a more complicaed ieraion procedure (Chafield 004). In pracice, esimaion is fairly ransparen o he user, as i accomplished auomaically by a compuer program wih lile or no user ineracion. The hird sep is o chec he model. This sep is also called diagnosic checing, or verificaion (Anderson 976). Two imporan elemens of checing are o ensure ha he residuals of he model are random, and o ensure ha he esimaed parameers are saisically significan. Usually he fiing process is guided by he principal of parsimony, by which he bes model is he simples possible model he model wih he fewes parameers -- ha adequaely describes he daa. Noes_5, GEOS 585A, Spring 05
Idenificaion by visual inspecion of acf and pacf. The classical mehod of model idenificaion as described by Box and Jenins (970) is judge he appropriae model srucure and order from he appearance of he ploed acf and pacf. We have already discussed he acf, and now ha he acf a lag measures he correlaion of he series wih iself lagged years. The parial auocorrelaion funcion (pacf) a lag is he auocorrelaion a lag afer firs removing auocorrelaion wih a A R model. If he A R model effecively whiens he ime series, he pacf a lag will be zero. The idenificaion of ARMA models from he acf and pacf plos is difficul and requires much experience for all bu he simples models. Le s loo a he diagnosic paerns for he wo simples models: AR() and MA(). The acf of an AR() model declines geomerically as funcion of lag. For example, he acf of 3 4 series ha follows an AR() model wih coefficien a 0.5 is {0.5, 0.5, 0.5, 0.5 } a lags -4. The pacf of he AR() process a lags is zero, because if he model is AR(), all auocorrelaion is removed by he AR() model. In summary, he diagnosic paerns of acf and pacf for an AR() model are: Acf: declines in geomeric progression from is highes value a lag Pacf: cus off abruply afer lag The opposie ypes of paerns apply o an MA() process: Acf: cus off abruply afer lag Pacf: declines in geomeric progression from is highes value a lag Theoreical acf and pacf for wo AR() processes wih large and small auoregressive coefficiens are shown in Figure 5.. The acf and pacf in boh cases follows he diagnosic paerns described above. The persisence is specified by he size of he coefficien. For an AR() model, he square of he auoregressive coefficien is analogous o R in regression. Accordingly, for a model wih a =-0.9 persisence explains 8% of he variance, and for a model wih a =-0.3 persisence explains 9% of he variance. Noe ha he lower he coefficien for he AR() model he more quicly geomeric decay leads o an acf approaching zero. Figure 5.. Theoreical auocorrelaion funcion (acf) and parial auocorrelaion funcion (pacf) of an AR() processes wih high and low amouns of posiive auocorrelaion. Noes_5, GEOS 585A, Spring 05 3
The heroreical paerns of decay of he acf and pacf can be visually compared wih he esimaed acf and pacf of a ime series o decide wheher he series was liely generaed by an AR() process. The heoreical decay paerns can be difficul o discern for acual ime series, especially if he series is shor. Higher-order AR, MA, and ARMA processes also have characerisic heoreical paerns of decay of he acf and pacf. The decay paerns are more complicaed ha hose illusraed for he AR() model. Generally, however, he pacf cus off afer lag p for an AR(p) process, and he acf cus of abruply afer lag q for an MA(q) process. Shor memory and long memory processes. Though he acf may die off very slowly for an AR() process wih a high AR coefficien (Figure 5.), he AR() process, and indeed all saionary AR, MA and ARMA processes are ermed shor memory processes. Such processes saisfy a condiion of evenual dying off of he acf. The condiion is given by C, where is he heoreical auocorrelaion a lag, and C and are consans wih 0 (Chafield 004, p. 6). For such processes, 0 converges. On he oher hand, here is anoher class of processes, called long-memory processes, for which does no converge. See Chafield (004, p. 60) for discussion of long-memory processes. Auomaed idenificaion by he FPE crierion. A differen way of idenifying ARMA models is by rial and error and use of a goodness-of-fi saisic. In his approach, a suie of candidae models are fi, and goodness-of-fi saisics are compued ha penalize appropriaely for excessive complexiy. (Thin of adjused R in regression.) Aaie s Final Predicion Error (FPE) and Informaion Theoreic Crierion (AIC) are wo closely relaed alernaive saisical measures of goodness-of-fi of an ARMA(p,q) model. Goodness of fi migh be expeced o be measured by some funcion of he variance of he model residuals: he fi improves as he residuals become smaller. Boh he FPE and AIC are funcions of he variance of residuals. Anoher facor ha mus be considered, however, is he number of esimaed parameers n p q. This is so because by including enough parameers we can force a model o perfecly fi any daa se. Measures of goodness of fi mus herefore compensae for he arificial improvemen in fi ha comes from increasing complexiy of model srucure. The FPE is given by F P E n N n / N where V is he variance of model residuals, N is he lengh of he ime series, and n p q is he number of esimaed parameers in he ARMA model. The FPE is compued for various candidae models, and he model wih he lowes FPE is seleced as he bes-fi model. The AIC (Aaie Informaion Crierion) is anoher widely used goodness-of-fi measure, and is given by A IC lo g V * V 0 n N (8) (9) Noes_5, GEOS 585A, Spring 05 4
As wih he FPE, he bes-fi model has minimum value of AIC. Neiher he FPE nor he AIC direcly addresses he quesion of he model residuals being wihou auocorrelaion, as hey ideally should be if he model has removed he persisence. A sraegy for model idenificaion by he FPE is o ) ieraively fi several differen models and find he model ha gives approximaely minimum FPE and ) apply diagnosic checing (see below) o assure ha he model does a good job of producing random residuals. Checing he model are he residuals random? A ey quesion in ARMA modeling is does he model effecively describe he persisence? If so, he model residuals should be random or uncorrelaed in ime and he auocorrelaion funcion (acf) of residuals should be zero a all lags excep lag zero. Of course, for sample series, he acf will no be exacly zero, bu should flucuae close o zero. The acf of he residuals can be examined in wo ways. Firs, he acf can be scanned o see if any individual coefficiens fall ouside some specified confidence inerval around zero. Approximae confidence inervals can be compued. The correlogram of he rue residuals (which are unnown) is such ha r is normally disribued wih mean E ( r ) 0 (0) and variance v a r( r ) () where r is he auocorrelaion coefficien of he ARMA residuals a lag. The appropriae confidence inerval for r can be found by referring o a normal disribuion cdf. For example, he 0.975 probabiliy poin of he sandard normal disribuion is.96 3. The 95% confidence inerval for r is herefore.9 6 / N. For he 99% confidence inerval, he 0.995 probabiliy poin of he normal cdf is.57. The 99% CI is herefore.5 7 / N. An r ouside his CI is evidence ha he model residuals are no random. A suble poin ha should be menioned is ha he correlogram of he esimaed residuals of a fied ARMA model has somewha differen properies han he correlogram of he rue residuals which are unnown because he rue model is unnown. As a resul, he above approximaion () overesimaes he widh of he CI a low lags when applied o he acf of he residuals of a fied model (Chafield, 004, p. 68). There is consequenly some bias oward concluding ha he model has effecively removed persisence. A large lags, however, he approximaion is close. A differen approach o evaluaing he randomness of he ARMA residuals is o loo a he acf as a whole raher han a he individual r ' s separaely (Chafield, 004). The es is called he pormaneau lac-of-fi es, and he es saisic is N K () Q N r This saisic is referred o as he pormaneau saisic, or Q saisic. The Q saisic, compued from he lowes K auocorrelaions, say a lags,, 0, follows a disribuion wih degrees of freedom, where p and q are he AR and MA orders of he model and K p q N is he lengh of he ime series. If he compued Q exceeds he value from he able for The Sysem Idenificaion Toolbox in MATLAB has funcions for he FPE and he AIC. (Compuaional noe: MATLAB compues he variance V used in he above equaions wih N- raher han N in he denominaor of he sum-of-squares erm.) 3 The MATLAB disool funcion is a handy ineracive graphics ool for geing probabiliy poins of he cdf for various disribuions Noes_5, GEOS 585A, Spring 05 5
some specified significance level, he null hypohesis ha he series of auocorrelaions represens a random series is rejeced a ha level. The p-value gives he probabiliy of exceeding he compued Q by chance alone, given a random series of residuals. Thus non-random residuals give high Q and small p-value. The significance level is relaed o he p-value by sig n ifican ce level (% ) = 0 0 ( - p ) (3) A significance level greaer han 99%, for example, corresponds o a p-value smaller han 0.0. Checing he model are he esimaed coefficiens significanly differen from zero? Besides he randomness of he residuals, we are concerned wih he saisical significance of he model coefficiens. The esimaed coefficiens should be significanly differen han zero. If no, he model should probably be simplified, say, by reducing he model order. For example, an AR() model for which he second-order coefficien is no significanly differen from zero migh be discarded in favor of an AR() model. Significance of he ARMA coefficiens can be evaluaed by comparing esimaed parameers wih he sandard deviaions. For an AR() model, he esimaed firs-order auoregressive coefficien, â, is normally disribued wih variance v a r( aˆ ) aˆ (4) where N is he lengh of he ime series. The approximae 95% confidence inerval for â is herefore wo sandard deviaions around â : aˆ N v a r( aˆ ) (5) For example, if he ime series has lengh 300 years, and he esimaed AR() coefficien is a, he 95% confidence inerval is ˆ 0.6 0 9 5 % C I = aˆ v a r( aˆ ) 0.6 0 v a r( aˆ ) 0.6 0 aˆ N 0.6 0 0.6 0 0.6 0 0.6 4 3 0 0 3 0 0 (6) 0.6 0.0 0 3 0.6 0 0.0 9 The esimaed parameer aˆ 0.6 0 is herefore significan as he confidence band does no include zero and in fac is highly significan as he confidence band is far from zero. Equaions are also available for he confidence bands around esimaed parameers of an MA() model and higher-order AR, MA, and ARMA models (e.g., Anderson 975, p. 70). The esimaed parameers should be compared wih heir sandard deviaions o chec ha he parameers are significanly differen from zero. 4 4 The presen funcion in he MATLAB Sysem Idenificaion Toolbox is convenien for geing he sandard deviaions of esimaed ARMA parameers Noes_5, GEOS 585A, Spring 05 6
5.4 Pracical vs saisical significance of persisence Noe from equaion (4) ha he variance of he esimaed auoregressive coefficien for an AR() model is inversely proporional o he sample lengh. For long ime series (e.g., many hundreds of observaions), ARMA modeling may yield a model whose esimaed parameers are significanly differen from zero bu very small. The persisence described by such a model migh acually accoun for a iny percenage of he variance of he original ime series. A measure of he pracical significance of he auocorrelaion, or persisence, in a ime series is he percenage of he series variance ha is reduced by fiing he series o an ARMA model. If he variance of he model residuals is much smaller han he variance of he original series, he ARMA model accouns for a large fracion of he variance, and a large par of he variance of he series is due o persisence. In conras, if he variance of he residuals is almos as large as he original variance, hen lile variance has been removed by ARMA modeling, and he variance due o persisence is small. A simple measure of fracional variance due o persisence: v a r( e ) R p v a r( y ) where var( y ) is he variance of he original series, and v a r( e ) he ARMA model. 5 Wheher any given value of (7) R p is he variance of he residuals of is pracically significan is a maer of subjecive judgmen and depends on he problem. For example, in a ime series of ree-ring index, R 0.5 0 would liely be considered pracically significan, as half he variance of he p original ime series is explained by he modeled persisence. On he oher hand, R 0.0 migh well be dismissed as pracically insignifican. p Example. To illusrae he seps in ARMA modeling consider he fiing of a model o a ime series of ree-ring index (Figure 5.). The ree-ring index covers several hundred years, bu for illusraive purposes, only a porion of he record has been used in he modeling. The ime series plo srongly suggess posiive auocorrelaion, as successive observaions end o persis above or below he sample mean. The series has a low-frequency specrum, wih much of he variance a wavelenghs longer han 0 years. This shape of specrum is broadly consisen wih a posiively auocorrelaed series. The acf indicaes significan posiive auocorrelaion ou o a lag of 4 years, and he pacf cus off afer lag 4. These acf and pacf paerns alone are enough o sugges an AR(4) model. Fiing an AR(4) model o he daa resuls in he equaion y 0.3 7 5 4 y 0. 9 y 0. 9 9 y 0. 5 4 7 y e (8) 3 4 which is in he form of equaion (4) exended o auoregressive order p=4. Wheher he coefficiens are significanly differen from zero can be evaluaed by comparing he esimaed coefficiens wih heir sandard deviaions: a: -0.3754 (±0.0956) a: -0.9 (±0.036) a3: 0.99 (±0.035) a4: -0.547 (±0.0968) 5 The equaion for he fracional percenage of variance due o persisence uses he one-sep-ahead residuals Noes_5, GEOS 585A, Spring 05 7
Since wice he sandard deviaion is an approximae 95% confidence inerval for he esimaed coefficiens, all excep a3 are significanly differen from zero. The AR(4) canno herefore be ruled ou because of insignifican coefficiens, especially since he highes-order coefficien, a4, is significan. Figure 5.. Diagnosic plos for ARMA modeling of a 08-year segmen of a ree-ring index. Time series is for a Pinus srobiformis (Bear Canyon, Jemez Mns, New Mexico) sandard chronology. Top: ime plo, 900-007. Boom lef: auocorrelaion funcion and parial auocorrelaion funcion wih 95% confidence inerveal. Boom righ: specrum. Daa: Pinus srobiformis (Bear Canyon, Jemez Mns, New Mexico) sandard chronology. Of ey imporance is he whieness of he residuals are hey non-auocorrelaed? A acf plo for he ree-ring example reveals ha none of he auocorrelaions of he AR(4) model residuals are ouside he 99% confidence inerval around zero (Figure 5.3). This is a desired resul, as effecive ARMA modeling should explain he persisence and yield random residuals. Whieness of residuals can alernaively be checed wih he Pormaneau es (see equaion ()). The null hypohesis for he es is all he auocorrelaions of residuals for lags o K are zero, where choice of K is up o he user. As annoaed on Figure 5.3, for K=5 we canno rejec he null hypohesis. The p-value for he es is greaer han 0.05 (p=0.3749), meaning he es is no significan a he 0.05 α-level. In summary, review of he individual auocorrelaions of residuals, he Pormaneau es on hose residuals, and he error bars for he esimaed AR parameers favor acceping he AR(4) model as an effecive model for he persisence in he ree-ring series. Noes_5, GEOS 585A, Spring 05 8
Figure 5.3. Auocorrelaion funcion of residuals of AR(4) model fi o 08-year ree-ring index. Annoaed are he resuls of Pormaneau es and he percenage of variance due o modeled persisence. 5.5 Prewhiening Prewhiening refers o he removal of auocorrelaion from a ime series prior o using he ime series in some applicaion. In dendroclimaology sandard indices of individual cores are prewhiened as a sep in generaing residual chronologies (Coo 985). Prewhiening can also be applied a he level of he sie chronology o remove he persisence (Monserud 986). In he conex of ARMA modeling, he prewhiened series is equivalen o he ARMA residuals 6. As prewhiening aims a removal of persisence, here is an expeced effec on he specrum. Specifically, for a persisen (posiively auocorrelaed) series, he specrum of he prewhiened series should be flaened relaive o he specrum of he original series. A posiively auocorrelaed series has a low-frequency specrum, while whie noise he objecive of ARMA modeling has a fla specrum. The effec of prewhiening on he specrum is clear for our ree-ring example (Figure 5.4). The original series has a low-frequency specrum and he prewhiened series has a nearly fla specrum. Noe also ha he specrum of he prewhiened specrum is lower overall han ha of he original series. The area under he specrum is proporional o variance, and he removal of variance due o persisence will resul in a specrum wih a smaller area. For he ree-ring series, a subsanial fracion of he variance (more han /3) is due o persisence. The flaening of he specrum, a frequency-domain effec, is refleced by changes in he ime domain. For a series wih posiive auocorrelaion, prewhiening acs o damp hose ime series feaures ha are characerisic of persisence. Thus we expec ha broad swings above and below 6 In pracice, he original mean of he ree-ring index is usually resored, so he prewhiened chronology has he same mean as he sandard chronology, raher han a mean of zero Noes_5, GEOS 585A, Spring 05 9
he mean (broad peas and roughs) will be reduced in ampliude, or damped. The damping is eviden in a zoomed porion of he ime series for he ree-ring example (Figure 5.5). For example, he swing above he mean for he period 983-995 has been lessened in ampliude, and posiive deparures in 989 and 994 have been convered o negaive deparures. Specific differences in an original and prewhiened series can be readily explained by referring o he equaion of he fied ARMA model used o prewhien. For he ree-ring example, he fied AR(4) model is equaion (8). Differences in he original and prewhiened index (Figure 5.5) derive direcly from his equaion. Recall firs of all ha y in he equaion is a deparure from he mean (red line in Figure 5.5). The residual is compued by summing over he deparures of he original series from he mean for he previous 4 years, afer muliplying hose residuals by he esimaed ARMA coefficiens. The prewhiened index for 988 is herefore drawn closer o he mean because negaive coefficiens are applied o fairly large posiive anomalies 987, 986, and 984. Figure 5.4 Specra of original and prewhiened ree-ring index. Time series is a 08-yr ree-ring from New Mexico (see previous example). Dashed line is 95% confidence inerval. Figure 5.5. Zoomed ime series segmens of original and prewhiened ree-ring index. The full original series isploed in Figure 5. (op). 5.6 Simulaion and Predicion In addiion o helping o objecively describe persisence and providing an objecive way o remove i, ARMA modeling can also be used in simulaion and predicion. A brief inroducion o hese opics is given here. More exensive reamen can be found elsewhere (e.g., Chafield 004,Wils 995). Simulaion is he generaion of synheic ime series wih he same persisence properies as he observed series. Predicion is he exension of he observed series ino he fuure based on pas and presen values. The AR() model y a y e (9) Where y is he deparure of a ime series from is mean, a is he auoregressive coefficien, and e is he noise erm, can be rearranged as y a y e. (0) A simulaion of y can be generaed from equaion (0) by he following seps: ) esimae he auoregressive parameer by modeling he ime series as an AR() process, ) generae a ime Noes_5, GEOS 585A, Spring 05 0
series of random noise, e, by sampling from an appropriae disribuion, 3) assume some saring value for y, and 4) recursively generae a ime series of y. The appropriae disribuion for he noise is ypically a normal disribuion wih mean 0 and variance equal o he variance of he residuals from fiing he AR() model o he daa. For example, five simulaions of he ree-ring index are ploed along wih original ime series in Figure 5.6. The simulaions appear o effecively mimic he low-frequency behavior of he observed series. Any synchrony in iming of variaions in he various series is due o chance alone, as he simulaions have been randomly generaed. A possible applicaion of such series is o generae empirical (as opposed o heoreical) confidence inervals of he relaionship beween he observed ime series and a climae variable. For such an applicaion, many (e.g., 0,000) simulaed ree-ring series of lengh 08 years can be generaed, correlaion coefficiens compued beween each simulaion and he climae series, and 95% confidence inerval for significan correlaion esablished as he 0.05 and 0.975 probabiliy poins of he 0,000 sample correlaions. If he correlaion beween he observed ime series and he climae series is ouside he confidence inerval, a significan saisical relaionship is inferred. Figure 5.6. Observed ree-ring index and five simulaion by AR(4) model. Top plo is idenical o ha a op of Figure 5.. Predicion differs from simulaion in ha he objecive of predicion is o esimae he fuure value of he ime series as accuraely as possible from he curren and pas values. Unlie simulaions, predicions uilize pas values of he observed ime series. A predicion form of he AR() model is yˆ ˆ () a y Noes_5, GEOS 585A, Spring 05
where he ^ indicaes an esimae. The equaion can be applied one sep ahead o ge esimae yˆ from observed y. A -sep-ahead AR() predicion can be made by recursive applicaion of equaion (). In recursive applicaion, he observed y a ime is used o generae he esimaed ŷ a ime. Tha esimae is hen subsiued as y o ge he esimaed ŷ a ime 3, and so on. The -sep-ahead predicions evenually converge o zero as he predicion horizon,, increases 7. Predicion is illusraed in Figure 5.7 for he ree-ring example. Recall ha he index for he 08-year period 900-007 was fi wih an AR(4) model, which explained abou /3 he variance of he series. The segmens of observed and prediced index beginning wih 990 are ploed in Figure 5.7. The observed index ends in 007, and for he years 990-007 he prediced values ploed are one-sep-ahead predicions. For he AR(4) model, his means ha he predicion for year is made from observed index in he preceding four years. One-sep-ahead predicions in general mae use of observed daa for imes ( ) o mae he predicion for year. The predicions ploed in Figure 5.7 for years beginning wih 009 are -sep-ahead predicions. These predicions use observed index for imes ( ) and prediced index for laer imes in maing a predicion for ime. The prediced index for 008 is a one-sep-ahead predicion, for 009 a -sep-ahead predicion, for 00 a 3-sep-ahead predicion and so forh. The persisence in he daa combined wih he low-growh period saring abou 000 leads o predicions of below normal ree-ring index for he 0 years of predicion horizon beyond he end of he observed daa. The predicions converge oward he mean as he horizon lenghens, however, because he influence of he pas gradually fades. 7 Because he modeling assumes y is a deparure from he mean, his convergence in erms of he original ime series is convergence oward he mean Noes_5, GEOS 585A, Spring 05
Figure 5.7. Predicion of ree-ring index by AR(4) model. Predicions generaed by model given by equaion 8 in ex. Time series described in capion o Figure 5.. 5.7 Exension o nonsaionary ime series ARMA modeling assumes he ime series is wealy saionary. Wih he appropriae modificaion, nonsaionary series can also be sudied wih ARMA modeling. Trend defined as a deerminisic funcion of ime can be removed by curve-fiing prior o ARMA modeling. Tha is he approach in dendrochronology, where a smooh curve describing he growh rend is removed in convering ring widhs o ring indices. Periodic ime series are a special case in which he rend is cyclical. An example of a periodic series is a monhly ime series of air emperaure wih is annual cycle. The mean is clearly nonsaionariy in ha i varies in a regular paern depending on monh. One way of handling such a series wih ARMA modeling is o remove he annual cycle by expressing he monhly daa as deparures from heir long-erm monhly means. Anoher way is by applying periodic ARMA models, in which separae parameers are simulaneously esimaed for each monh of he year. Periodic ARMA models are discussed by Salas e al. (980). A rend iself migh be a sochasic feaure of a ime series. For example, a random wal ime series wanders wih a ime varying populaion mean. The series does no end o reurn o any specific preferred level. Such series can be derended by firs-differencing before ARMA modeling a modeling approach called auoregressive-inegraed-moving-average (ARIMA) modeling. Furher discussion of ARIMA modeling can be found elsewhere (Anderson 976; Box and Jenins 976; Salas e al. 980) Noes_5, GEOS 585A, Spring 05 3
5.8 References Anderson, O., 976, Time series analysis and forecasing: he Box-Jenins approach: London, Buerworhs, p. 8 pp. Box, G.E.P., and Jenins, G.M., 976, Time series analysis: forecasing and conrol: San Francisco, Holden Day, p. 575 pp. Chafield, C., 004, The analysis of ime series, an inroducion, sixh ediion: New Yor, Chapman & Hall/CRC. Coo, E.R., 985, A ime series approach o ree-ring sandardizaion, Ph. D. Diss., Tucson, Universiy of Arizona. -----, Shiyaov, S., and Mazepa, V., 990, Esimaion of he mean chronology, in Coo, E.R., and Kairiusis, L.A., eds., Mehods of dendrochronology, applicaions in he environmenal sciences: In:,: Kluwer Academic Publishers, p. 3-3. Ljung, L., 995, Sysem Idenificaion Toolbox, for Use wih MATLAB, User's Guide, The MahWors, Inc., 4 Prime Par Way, Naic, Mass. 0760. Monserud, R., 986, Time series analyses of ree-ring chronologies, Fores Science 3, 349-37. Salas, J.D., Delleur, J.W., Yevjevich, V.M., and Lane, W.L., 980, Applied modeling of hydrologic ime series: Lileon, Colorado, Waer Resources Publicaions, p. 484 pp. Wils, D.S., 995, Saisical mehods in he amospheric sciences: Academic Press, 467 p. Noes_5, GEOS 585A, Spring 05 4