7 Detrending. 7.1 Identifying trend: a frequency-domain approach

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1 7 Derending Trend in a ime series is a slow, gradual change in some propery of he series over he whole inerval under invesigaion. Trend is someimes loosely defined as a long erm change in he mean, bu can also refer o change in oher saisical properies. For example, ree-ring series of measured ring widh frequenly have a rend in variance as well as mean. In radiional ime series analysis, a ime series was decomposed ino rend, seasonal or periodic componens, and irregular flucuaions, and he various pars were sudied separaely. Modern analysis echniques frequenly rea he series wihou such rouine decomposiion, bu separae consideraion of rend is sill ofen required. Derending is he saisical or mahemaical operaion of removing rend from he series. Derending is ofen applied o remove a feaure hough o disor or obscure he relaionships of ineres. In climaology, for example, a emperaure rend due o urban warming migh obscure a relaionship beween cloudiness an air emperaure. Derending is also someimes used as a preprocessing sep o prepare ime series for analysis by mehods ha assume saionariy. Many alernaive mehods are available for derending. Simple linear rend in mean can be removed by subracing a leas-squares-fi sraigh line. More complicaed rends migh require differen procedures. For example, he cubic smoohing spline is commonly used in dendrochronology o fi and remove ring-widh rend ha migh no be linear, or no even monoonically increasing or decreasing over ime. In sudying and removing rend, i is imporan o undersand he effec of derending on he specral properies of he ime series. This effec can be summarized by he frequency response of he derending funcion. 7.1 Idenifying rend: a frequency-domain approach Idenificaion of rend in a ime series is subjecive because rend in a sample canno be unequivocally disinguished from low frequency flucuaions. Wha looks like rend in a shor ime series segmen ofen proves o be a low-frequency flucuaion perhaps par of a cycle -- in he longer series. Someimes knowledge of he physical sysem helps in idenifying rend. For example, a decrease of ring widh of a ree wih ime is expeced parly on geomerical grounds: he annual incremen of wood is being laid down on an ever-increasing circumference. If he amoun of wood produced annually levels off as he ree ages, he ring widh would be expeced o decline. The modified negaive exponenial derending curve for ring widh is suppored by his knowledge of he physical sysem (Fris 1976). Lacking a physical basis for idenifying rend, we need o rely on saisical mehods. The frequency domain is paricularly useful here. Granger and Haanaka (1964 p. 130) give some insigh ino specral inerpreaion of rend. They conclude ha we are unable o differeniae beween a rue rend and a very low frequency flucuaion, and give he following advice: I has been found useful by he auhor o consider as rend in a sample of size n all frequencies less han 1( n ) as hese will all be monoonic increasing if he phase is zero, bu i mus be emphasized ha his is an arbirary rule. I may also be noed ha i is impossible o es wheher a series is saionary or no, given only a finie sample as any apparen rend in mean could arise from an exremely low frequency. If we apply he above reasoning o a 500-year ree-ring series, we would say ha variaion wih period longer han wice he sample size, or 1000 years, should be regarded as rend. In anoher paper, Granger (1966) defines rend in mean as comprising all frequency componens whose wavelengh exceeds he lengh of he observed ime series. Cook e al. (1990) refer o Granger s (1966) rend in mean concep in giving suggesions for derending ree-ring daa: Noes_7, GEOS 585A, Spring 005 1

2 Given he above definiion of rend in mean, anoher objecive crierion for selecing he opimal frequency response of a digial filer is as follows. Selec a 50% frequency-response cuoff in years for he filer ha equals some large percenage of he series lengh, n. This is he %n crierion described in Cook (1985). The resuls of Cook (1985) sugges ha he percenage is 67%n o 75%n based on using he cubic smoohing spline as a digial filer. The %n crierion ensures ha lile low-frequency variance, which is resolvable in he sandardized ree rings, will be los in esimaing and removing he growh rend. This crierion also has a bias of sors because of he siff characer of he low-pass filer esimaes of he growh rend. I will no necessarily guaranee and, in fac, will rarely possess any kind of opimal goodness-of-fi. 7. Fiing he rend Four alernaive approaches o derending are: 1) firs differencing, ) curve-fiing, 3) digial filering and 4) piecewise polynomials. This secion is weighed heavily oward he piecewise polynomials approach, which is widely used in dendrochronology. Firs differencing. A ime series ha is non-saionary in mean (e.g., rend in mean) can be made saionary by aking he firs difference. The firs-difference is simply he difference of he value of he series a imes and 1 : w = x x 1 (1) where x is he original ime series and w is he firs-differenced series. If he series is nonsaionariy in no jus he mean bu in he rae of change of he mean (he slope), saionariy can be induced by aking he second difference, or he firs difference of he firs difference: u = w w 1 () Differencing has been applied in hydrology in he conex of ARIMA (Auoregressive- Inegraed-Moving-Average) modeling of sreamflow series (Salas e al. 1980). As wih any derending mehod, firs differencing can be expeced o srongly aenuae he variance a he lowes frequencies in a ime series. Salas e al. (1980) repor ha firs differencing can be problemaic in hydrology because i ends o inroduce spurious high-frequency variaion ino a series. Anderson (1975) describes differencing as a way o remove nonsaionariy from ime series in general. According o Anderson (1975), each successive differencing will decrease he variance of he series, bu a some poin, higher-order differencing will lead o an increase in variance. When variance increases, he series has been over-differenced. Firs differencing is unsuiable for ime series whose level iself has imporance, as he differenced series essenially is jus change in level from one observaion o he nex, regardless of he level iself. Curve-fiing. If a ime series changes in level gradually over ime, i makes sense o consider as rend some simple funcion of ime iself. Someimes he mahemaical form of he rend funcion has physical basis. For example, a modified negaive exponenial curve wih concepual basis in he change of ree geomery wih ime has been used o remove he age rend from ring-widh series (Fris 1976). The simples and mos widely used funcion of ime used in derending is he leas-squares-fi sraigh line, which reas linear rend. Simple linear regression is used o fi he model x = a+ b+ e (3) where x is he original ime series a ime, a is he regression consan, b is he regression coefficien, and e are he regression residuals. The rend is hen described by Noes_7, GEOS 585A, Spring 005

3 g ˆ ˆ = a+ b (4) where g is he rend, â is he esimaed regression consan, and ˆb is he esimaed regression coefficien. The advanage of he sraigh-line mehod is simpliciy. The sraigh line may unrealisic, however, in resricing he funcional form of he rend. Oher funcions of (e.g., quadraic) migh be beer depending on he ype of daa. Digial filering. Anoher procedure for dealing wih rend is o describe he rend as a linearly filered version of he original series. The original series is convered o a smooh rend line by weighing he individual observaions s g = ax r + r (5) r= q where a r is a se of filer weighs (summing o 1.0), and g is he smooh rend line. The weighs are ofen symmeric, wih s = q and aj = a j. If he weighs are symmeric and all equal, he filer is a simple moving average, which generally is no recommended for measuring rend (Chafield 1975). x is he original series, { } Piecewise polynomials. An alernaive o fiing a curve o he enire ime series (curve fiing) is o fi polynomials of ime o differen pars of he ime series. Polynomials used his way are called piecewise polynomials. The cubic smoohing spline is a piecewise polynomial of ime,, wih he following properies: The polynomial is cubic ( raised o hird power) A separae polynomial is fi o every sequence of hree poins in he series The firs and second derivaives are coninuous a each poin The spline parameer specifies he flexibiliy and depends on he relaive imporance given o smoohness of he fied curve, and closeness of fi, or how close he fied curve passes o he individual daa poins Given he approximae values yi = g( xi) + εiof some supposedly smooh funcion g a daa poins x1,, xn and an esimae δ yi of he sandard deviaion of y i, he problem is o recover he smooh funcion from he daa. (Noe here he use of x for ime, which is no consisen wih he noaion in previous secions.) Le s( xi ) be he spline curve, or he approximaion o he smooh funcion g. Following De Boor (1978, p. 35), he spline curve is derived by minimizing he quaniy N x y ( ) N i s xi p + (1 p) D s i= 1 δ y (6) i x1 over all funcions s for a given spline parameer, p, where D s refers o he second derivaive of s wih respec o ime. The firs erm is similar o a sum-of-squares of deviaions. The second erm inegraes curvaure conribuions (second derivaive). Minimizing (6) esablishes a compromise beween saying close o he given daa (firs erm) and obaining he smoohes possible curve (second erm). The choice of p, where p can range from 0 o 1, depends on which of hose wo goals is given he greaer imporance. For p = 0, s is he leas squares sraigh-line fi o he daa. A he oher exreme, p = 1, s is he cubic spline inerpolan, and passes hrough each daa poin. As p ranges from 0 o 1, he smoohing spline changes from one exreme o he Noes_7, GEOS 585A, Spring 005 3

4 oher. The erm δ yi allows for differenial weighing of daa poins. Following recommendaions of Cook and Peers (1981) we use he defaul MATLAB weighing (1 for all poins). 7.3 Frequency response The frequency response funcion describes how a linear sysem responds o sinusoidal inpus differen frequencies (Chafield 1975, p. 198). The frequency response funcion has wo componens -- he gain and he phase. The gain a a given frequency describes how he ampliude of a sinusoid a ha frequency is damped or amplified by he sysem. The phase describes how a wave a ha frequency is shifed in absolue ime. In reference o a spline curve, he phase is zero, and he "frequency response" merely describes he gain, or he ampliude, of he response funcion. The inpu o he "sysem" in his case is he original ime series; he oupu is he smoohed curve purpored o represen he rend. The frequency response measures how srongly he spline curve would respond o or rack a periodic componen of a given frequency, should he ime series have such a componen. The ampliude of frequency response a a given frequency is he raio of he ampliude of he sinusoidal componen in he smoohed series (he spline curve) o he ampliude in he original ime series. Relaion of frequency response o spline parameer The cubic smoohing spline has become increasingly popular as a derending mehod in dendrochronology because he spline is adapable and easily applied o a wide range of ypes of age rend or growh rend found in ree-ring daa. Applicaion of he spline o dendrochronology was firs proposed by Cook and Peers (1981), who derived a mahemaical relaionship beween he frequency response of he spline and he spline parameer, p. The relaionship is given by 1 u( f) = 1 (7) p(cos π f + ) 1 + 6(cos 1) π f where u( f) is he ampliude of frequency response a frequency f, and p is he spline parameer as defined earlier. A plo of u( f ) agains f shows he response of he spline o variaions a differen frequencies. For rend lines, such as he smoohing spline, his response is higher oward he low-frequency end of he specrum. Cook and Peers (1981) furher derived a useful relaionship for he spline parameer p as a funcion of he 50% frequency response, he frequency a which he ampliude of he frequency response of he spline is This relaionship is given by: 6( cosπ f 1) p = (8) cos π f + ( ) Given a desired 50% frequency response, you can use (8) o compue he spline parameer. For example, say he desired 50% frequency response is a a period 100 years, corresponding o a frequency of 1/100. The desired spline parameer is ( 1) p= = 7.797E 6 (9) ( ) Noes_7, GEOS 585A, Spring 005 4

5 For a given ime series, calling MATLAB funcion casps() wih above spline parameer value will yield he suiably smoohed ime series. 7.4 Removal of rend Once a rend line has been fi o he daa, we can regard ha line as represening he rend. The quesion remains, how o remove he rend? If he rend-idenificaion mehod has idenified a rend line, wo opions are available. Firs is o subrac he value of he rend line from he original daa, giving a ime series of residuals from he rend. This difference opion is aracive for simpliciy, and for giving a convenien breakdown of he variance: he residual series is in he same unis as he original series, and he oal sum of squares of he original daa can be expressed as he rend sum-of-squares plus he residual sum-of-squares. The raio opion is aracive for some kinds of daa because he raio is dimensionless, and he raio operaion ends o remove rend in variance ha migh accompany rend in mean. Treering widh is one such daa ype: variance of ring widh ends o be high when mean ring widh is high, and low when mean ring widh is low. Raio-derending generally is feasible for nonnegaive ime series only, and runs he risk of explosion of he derended series o very high values if he fied rend line approaches close o zero. 7.5 Effec of derending on specrum Derending has he same effec on he specrum as high-pass filering, in ha he variance a low frequencies is diminished relaive o variance a high frequencies. The frequency response of he spline is high for hose frequencies racked closely by he spline. In he subsequen removal of he rend line, hese frequencies are mosly removed. Frequencies a which he frequency response of he spline is high are herefore hose a which he specrum of he derended series is low. In general, a he lowes frequencies, he specrum of he derended series will be diminished relaive o he specrum of he original daa. The more flexible he spline, he higher he frequency-range affeced by he derending. Normalized specrum. In comparing specra of series derended by differen mehods, i is convenien o plo he specral of several derended series on he same pair of axes. To faciliae inerpreaion of he relaive effecs of derending on differen wavelenghs, he specra are bes ploed as normalized specrum for such comparison. The normalized specrum is he specrum sandardized (or divided) by he variance of he series. Recall ha he area under he specrum equals or is proporional o he variance. The area under he normalized specrum is equal o one. Thus he disribuion of variance wih frequency for series wih differen variances can be readily compared wih normalized specra ploed on he same figure. 7.6 Quanifying he imporance of rend A simple measure of he pracical imporance of rend in a ime series is he fracion of original variance of he series accouned for by he fied rend line, which can be compued by var( e ) R = 1 (10) var( x ) where var( x ) is he variance of he original ime series, and var( e ) is he variance of he residuals from he rend line. Equaion (10)measures he imporance of he end componen in a ime series ime series, and can range from 0 for no imporance o 1 if he series is pure rend. Noe ha for raio derending, Noes_7, GEOS 585A, Spring 005 5

6 he oal variance of he original series canno be decomposed ino variance due o rend and residual variance because he derended series is NOT a residual. 7.7 References Anderson, O., 1976, Time series analysis and forecasing: he Box-Jenkins approach: London, Buerworhs, p. 18 pp. Chafield, C., 1975, The analysis of ime series: Theory and pracice, Chapman and Hall, London, 63 pp. Cook, E.R., 1985, A ime series approach o ree-ring sandardizaion, Ph. D. Diss., Tucson, Universiy of Arizona , Briffa, K., Shiyaov, S., and Mazepa, V., 1990, Tree-ring sandardizaion and growh-rend esimaion, in Cook, E.R., and Kairiuksis, L.A., eds., Mehods of Dendrochronology, Applicaions in he Environmenal Sciences: Kluwer Academic Publishers, p , and Peers, K., 1981, The smoohing spline: A new approach o sandardizing fores inerior ree-ring widh series for dendroclimaic sudies, Tree-Ring Bullein 41, de Boor, C., 1978, A pracical guide o splines: New York, Springer-Verlag, 39 p , 1999, Spline oolbox for use wih MATLAB, user's guide, version : Naick, MA, The MahWorks, Inc. Fris, H.C., 1976, Tree rings and climae: London, Academic Press, 567 p. Granger, C.W.J., 1966, On he ypical shape of an economeric variable: Economerics, v. 34, p Granger, C.W.J., and Haanaka, M., 1964, Specral analysis of economic ime series: Princeon, New Jersey, Princeon Universiy Press. Panofsky, H.A., and Brier, G.W., 1958, Some applicaions of saisics o meeorology: The Pennsylvania Sae Universiy Press, 4 p. Salas, J.D., Delleur, J.W., Yevjevich, V.M., and Lane, W.L., 1980, Applied modeling of hydrologic ime series: Lileon, Colorado, Waer Resources Publicaions, p. 484 pp. World Meerorological Organizaion, 1966, Technical Noe No. 79: Climaic Change, WMO-No, 195.TP.100, Geneva, 80 pp. Noes_7, GEOS 585A, Spring 005 6