6 Specral Analysis -- Smoohed Periodogram Mehod 6. Hisorical background The specrum of a ime series can be esimaed by a variey of mehods. In lesson 4 we looked a he Blackman-Tukey mehod, which uses a Fourier ransform of he smoohed, runcaed auocovariance funcion. In conras, he smoohed periodogram mehod uses a Fourier ransform of he ime series iself. The ransform yields a quaniy called he raw periodogram, which in essence is a highly resolved specral esimae. The raw periodogram was inroduced in he lae800s for sudy of periodiciy in ime series. Unforunaely, he raw periodogram is a crude specral esimae wih high variance (grea uncerainy). Smoohing he raw periodogram produces a beer esimae of he specrum one wih a smaller variance (igher confidence inerval). Bu smoohing reduces he abiliy of he specrum o resolve periodic feaures ha may be very close in wavelengh. The smoohness, resoluion and variance of specral esimaes is conrolled by he choice of filers in smoohed periodogram analysis. Exremely broad smoohing produces an underlying smoohly varying specrum, or null coninuum, agains which specral peaks in less-severely smoohed periodogram can be esed for significance. This approach is an alernaive o he specificaion of a funcional form of he null coninuum (e.g., red noise). The periodogram was one of he earlies saisical ools for sudying periodic endencies in ime series (Figure 6.). Prior o developmen of he periodogram such analysis was edious and generally feasible only when he periods of ineres covered a whole number of observaions. Figure 6.. Timeline of developmens in specral analysis of ime series. Timeline based on informaion in Bloomfield (000, p. 5) and Hayes (996). Noes_6, GEOS 585A, Spring 05
Schuser (897) showed ha he periodogram could yield informaion on periodic componens of a ime series and could be applied even when he periods are no known beforehand. Following he developmen of saisical heory of he specrum in he 90s and 930s, he smoohed periodogram was proposed as a esimaor of he specrum (Daniell 946). Bloomfield (000) noes ha use of he smoohed periodogram in his sense had acually been anicipaed much earlier in 94 -- by Alber Einsein. The smoohed periodogram enjoyed a brief period of populariy as a specral esimaor. Anoher mehod, Fourier ransformaion of he runcaed and smoohed auocorrelaion funcion (e.g., he Blackman-Tukey mehod), gained prevalence over he smoohed periodogram in he 950s because of compuaional advanages. The smoohed periodogram has become popular again recenly. According o Chafield (004, p. 36), wo facors have led o increasing use of he smoohed periodogram. Firs is he adven of high-speed compuers. Second is he discovery of he fas Fourier Transform (FFT), which grealy speeded up compuaions (Cooley and Tukey 965). Today he smoohed periodogram is one of many alernaive mehods available for esimaing he specrum. Some of hese mehods are lised in Table. Pros and cons of he various mehods excep MTM -- are discussed in Hayes (996). Table. Alernaive mehods for specrum Mehod Summary Reference Blackman-Tukey Fourier ransformaion of smoohed, runcaed auocovariance funcion Chafield, 975 Smoohed periodogram Welch s mehod Muli-aper mehod (MTM) Singular specrum analysis (SSA) Maximum enropy (MEM) Esimae periodogram by DFT of ime series; Smooh periodogram wih modified Daniell filer Averaged periodograms of overlapped, windowed segmens of a ime series Use orhogonal windows ( apers ) o ge approximaely independen esimaes of specrum; combine esimaes Eigenvecor analysis of auocorrelaion marix o eliminae noise prior o ransformaion o specral esimaes Parameric mehod: esimae acf and solve for AR model parameers; AR model has heoreical specrum Bloomfield, 000 Welch, 967 Percival and Walden, 993 Vauard and Ghil, 989 Kay, 988 6. Seps in smoohed-periodogram mehod The main seps in esimaing he specrum by he smoohed periodogram mehod are:. Subrac mean and derend ime series. Compue discree Fourier ransform (DFT) 3. Compue (raw) periodogram 4. Smooh he periodogram o ge he esimaed specrum Noes_6, GEOS 585A, Spring 05
The firs sep in esimaion of he specrum by he smoohed periodogram mehod is subracion of he sample mean. This operaion has no effec on he variance, and amouns o a shif of he series along he verical axis (Figures 6., 6.3). The mos obvious problem wih no subracing he mean is ha an abrup offse is inroduced when he series is padded wih zeros in a laer sep in he analysis. Figure 6.. Time plo of Wolf Sunspo Number, 700-007. This ime series is known o have an irregular cycle wih period near years. The long-erm mean is 49.9. Daa source: hp://sidc.oma.be/sunspo-daa/ Figure 6.3. Sunspo series as deparure from long-erm mean. Noes_6, GEOS 585A, Spring 05 3
Any obvious rend should also be removed prior o specral esimaion. Trend produces a specral peak a zero frequency, and his peak can dominae he specrum such ha oher imporan feaures are obscured. Afer derending, he nex seps are compuaion of he Fourier ransform, compuaion of he raw periodogram, and smoohing of he periodogram. Discree Fourier ransform. Say x, x,, x 0 n is an arbirary ime series of lengh n. The ime series can be expressed as he sum of sinusoids a he Fourier frequencies of he series: x A ( 0 ) ( ) c o s ( ) s in A f f B f f j j j j 0 j n / A ( f ) c o s f, 0,,, n n / n / where he summaion is over Fourier frequencies j f, j,, ( n ) /, j n and he las erm in braces is included only if n is even (Bloomfield 000, p. 38). Noe ha he oal number of coefficiens is n wheher n is even or odd. The coefficiens in () are given by n A ( f ) x c o s f n 0 n B ( f ) x s in f. n 0 Equaions () are sine and cosine ransforms ha ransform he ime series x ino wo series of coefficiens of sinusoids. The relaionships in () can be more succincly expressed in complex noaion by making use of he Euler relaion ix e co s x i sin x (3) and is inverse ix ix ix ix co s x e e, sin x e e (4) In general, observed daa are sricly real-valued, bu hey may be regarded as complex numbers wih zero imaginary pars. Suppose x, x,, x 0 n is such a real-valued ime series expressed as complex numbers. The discree Fourier ransform (DFT) of x is given in complex noaion by () () n if (5) d ( f ) x e n 0 Periodogram. The relaionships () ransform he ime series ino a series of coefficiens a is Fourier frequencies. The discree Fourier ransform is he complex expression of hese coefficiens d ( f ) A ( f ) B ( f ) i (6) where A and B are idenical o he quaniies defined in (). The original daa can be recovered from he DFT using he inverse ransform if j x d ( f ) e j which is he complex equivalen of equaion (). (7) j Noes_6, GEOS 585A, Spring 05 4
The discree Fourier ransform has wo represenaions. The firs is in erms of is real and imaginary pars, A( f ) / and B( f ) /. The second is in erms of is magniude R( f ) and phase ( f ) The magniude, given by i ( f ) (8) d ( f ) R ( f ) e R ( f ) d ( f ) (9) measures how srongly he oscillaion a frequency f is represened in he daa. The srengh of he periodic componen is more ofen represened by he periodogram defined as (0) I ( f ) n R ( f ) n d ( f ) The sine and cosine erms a he Fourier frequencies are orhogonal, and so he variance of he ime series x can be decomposed ino componens a he individual frequencies. For he sine and cosine ransforms, he sum of squares of he original daa can be expressed as n x n A (0 ) ( ) ( ) ( ) n A f B f n A f j j n / () 0 0 j n / where he las erm is included only if n is even. The analog for he discree Fourier ransform in complex noaion is n x n ( ) ( ) d f I f j () j 0 j j If x is a ime series expressed as deparures from is mean, he sums of squares in equaions () and () are simply n imes he variance. Equaion () herefore indicaes ha (a) he sum of he periodogram ordinaes equals he sum of squares of deparures of he ime series from is mean, (b) he sum of periodogram ordinaes divided by he series lengh equals he series variance, and (c) he periodogram ordinae a Fourier frequency f is proporional o he variance accouned for by ha frequency componen. The periodogram a his sage is a raw periodogram, meaning i has no ye been smoohed. The raw periodogram of he Wolf sunspo number is ploed in Figure 6.4. Each poin represens he relaive variance of he ime series conribued by a frequency range cenered a he poin. Wavelenghs near years make relaively large conribuion o he variance. j Noes_6, GEOS 585A, Spring 05 5
Frequency (cycles/year) Figure 6.4. Raw periodogram of Wolf sunspo number, 700-007. Periodogram ordinaes give he relaive variance conribued a differen frequency ranges cenered on fundamenal frequencies (afer padding) of he series. The number of poins in he plo is 56 because he series has been padded o lengh 5 before periodogram analysis (see Secion 6.6). Smoohing he periodogram. The periodogram is a wildly flucuaing esimae of he specrum wih high variance. For a sable esimae, he periodogram mus be smoohed. Bloomfield (000, p. 57) recommends he Daniell window as a smoohing filer for generaing an esimaed specrum from he periodogram. The modified Daniell window of span, or lengh, m, is defined as g, i o r i m i ( m ), i o h e rw is e m h where m is he number of weighs, or span of he filer, g is he i weigh of he filer, and i is an i index such ha i,, m. The Daniell filer differs from an evenly weighed moving average (recangular filer) only in ha he firs and las weighs are half as large as he oher weighs. A plo of he filer weighs herefore has he form of a rapezoid. Filer weighs of 5-weigh Daniell and recangular filers are ploed in Figure 6.5. The advanage of he Daniell filer over he recangular filer for smoohing he periodogram is ha he Daniell filer has less leakage, which refers o he influence of variance a non-fourier frequencies on he specrum a he Fourier frequencies. The leakage is relaed o sidelobes in he frequency response of he filer. Successive smoohing by Daniell filers wih differen spans gives an increasingly smooh specrum, and is equivalen o single applicaion of a resulan filer produced by convoluing he individual spans of he Daniell filers (Bloomfield 000, p. 57). A smoohed periodogram of he Wolf sunspo number is ploed in Figure 6.6 The smoohing for his example was done wih successive applicaion of Daniell filers of lengh 7 and. Broader (longer) filers would give a smooher specrum. Narrower filers would give a rougher specrum. The proper amoun of smoohing is somewha subjecive, and depends on he characerisics of he daa. If he naural periodiciy of a series is such ha peaks in he specrum (3) Noes_6, GEOS 585A, Spring 05 6
Weigh Weigh 0.35 0.3 0.5 0. Daniell 0.35 0.3 0.5 0. Recangular are closely spaced in frequency, use of oo broad a filer will merge he peaks. The radeoffs in smoohness, sabiliy and resoluion should be considered in slecing widhs of Daniell filers (see Secion 6.4). 0.5 0. 0.05 0-5 0 5 Index of weigh 6.3 Null coninuum Alhough he specrum of a ime series is innaely useful for describing he disribuion of variance as a funcion of frequency, ineres someimes ceners on how he sample specrum for a given ime series differs from ha of some known generaing process. Ineres also someimes ceners on he saisical significance of peaks in he specrum. Significance can be evaluaed only by reference o some sandard of comparison. The quesion is significanly differen han wha? A sandard for comparison is he null coninuum. The null coninuum is a general null hypohesis for he specrum. The null coninuum can be heoreically based, or daabased. The simples form of null coninuum is whie noise, which has an even disribuion of variance over frequency. The whie noise specrum is consequenly a horizonal line. Variance is no preferenially concenraed in any paricular frequency range. In esing for significance of specral peaks, he whie noise null coninuum may be inappropriae if i is known ha he series is persisen. Persisence, or posiive auocorrelaion, in a ime series skews is variance oward he low-frequency side of he specrum. One opion for dealing wih persisen processes is o use he heoreical specrum of an auoregressive process as he null coninuum. Theoreical AR specra. An auoregressive process has a characerisic specrum whose shape depends on he model order and he values of he model parameers. An AR() process can have a specrum ha ranges from red-noise (emphasizing low frequencies) o blue noise (emphasizing high frequencies). Higher-order AR process can have very complicaed specral shapes. The AR() model, for example, can represen a pseudo-periodic process. A variey of heoreical specra for various AR() and AR() processes are illusraed by Wilks (995, p. 353). The equaions for AR specra in his secion are aken from Wilks (995), wih one difference: signs on he AR coefficiens have been changed for consisency wih Malab s noaion for AR models. Malab s Sysem Idenificaion Toolbox wries he AR() model as and he AR() model by 0.5 0.05 0-5 0 5 Index of weigh Figure 6.5. Daniell filer and recangular filer of span (lengh) 5. 0. y a y e (4) y a y a y e (5) where y is he ime series expressed as deparures from is mean, a and a are auoregressive parameers, and e is he noise erm. A posiively auocorrelaed AR() series has a 0 by his convenion. AR() model. For he simple case of he AR() process, negaive values of he auoregressive parameer build a memory ino he sysem ha ends o smooh over shor-erm (high-frequency) variaions and emphasize he slower variaions. The effecs are progressively sronger as he parameer ges closer o -. Noes_6, GEOS 585A, Spring 05 7
The heoreical specrum of an AR() process can be wrien in erms of he AR parameer, he variance of he residuals and he sample size. For an AR() model, 4 N S ( f ), 0 f / a a c o s( f ) where is he variance of he residuals, N is he number of observaions, a is he auoregressive parameer, f is frequency in cycles per year (or ime uni), and S ( f ) is he heoreical specrum. The shape is enirely deermined by he AR parameer; and N ac o scale he specrum higher or lower, bu do no change he relaive disribuion of variance over frequency. Equaion (6) can be used o generae a heoreical specrum for any observed ime series. The series is firs modeled as an AR() process. The auoregressive parameer and variance of residuals are esimaed from he daa. Plos of specra for differen values of a show how he specrum varies as a funcion of auoregressive parameer. The special case of a 0 corresponds o a whie noise process. By analogy wih visible ligh, whie noise conains an equal mixure of variance a all frequencies. The heoreical specum of whie noise is a horizonal line. For a 0, he specrum is enhanced a he low frequencies and depleed a he higher frequencies. By analogy wih he ligh, he specrum is called red noise. For a 0, he process ends o creae erraic shor-erm variaions, wih posiive auocorrelaion a even lags and negaive auocorrelaion a odd lags. The specrum is enriched a he high frequencies, and depleed a he low frequencies. Such series are someimes called blue noise. (6) Noes_6, GEOS 585A, Spring 05 8
AR () model. Theoreical specra can also be expressed for oher AR models. For he special case of he AR() model, he specrum is : 4 N S ( f ), 0 f / a a a a c o s f a c o s 4 f (7) The specrum of an AR() process is paricularly ineresing because i can express pseudoperiodiciy, wih he specral deails depending on he size of he wo parameers. For example, for a 0.9, a 0.6, he specrum has a srong peak near a frequency of 0.5. For annual ime series, his frequency corresponds o a wavelengh of 6.7 yr. I is herefore possible ha a ime series ha exhibis quasi-periodic behavior wih a variance peak a 6.7 years resuls from a shor memory process wih dependence on he pas resriced o he mos recen wo years. Theoreical AR specra can be useful in daa inerpreaion. Similariy of he gross shape of he esimaed specrum wih ha of an AR process may sugges he process as a simple generaing mechanism. Allowable ranges of AR parameers. AR parameers mus say wihin a cerain range for a process o be saionary. For he AR() model i is required ha a (8) For an AR() process, several condiions mus be saisfied: a a a a a The allowable ranges of AR() parameers and he porions of he bivariae range corresponding o pseudo-periodic behavior in a ime series are discussed by Anderson (976). Whie noise, AR() and AR() heoreical specra are skeched in figure 6.7. I should be noed ha higher-order AR process can have very complicaed specra. The maximum enropy mehod of specral esimaion fis a high-order AR process o he series and uses he heoreical AR specrum as he specral esimae. Anoher approach o a null coninnum is empirically based and does no aemp o assign any paricular heoreical generaing model as a null hypohesis (Bloomfield 000). This approach uses a grealy smoohed version of he raw periodogram as he null hypohesis. Figure 6.8 illusraes a smoohed-periodogram null coninuum for he sunspo series using successive applicaions of Daniell filers of spans 43, 6 and 77. The esimaion of a null coninuum by smoohing he periodogram relies on subjecive judgemen and rial-and-error. In paricular he null coninuum should follow jus he smooh underlying shape of he disribuion of variance over frequency. If smoohed insufficienly, he null coninuum will bulge a he imporan peaks in he specrum. This would clearly be undesirable as he es of significance of he peak is ha i is differen from he null coninuum. (9) Noes_6, GEOS 585A, Spring 05 9
Figure 6.6. Smoohed periodogram esimae of specrum of Wolf sunspo number. Raw periodogram (poins) smoohed by Daniell filers of lengh 7 and. Bandwidh gives resoluion of he specral esimae. Specral peak a 0.6 years. Figure 6.7. Skech of general shapes of whie noise, AR() and AR() null coninua. Noes_6, GEOS 585A, Spring 05 0
Figure 6.8. Specrum and empirically-based null coninuum of Wolf sunspo series. Daniell filers of spans 7 and were applied o smooh he raw periodogram ino he specrum. Spans of lengh 43, 6 and 77 were used o smooh he raw periodogram ino he null coninuum. In evaluaing he significance of he specral peak, he null hypohesis is ha he variance conribuion a he frequency of he peak is no differen ha he variance conribue a he frequency in he null coninuum. 6.4 Specral properies Smoohness. The raw (unsmoohed) periodogram is a rough esimae of he specrum. The periodogram is proporional o variance conribued a he fundamenal frequencies. Unforunaely, he raw periodogram is of lile direc usefulness because of he high variance of he specral esimaes. Flucuaions in he raw periodogram can be driven by sampling variabiliy, a problem ha becomes more severe he shorer he series. Smoohing he raw periodogram reduces his problem. If he raw periodogram is smoohed wih Daniell filers of various spans, he resul is a specrum smooher in appearance, and wih a igher confidence inerval. Bu excessive smoohing obscures imporan specral deail, and insufficien smoohing leaves erraic unimporan deail in he specrum. Smoohness is closely relaed o bias, as discussed in he lecure on he Blackman-Tukey mehod of specral esimaion. As a specrum is smoohed more and more, he esimaed specrum evenually approaches a feaureless curve, biased owards he local mean. Sabiliy. The sabiliy of he specral esimae is he exen o which esimaes compued from differen segmens of a series agree, or, in oher words, he exen o which irrelevan fine srucure in he periodogram is eliminaed (Bloomfield 000, p. 56). High sabiliy corresponds o low variance of he esimae, and is aained by averaging over many periodogram ordinaes. The number of periodogram ordinaes averaged over in he smoohed periodogram mehod as Noes_6, GEOS 585A, Spring 05
described by Bloomfield (000) is defined by he span of he Daniell filer. If he ime series has no been padded or apered, he variance of he specral esimae is given by sˆ f s f g (0) var ( ) ( ) u where sˆ( f ) is he specral esimae a frequency f, s( f ) is he rue (unknown) value of he specrum, assumed o be approximaely consan over he inerval of averaging, and he summaion g is he sum of squared weighs of he Daniell filer used o smooh he u u periodogram. The sum of periodogram weighs mus equal for he specral esimae o be an unbiased esimae of he rue specrum (Bloomfield 000, p. 78). The broader he Daniell filer, he lower he sum of squares of weighs and he lower he variance of he specral esimae. For example, for he 3-weigh Daniell filer. 5,.5 0,. 5 he sum of squares of weighs is 0.375, while for he 5-weigh filer. 5,. 5,. 5,. 5,. 5 he sum of squares is 0.88. An approximae confidence inerval for he specral esimae can be derived by considering ha he periodogram esimaes are independen and exponenially disribued. The specral esimae, as a sum of independen exponenially disribued quaniies, is approximaely disribued. The disribuion of ˆ S ( f ) can be shown o be approximaely wih degrees of freedom where g g u u v g u () is he sum of squared Daniell weighs. The relaionship in () can be used o place a confidence inerval around he specral esimaes. For example, a 95% confidence inerval for sˆ( f ) is given by v sˆ( f ) v sˆ( f ) s( f ) (0.9 7 5 ) (0.0 5 ) where (0.0 5 ) and (0.9 7 5 ) are he.5% and 97.5% poins of he v v v degrees of freedom. v () v disribuion wih Resoluion. Resoluion is he abiliy of he specrum o represen he fine srucure of he frequency properies of he series. The fine srucure is he variaion in he specrum beween closely spaced frequencies. For example, narrow peaks are par of he fine srucure of he specrum. The raw periodogram measures he variance conribuions a he Fourier frequencies, or he fines possible srucure. Smoohing he periodogram, for example wih a Daniell filer, averages over adjacen periodogram esimaes, and consequenly lowers he resoluion of he specrum. The wider he Daniell filer, he greaer he smoohing and he greaer he decrease in resoluion. If wo periodic componens in he series are close o he same frequency, he smoohed specrum migh be incapable of idenifying, or resolving, he individual peaks. The widh of he frequency inerval applicable o a specral esimae is called he bandwidh of he esimae. If a hypoheical periodogram were o have jus a single peak a a paricular Fourier frequency, he smoohed specrum is roughly he image of he Daniell filer used o smooh he periodogram, and he peak in he specrum is spread ou over several Fourier frequencies. How many Fourier frequencies he peak covers depends on he spans of he filer. A reasonable measure of he bandwidh of he specral esimae is herefore he widh of he resulan Daniell filer used o smooh he periodogram. Depending on how he resulan Daniell filer has been consruced, he shape of he filer also varies. Thus one filer may have only a few weighs appreciably differen from zero, while anoher filer of he same lengh may have fewer or more appreciably non-zero Noes_6, GEOS 585A, Spring 05
weighs. Raher han he widh of he Daniell filer, herefore, a more effecive measure of bandwidh also akes ino accoun he values of he Daniell filer weighs. One such measure of bandwidh is he widh of he recangular filer ha has he same variance as he Daniell filer. The variance of he esimaor is proporional o he sum of squares of he filer weighs. The bandwidh for a given Daniell filer can herefore be compued as follows:. Compue he sum of squares of he Daniell filer weighs. Compue he number of weighs n of he evenly weighed moving average ha has w he same sum of squares as compued in () 3. Compue he bandwidh as b w n f, where f is he spacing of he Fourier w frequencies. (Noe ha if he series has been padded o lengh N ', he spacing is aken as / N ' ) Differences in smoohness, sabiliy and resoluion are illusraed for specra of he Wolf sunspo series in Figure 6.9. A lesser amoun of smoohing of he raw periodogram yields he specrum in Figure 6.9A. A greaer amoun of smoohing yields he specrum in Figure 6.9B. The bandwidhs indicae he differences in resoluion of he wo versions of he specral. Boh versions clearly show he main specral peak near years, bu he peak is narrower and much higher for he specrum wih less smoohing. On he oher hand, he confidence inerval around he specrum is much igher for he specrum wih greaer smoohing. Trial and error compuaing and ploing of specral wih differen degrees of smoohing for he smoohed periodogram mehod is a analogous o he window smoohing approach described in lesson 4 for he Blackman-Tukey mehod of specral esimaion. Noe ha he sunspo series also exhibis a specral peak a near frequency 0.0 (wavelengh 00 years). This lower-frequency flucuaion is eviden also in he ime plo of he series (Figure 6.). Too much smoohing (e.g., Figure 6.9B) makes i impossible o resolve his peak from rend (peak a zero frequency). 6.5 Tesing for periodiciy A peak in he esimaed specrum can be esed for significance by comparing he specral esimae a a given frequency wih he confidence inerval for he esimae. Two consideraions for he esing are:. A significance es requires a null hypohesis. For he specrum, he null hypohesis is ha he specrum a he specified frequency is no differen from some null specrum, or null coninuum. An earlier secion described a whie noise null coninuum, an auoregressive null coninuum and a null coninuum based on a grealy smoohed raw periodogram. The null hypohesis is hen ha he esimaed specrum is no differen han his underlying specrum.. The confidence bands developed above (equaion ) are no simulaneous. In oher words, he bands should be used sricly o es for significance of a peak a a specified frequency, and ha frequency should be specified before running he specral analysis. This approach can be conrased wih a fishing expediion, in which he specrum is esimaed and hen browsed o idenify significan peaks. Simulaneous confidence bands, which would be much wider han hose given by equaion, are needed if he specrum is o be in such an exploraory mode o pick ou significan peaks. Noes_6, GEOS 585A, Spring 05 3
Figure 6.9. Specra of Wolf sunspo number using wo levels of smoohing of raw periodogram. (A) Smoohing wih Daniell filer spans [3 5 7]. (B) Smoohing wih Daniell filer spans [ 5 3]. Noe he difference in range of y-axis. Noes_6, GEOS 585A, Spring 05 4
To summarize, he es for periodiciy begins wih specificaion of a period or frequency of ineres. Second, he specrum and is confidence inerval are esimaed, possibly using a window-closing procedure. Third, a null coninuum is drawn so ha he peaks in he specrum can be compared o a null specrum wihou hose peaks bu wih he same broad underlying specral shape. Finally, he peak is judged significan a 95% if he lower CI does no include he null coninuum. 6.6 Addiional consideraions: apering, padding and leakage Tapering and padding. Tapering and padding are mahemaical manipulaions someimes performed on he ime series before periodogram analysis o improve he saisical properies of he specral esimaes or o speed up he compuaions. In specral analysis, a ime series is regarded as a finie sample of an infiniely long series, and he objecive is o infer he properies of he infiniely long series. If he observed ime series is viewed as repeaing iself an infinie number of imes, he sample can be considered as resuling from applying a daa window o he infinie series. The daa window is a series of weighs equal o for he N observaions of he ime series and zero elsewhere. This daa window is recangular in appearance. The effec of he recangular daa window on specral esimaion is o disor he esimaed specrum of he unknown infinie-lengh series by inroducing leakage. Leakage refers o he phenomenon by which variance a an imporan frequency (say a frequency of a srong periodiciy) leaks ino oher frequencies in he esimaed specrum. The ne effec is o produce misleading peaks in he esimaed specrum. The objecive of apering is o reduce leakage. Tapering consiss of alering he ends of he mean-adjused ime series so ha hey aper gradually down o zero. Before apering, he mean is subraced so ha he series has mean zero. A mahemaical aper is hen applied. A frequenly used aper funcion is he spli cosine bell, given by c o s / p, 0 p /, w ( ), / / p p p c o s ( ) / p, p / where p is he proporion of daa desired o be apered, is he ime index, and w ( ) are he aper p weighs. A suggesed proporion is 0%, or p 0. 0, which means ha 5% is apered on each end (Bloomfield 000, p. 69). Padding. The Fas Fourier Transform (FFT), inroduced by Cooley and Tukey (965), is a compuaional algorihm ha can grealy speed up compuaion of he Fourier ransform and specral analysis. The FFT is mos effecive if he lengh of ime series, n, has small prime numbers. One way of achieving his is o pad he ime series wih zeros unil he lengh of he series is a power of before compuing he Fourier ransform. The padded daa are defined as x x n 0 n n ' 0 ' where x is he original ime series, afer subracing he mean. I can be shown (Bloomfield 000, p. 6) ha he discree Fourier ransform of he padded series differs rivially from ha of he original series As a side effec of padding, he grid of frequencies on which he ransform is calculaed is changed o a finer spacing. This change suggess ha padding wih zeros can also be used o aler (3) (4) Noes_6, GEOS 585A, Spring 05 5
he Fourier frequencies such ha some period of a-priori ineres falls near a Fourier frequency. This is an accepable procedure (e.g., Michell e al. 966). The finer spacing of Fourier frequencies for a given span of Daniell filer gives a specral esimae wih a narrower bandwidh (see #3 under Resoluion above), bu he increase in resoluion comes a he expense of a decrease in sabiliy of he specral esimae (see eqn (6) below). Effec of padding and apering on sabiliy. Tapering and padding boh have he effec of increasing he variance of he specral esimae. If he ime series is apered by he spli cosine bell aper and he oal proporion of he series apered is p, he variance of he specral esimae (see eqn (0)) is increased by a facor of c T 8 9 3 p 8 5 p If he ime series is padded from an iniial lengh of N o a padded lengh of variance is increased by a facor of c P N ' N N ', he (5) (6) If a ime series has been padded and apered, an equaion of form () can sill be used for he confidence inerval for he specrum, excep wih an effecive degrees of freedom defined as where v g (7) g c c g * T P u * (8) A simple example will serve o illusrae he compuaion of a confidence inerval when he series has been padded and apered before compuaion of he specrum. Say he original ime series has a lengh 300 years, a oal of 0% of he series has been apered, and ha he apered series has hen been padded o lengh 5 by appending zeros. Equaions (5) and (6) give variance inflaion facors and c T 8 9 3 p 8 9 3(. ) 8 5 8 5 (. ) p u.63 (9) 5 c.7 0 6 7 (30) P 300 If he periodogram is smoohed by a 5-weigh Daniell filer, {.5.5.5.5.5}, he quaniy g is given by * equivalen degrees of freedom are and he 95% confidence inerval is (3) g c c g. 6 3(.7 0 6 7 )(. 8 8) 0.4 6 9, * T P u u v 4.8 0 5, (3) g 0.4 6 9 * 5 sˆ( f ) s( f ) 5 sˆ( f ) o r.8 3.8 3 0.3 9 sˆ( f ) s ( f ) 6.0 sˆ( f ) (33) Noes_6, GEOS 585A, Spring 05 6
6.7 References Anderson, O.D.. 976. Time series analysis and forecasing: he Box-Jenkins approach. Buerworhs, London, 8 p. Blackman, R.B., and Tukey, J.W., 959, The measuremen of power specra, from he poin of view of communicaions engineering: New York, Dover. Bloomfield, P., 000, Fourier analysis of ime series: an inroducion, second ediion: New York, John Wiley & Sons, Inc., 6 p. Chafield, C., 004, The analysis of ime series, an inroducion, sixh ediion: New York, Chapman & Hall/CRC. Cooley, J.W., and Tukey, J.W., 965, An algorihm for he machine compuaion of complex Fourier series: Mah. Compu., v. 9, p. 97-30. Daniell, P.J., 946, Discussion on he symposium on auocorrelaion in ime series: J. Roy. Sais. Soc. (Suppl.), v. 8, p. 88-90. Einsein, A., 94, Arch. Sci. Phys. Naur., v. 4.37, p. 54-56. Hamming, R.W., and Tukey, J.W., 949, Measuring noise color, Bell Telephone Laboraories Memorandum. Hayes, M.H., 996, Saisical digial signal processing and modeling: New York, John Wiley & Sons, Inc. Kay, S.M., 988, Modern specral esimaion: Engelwood Cliffs, NJ, Prenice Hall. Lagrange, 873, Recherches sur la manière de former des ables des planèes d'apres les seules observaions, in Oevres de Lagrange, v. VI, p. 507-67. Michell, J.M., Jr., Dzerdzeevskii, B., Flohn, H., Hofmeyr, W.L., Lamb, H.H., Rao, K.N., and Wallén, C.C., 966, Climaic change: Technicall Noe No. 79, repor of a working group of he Commission for Climaology; WMO No. 95 TP 00: Geneva, Swizerland, World Meeorological Organizaon, 8 p. Percival, D.B., and Walden, A.T., 993, Specral analysis for physical applicaions: Cambridge Universiy Press. Schuser, A., 897, On lunar and solar periodiciies of earhquakes: Proc. Roy. Soc., p. 455-465. The MahWorks, I., 996, Malab signal processing oolbox: Naick, MA, The MahWorks, Inc. Thomson, W., 876, On an insrumen for calculaing he inegral of e produc of wo given funcions: Proc. Roy. Soc., v. 4, p. 66-68. Vauard, R., and Ghil, M., 989, Singular specrum analysis in nonlinear dynamics, wih applicaions o paleoclimaic ime series, Physica D 35, 395-44. Welch, P.D., 967, The use of fas Fourier ransform for he esimaion of power specra: A mehod based on ime averaging over shor, modified periodograms: IEEE Trans. Audio Elecroacous., v. AU-5, p. 70-73. Wilks, Daniel S., 995. Saisical mehods in he amospheric sciences. Academic Press, New York. 467p. Noes_6, GEOS 585A, Spring 05 7