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# Class Note for GEOS 585A with Professor Meko at UA

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Date Created: 02/06/15

6 Spectral Analysis Smoothed Periodogram Method 61 Historical background There are many available methods for estimating the spectrum of a time series In lesson 4 we looked at the Blackman Tukey method which is based on Fourier transformation of the smoothed truncated autocovariance function The smoothed periodogram method circumvents the transformation of the acf by direct Fourier transformation of the time series and computation of the raw periodogram a function rst introduced in the 1800s for study of time series The raw periodogram is smoothed by applying combinations or spans of one or more lters to produce the estimated spectrum The smoothness resolution and variance of the spectral estimates is controlled by the choice of lters A more accentuated smoothing of the raw periodogram produces an underlying smoothly varying spectrum or null continuum against which spectral peaks can be tested for signi cance This approach is an alternative to the speci cation of a functional form of the null continuum e g AR spectrum The periodogram was one of the earliest statistical tools for studying periodic tendencies in time series Figure 61 Prior to development of the periodogram such analysis was tedious and generally feasible only when the periods of interest covered a whole number of observations Historical Background Figure 61 Timeline of developments in spectral analysis of time series Timeline based on information in Bloom eld 2000 p 5 and Hayes 1996 Notes76 GEOS 585A Spring 2009 1 Schuster 1897 showed that the periodogram could yield information on periodic components of a time series and could be applied even when the periods are not known beforehand Following the development of statistical theory of the spectrum in the 1920s and 1930s the smoothed periodogram was proposed as a estimator of the spectrum Daniell 1946 Bloom eld 2000 notes that use of the smoothed periodogram in this sense had actually been anticipated much earlier by Einstein in1914 The smoothed periodogram enjoyed a brief period of popularity as a spectral estimator Another method Fourier transformation of the truncated and smoothed autocorrelation function eg the BlackmanTukey method gained prevalence over the smoothed periodogram in the 1950s because of computational advantages The smoothed periodogram has become popular again recently According to Chat eld 2004 p 136 two factors have led to increasing use of the smoothed periodogram First is the advent of highspeed computers Second is the discovery of the fast Fourier Transform FFT which greatly speeded up computations Cooley and Tukey 1965 Today the smoothed periodogram is one of many alternative methods available for estimating the spectrum Some of these methods are listed in Table 1 Pros and cons of the various methods 7 except MTM are discussed in Hayes 1996 Table 1 Alternative methods for spectrum Method Summary Reference Blackman Tukey Fourier transformation of smoothed Chat eld 1975 truncated autocovariance function Smoothed Estimate periodogram by DFT of time series Bloom eld 2000 periodogram Smooth periodogram with modi ed Daniell lter Welch s method Averaged periodograms of overlapped Welch 1967 windowed segments of a time series Multi taper method Use orthogonal windows tapers to get Percival and MTM approximately independent estimates of spectrum Walden 1993 combine estimates Singular spectrum Eigenvector analysis of autocorrelation matrix to Vautard and Ghil analysis SSA eliminate noise prior to transformation to spectral 1989 estimates Maximum entropy Parametric method estimate acf and solve for AR Kay 1988 MEM model parameters AR model has theoretical spectrum 62 Steps in smoothedperiodogram method The main steps in estimating the spectrum by the smoothed periodogram method are Subtract mean and detrend time series Compute discrete Fourier transform DFT Compute raw periodogram Smooth the periodogram to get the estimated spectrum 5wa Notes76 GEOS 585A Spring 2009 2 The rst step in estimation ofthe spectrum by the smoothed periodogram method is subtraction of the sample mean This operation has no effect on the Variance and amounts to a shi ofthe series along the Vertical axis Figures 62 63 The most obvious problem With not subtracting the mean is that an abrupt offset is introduced When the series is padded With zeros in a later step in the analysis Number W WWWWVUUWUUWWU VWW n w w w 1700 1750 mm 1350 1900 1950 2000 Year Figure 62 Time plot of Wolf Sunspot Number 17002007 This time series is known to have an irregular cycle with period near 11 years The longiterm mean is 499 The Source httpwwwngdcnoaagovstpSOLARitpsunspotnumberhtml 150 Deparlure i Mi quoti r milquot ll WWWMlvgwwllwlvwwt 0 x x 31700 i750 law 1850 woo win 2000 Year Figure 63 Sunspot series as departure from longiterm mean Notes76 GEOS 585A Spring 2009 Any obvious trend should also be removed prior to spectral estimation Trend produces a spectral peak at zero frequency and this peak can dominate the spectrum such that other important features are obscured The analysis then continues to computation of the Fourier transform and raw periodogram and to smoothing of the periodogram Discrete Fourier transform Say x0x1 le is an arbitrary time series oflength n The time series can be expressed as the sum of sinusoids at the Fourier frequencies of the series xt A02 Z Agcos2n tBjsin2njt 0ltltVl2 1 Afn2cos27rfnZt t0lnil where the summation is over Fourier frequencies f 1 112n712 71 and the last term in braces is included only if n is even Bloomfield 2000 p 38 Note that the total number of coef cients is n whether n is even or odd The coefficients in l are given by anl Af7xtcos27rft 2 2r1 Bf7xtsm27rft Equations 2 are sine and cosine transforms that transform the time series X into two series of coef cients of sinusoids The relationships in 2 can be more succinctly expressed in complex notation by making use of the Euler relation e cosxisinx 3 and its inverse l cosx e e39 sinx e 7e39 4 2 0 In general observed data are strictly realvalued but they may be regarded as complex numbers with zero imaginary parts Suppose x0x1xn1 is such a realvalued time series expressed as complex numbers The discrete Fourier transform DFT of xt is given in complex notation by nefite w 5 Periodogram The relationships 2 transform the time series into a series of coefficients at its Fourier frequencies The discrete Fourier transform is the complex expression of these coef cients df whereA and B are identical to the quantities de ned in 2 The original data can be recovered from the DFT using the inverse transform x 2 do 2 1 7 which is the complex equivalent of equation 1 if iLf 6 2 2 Notes76 GEOS 585A Spring 2009 4 The discrete Fourier transform has two representations The first is in terms of its real and imaginary parts Af2and iBf 2 The second is in terms of its magnitude Rf and Phase f df Rf8 f 8 The magnitude given by Rm ldf 9 measures how strongly the oscillation at frequency f is represented in the data The strength of the periodic component is more often represented by the periodogram defined as 1f quotRf2 nldmlz 10 The sine and cosine terms at the Fourier frequencies are orthogonal and so the variance of the time series xt can be decomposed into components at the individual frequencies For the sine and cosine transforms the sum of squares of the original data can be expressed as n71 th2nA0z2n Z Af2Bf2nA ZZ 11 r0 0lt 1012 where the last term is included only if n is even The analog for the discrete Fourier transform in complex notation is ilxrlznzld lzZI 12 If xt is a time series expressed as departures from its mean the sums of squares in equations 11 and 12 are simply n times the variance Equation 12 therefore indicates that a the sum of the periodogram ordinates equals the sum of squares of departures of the time series from its mean b the sum of periodogram ordinates divided by the series length equals the series variance and c the periodogram ordinate at Fourier frequency f is proportional to the variance accounted for by that frequency component The periodogram at this stage is a raw periodogram meaning it has not yet been smoothed The raw periodogram of the Wolf sunspot number is plotted in Figure 64 Each point represents the relative variance of the time series contributed by a frequency range centered at the point Wavelengths near 11 years make relatively large contribution to the variance Notes76 GEOS 585A Spring 2009 5 RelahveVarlance n 005 01 015 02 025 03 035 0A 045 05 ammo mt Figure 64 Raw periodogram oi Wolf sunspot number 170072007 Periodogram ordinates give the relative variance contributed at dilTerent irequency ranges centered on hmdamental irequencies alter padding oi the series The number oipoints in the plot is 256 because the series has been padded to length 512 before periodogram analysis see Section 66 Smoothing the periodogram The periodogram is awildly fluctuating estimate ofthe spectrum with high variance For astable estimate the periodogram must be smoothed Bloomfield 2000 p 157 recommends the Daniell window as a smoothing filter for generating an estimated spectrum from the periodogram The modified Daniell window ofspan or length m is defined as g 1 72m717 1 i otherwise In 71 g i10rim 13 where m is the number ofweights orsptm ofthe filter and g is the imweight ofthe filter The Daniell filter differs from an evenly weighted moving average rectangular filter only in that the first and last weights are halfas large as the other weights A plot ofthe filter weights therefore has the form ofatrapezoid For example Figure 65 shows filter weights of5weight Daniell and rectangular filters The advantage ofthe Daniell filter over the rectangular filter for smoothing the periodogram is that the Daniell filterhas less leakage which refers to the in uence of variance at nonFourier frequencies on the spectrum at the Fourier frequencies The leakage is related to sidelobes in the frequency response ofthe filter Successive smoothing by Daniell filters with different spans gives an increasingly smooth spectrum and is equivalent to single application ofaresultant filter produced by convoluting the individual spans ofthe Daniell filters Bloomfield 2000 p 157 A smoothed periodogram ofthe Wolfsunspot number is plotted in Figure 66 The smoothing for this example ms done with successive application ofDaniell filters oflength 7 and 11 Broader longer filters would give a smoother spectrum Narrower filters would give a rougher spectrum The proper amount ofsmoothing is somewhat subjective and depends on the characteristics ofthe data lfthe natural periodicity ofaseries is such that peaks in the spectrum are closely spaced in frequency use oftoo broad afilterwill merge the peaks The tradeoffs in Notesg6 GEOS 585A Spring 2009 6 0 35 Dame 0 35 Re aquot9 39a smoothness stability and resolution should be considered in slecting widths 0 3 0 3 of Daniell lters see Section 64 0 25 0 25 E 0 2 E 0 2 g 0 15 g 0 15 0 l 0 l 0 05 0 05 05 0 5 05 0 5 39 Index ofWeigm Index ofWeigm 6393 continuum Figure 65 Daniell filter and rectangular lter of span Altthugh the Specmlm Of a my series is innately useful for describing length 5 the distribution of variance as a function of frequency sometimes interest centers on how the sample spectrum for a given time series differs from that of some known generating process Interest also sometimes centers on the statistical signi cance of peaks in the spectrum Significance can be evaluated only by reference to some standard of comparison The question is significantly different than what A standard for comparison is the null continuum The null continuum is a general null hypothesis for the spectrum The null continuum can be theoretically based or databased The simplest form of null continuum is white noise which has an even distribution of variance over frequency The white noise spectrum is consequently a horizontal line Variance is not preferentially concentrated in any particular frequency range In testing for signi cance of spectral peaks the white noise null continuum may be inappropriate if it is known that the series is persistent Persistence or positive autocorrelation in a time series can skew its frequency concentration toward the lowfrequency side of the spectrum One option for dealing with persistent processes is to use the theoretical spectrum of an autoregressive process as the null continuum Theoretical AR spectra An autoregressive process has a characteristic spectrum whose shape depends on the model order and the values of the model parameters An ARl process can have a spectrum that ranges from rednoise emphasizing low frequencies to blue noise emphasizing high frequencies Higherorder AR process can have very complicated spectral shapes The AR2 model for example can represent a pseudoperiodic process A variety of theoretical spectra for various ARl and AR2 processes are illustrated by Wilks 1995 p 353 The equations for AR spectral in this section are taken from Wilks 1995 Note that the equations below differ from those in Wilks 1995 so that the equations are consistent with the MATLAB sign convention for AR models MATLAB s System Identi cation Toolbox writes the AR 1 model as y 1344 8 14 and the AR2 model by y alytrl azyrrz 8 15 where yr is the time series expressed as departures from its mean 71 and 12 are autoregressive parameters and 8t is the noise term A positively autocorrelated ARl series has 11 lt 0 by this convention ARl model For the simple case of the ARl process negative values of the autoregressive parameter build a memory into the system that tends to smooth over shortterm highfrequency Notes76 GEOS 585A Spring 2009 7 variations and emphasize the slower variations The effects are progressively stronger as the parameter gets closer to l The theoretical spectrum can be written in terms of the AR parameter the variance of the residuals and the sample size For an ARl model 2 5f ts m 1 11 2111 cos27rf where a is the variance of the residuals N is the number of observations 71 is the ngglZ 16 autoregressive parameter f is frequency in cycles per year or time unit and S f is the theoretical spectrum The shape is entirely determined by the AR parameter aquot andN act to scale the spectrum higher or lower but do not change the relative distribution of variance over frequency Equation 16 can be used to generate a theoretical spectrum for any observed time series The series is first modeled as an ARl process The autoregressive parameter and variance of residuals are estimated from the data Plots of spectra for different values of 71 show how the spectrum varies as a function of autoregressive parameter The special case of a1 0 corresponds to a white noise process By analogy with visible light white noise contains an equal mixture of variance at all frequencies The theoretical spectum of white noise is a horizontal line For 71 lt 0 the spectrum is enhanced at the low frequencies and depleted at the higher frequencies By analogy with the light the spectrum is called red noise For 71 gt 0 the process tends to create erratic shortterm variations with positive autocorrelation at even lags and negative autocorrelation at odd lags The spectrum is enriched at the high frequencies and depleted at the low frequencies Such series are sometimes called blue noise Notes76 GEOS 585A Spring 2009 8 AR 2 model Theoretical spectra can also be expressed for other AR models For the special case of the ARMA2 model the spectrum is 40 N S f 2 2 1 71 a2 2a11 a2 cos27rf 2112 cos47rf nggUZ an The spectrum of an AR2 process is particularly interesting because it can express pseudoperiodicity with the spectral details depending on the sized of the two parameters For example for 71 709 12 06 the spectrum has a strong peak near a frequency of0 15 For annual time series this frequency corresponds to a wavelength of 67 yr It is therefore possible that a time series that exhibits quasiperiodic behavior with a variance peak at 67 years resulted from a short memory process with dependence on the past restricted to the most recent two years Theoretical AR spectra can be useful in data interpretation Similarity of the gross shape of the estimated spectrum with that of an AR process may suggest the process as a simple generating mechanism Allowable ranges of AR parameters AR parameters must stay within a certain range for a process to be stationary For the AR1 model it is required that 71 lt a1 lt 1 18 For an AR2 process several conditions must be satis ed 71lt 12 lt 1 a1 12 gt 71 19 a1 7 1 lt1 The allowable ranges of AR2 parameters and the portions of the bivariate range corresponding to pseudoperiodic behavior in a time series are discussed by Anderson 1976 White noise AR1 and AR2 theoretical spectra are sketched in gure 67 It should be noted that higherorder AR process can have very complicated spectra The maximum entropy method of spectral estimation ts a highorder AR process to the series and uses the theoretical AR spectrum as the spectral estimate Another approach to a null continnum is empirically based and does not attempt to assign any particular theoretical generating model as a null hypothesis Bloom eld 2000 This approach uses a greatly smoothed version of the raw periodogram as the null hypothesis Figure 68 illustrates a smoothedperiodogram null continuum for the sunspot series using successive applications of Daniell lters of spans 4361 and 77 The estimation of a null continnum by smoothing the periodogram relies on subjective judgement and trialanderror In particular the null continuum should follow just the smooth underlying shape of the distribution of variance over frequency If smoothed insufficiently the null continuum will bulge at the important peaks in the spectrum This would clearly be undesirable as the test of signi cance of the peak is that it is different from the null continuum Notes76 GEOS 585A Spring 2009 9 x10 Bandwidth 2 5 Smunmzd Penudngram spzcuum Re aiwe Variance as a V 7 u 005 U1 015 02 025 u 035 04 045 05 quuoncywr y Figure 66 Smoouied perioaograrn estimate of spectrum oi Wolf sunspot number Raw periodograni points snioouied by Danieu rmers oi length 7 and 11 Bandwidth gives resoiuu39on oi the spedral estimate Spectral peak at 106 years ARU AW WME Nmsa Ra atwa Variance Frequency Figure 67 Sketch oi general shapes ofwhiu noise AR1 and Aka null eonu39nua Notesi GEOS 5 85A Spring 2009 10 lAnnn r r r r r r r r ri lznnn Hartman tuunn g 5perttum 2 mun gt E snnn g Null cuntmuum Anna 2000 n nus ml 015 02 025 03 035 DA 045 ns Frequency tyr t Figure 58 sn ectnrrn and Etnpiriczllyrhzsed null ennu39nuurn nr Wnl sunsp nt series naniull lters nr spans 7and 11 were applied tn srnnnth theraw perlndngrarn lntn the spectrum Spans nflmglh 43 51 and 77were used tn srnnnth theraw perindngrarn lntn the null ennu39nuurn ln evaluating the signi cance n the spectral p mk the null hyp nthesls ls thatthe variance cnntrihlltilm at therrequeney nf39hepmk ls nn different that the wriance cnntrihllte at the frequency in the null ennu39nuurn 64 Spectral properties Smnnthness The raw unsmoothed penodogam IS a rough estlmate of the spectrum The penodogam IS prop ortlonal to Varlance contnbuted at the fundamental trequencres Unformnalely the raw penodogam 15 of llttle dlrect usefulness because of the hlgh Varlance of the spectral estrmates Smoothng the penodogam wth Danlell lters ofvanous spans results In a spectrum much smoother In appearancethan theraw penodogmm Excesslve smoothlng obscures the lmportant spectral detall lnsumclent smoothlng leaves erratrc ummp ortant detall tn the spectrum Smoothness IS closely related to bum as dlscussed In the lecture on the Blachnanr Tukey method ofspectral estlmatron As a spectmm IS smoothed more and more the estrmated spectrum eventually approaches a featureless curve blased towards the local mean Smbility The stablllty ofthe spectral estrmate ls the artent to whlch estlmates computed from dlfferent segments ofa senes agee or n otherwords the ertent to whlch urelevant ne structure tn the penodogam ls ellmlnatedquot Bloom eld 2000 p 156 ngh stablllty corresponds to low Varlance othe estlmate and ls attalned by av eraglng over many penodogmm ordlnates The number ofperlodogam ordlnates averaged over tn the smoothed perlodogmm method as desmbed by Bloom eld 2000 ls deflned by the span ofthe Danlell lter Ifthe tlme senes has not been padded or tapered the vanance othe spectral estlmate ls glven by MW 022g 20gt Notes GEOS 585A Spnng 2009 ll where f is the spectral estimate at frequency f s f is the true and unknown value of the spectrum assumed to be approximately constant over the interval of averaging and the summation 2 g is the sum of squared weights of the Daniell lter used to smooth the periodogram The sum of periodogram weights must equal 1 for the spectral estimate to be an unbiased estimate of the true spectrum Bloom eld 2000 p 178 The broader the Daniell lter the lower the sum of squares of weights and the lower the variance of the spectral estimate For example for the 3weight Daniell lter 255025 the sum of squares of weights is 0375 while for the 5weight lter 125252525 125 the sum of squares is 02188 An approximate con dence interval for the spectral estimate can be derived by considering that the periodogram estimates are independent and exponentially distributed The spectral estimate as a sum of independent exponentially distributed quantities is approximately 2 distributed The distribution of 839 f can be shown to be approximately 12 with degrees of freedom v i2 21 3 where g2 2 g is the sum of squared Daniell weights The relationship in 21 can be used to place a confidence interval around the spectral estimates For example a 95 confidence interval for f is given by V5f V f 130975 gsmg 130025 22 where I 0025 and I 0975 are the 25 and 975 points of the 12 distribution with v degrees of freedom Resolution Resolution is the ability of the spectrum to represent the ne structure of the frequency properties of the series The ne structure is the variation in the spectrum between closely spaced frequencies For example narrow peaks are part of the ne structure of the spectrum The raw periodogram measures the variance contributions at the Fourier frequencies or the nest possible structure Smoothing the periodogram for example with a Daniell filter averages over adjacent periodogram estimates and consequently lowers the resolution of the spectrum The wider the Daniell lter the greater the smoothing and the greater the decrease in resolution If two periodic components in the series were close to the same frequency the smoothed spectrum might be incapable of identifying or resolving the individual peaks The width of the frequency interval applicable to a spectral estimate is called the bandwidth of the estimate If a hypothetical periodogram were to have just a single peak at a particular Fourier frequency the smoothed spectrum is roughly the image of the Daniell lter used to smooth the periodogram and the peak in the spectrum is spread out over several Fourier frequencies How many Fourier frequencies the peak covers would depend on the spans of the lter A reasonable measure of the bandwidth of the spectral estimate is therefore the width of the resultant Daniell filter used to smooth the periodogram Depending on how the resultant Daniell filter has been constructed the shape of the lter also varies Thus one filter may have only a few weights appreciably different from zero while another lter of the same length may have fewer or more appreciably nonzero weights Rather than the width of the Daniell lter therefore a more effective measure of bandwidth also takes into account the values of the Daniell lter weights One such measure of bandwidth is the width of the rectangular lter that has the same variance as the Daniell lter Notes76 GEOS 585A Spring 2009 12 The variance of the estimator is proportional to the sum of squares of the lter weights The bandwidth for a given Daniell lter can therefore be computed as follows 1 Compute the sum of squares of the Daniell lter weights 2 Compute the number of weights nw of the evenly weighted moving average that has the same sum of squares as computed in l 3 Compute the bandwidth as bw nWAf where Af is the spacing of the Fourier frequencies Note that if the series has been padded to length N 39 the spacing is taken as l N 39 Differences in smoothness stability and resolution are illustrated for spectra of the Wolf sunspot series in Figure 69 A lesser amount of smoothing of the raw periodogram yields the spectrum in Figure 69A A greater amount of smoothing yields the spectrum in Figure 69B The bandwidths indicate the differences in resolution of the two versions of the spectral Both versions clearly show the main spectral peak near 11 years but the peak is narrower and much higher for the spectrum with less smoothing On the other hand the con dence interval around the spectrum is much tighter for the spectrum with greater smoothing Trial and error computating and plotting of spectral with different degrees of smoothing for the smoothed periodogram method is a analogous to the window smoothing approach described in lesson 4 for the BlackmanTukey method of spectral estimation Note that the sunspot series also exhibits a spectral peak at near frequency 001 wavelength 100 years This lowerfrequency uctuation is evident also in the time plot of the series Figure 62 Too much smoothing eg Figure 69B makes it impossible to resolve this peak from trend peak at zero frequency 65 Testing for periodicity A peak in the estimated spectrum can be tested for signi cance by comparing the spectral estimate at a given frequency with the con dence interval for the estimate Two considerations for the testing are l A signi cance test requires a null hypothesis For the spectrum the null hypothesis is that the spectrum at the speci ed frequency is not different from some null spectrum or null continuum An earlier section described a white noise null continuum an autoregressive null continuum and a null continuum based on a greatly smoothed raw periodogram The null hypothesis is then that the estimated spectrum is no different than this underlying spectrum 2 The con dence bands developed above equation 22 are not simultaneous In other words the bands should be used strictly to test for signi cance of a peak at a speci ed frequency and that frequency should be speci ed before running the spectral analysis This approach can be contrasted with a shing expedition in which the spectrum is estimated and then browsed to identify signi cant peaks Simultaneous con dence bands which would be much wider than those given by equation 22 are needed if the spectrum is to be in such an exploratory mode to pick out signi cant peaks To summarize the test for periodicity begins with speci cation of a period or frequency of interest Second the spectrum and its con dence interval are estimated possibly using a windowclosing procedure Third a null continuum is drawn so that the peaks in the spectrum can be compared to a null spectrum without those peaks but with the same broad underlying spectral shape Finally the peak is judged signi cant at 95 if the lower CI does not include the null continuum Notes76 GEOS 585A Spring 2009 13 A5 A A 51quot 3 5 95 Conneenee nterva m 3 g 25 Spectrum gt i 2 E 15 WME HE SE spectrum 0 005 01 015 02 025 03 035 DA 045 05 Frequency yr 12000 B 10000 SW 95 Cun dence intewai m 0000 E Specimm 0000 1 Anna WME HE SE spectrum 2000 0 005 01 015 02 025 03 035 DA 045 05 Frequency yr Figure 69 Spectra oI WolI sunspot number using two levels oi smoothing oi raw perioaogram A Smoothing with Danieu lter spans 3 5 7 B Smoothingwith Danieurilter spans 11 15 23 Note the dinerenee in range oi yeaxis Note576 GEOS 585A Spring 2009 14 66 Additional considerations tapering padding and leakage Tapering and padding Tapering and padding are mathematical manipulations sometimes performed on the time series before periodogram analysis to improve the statistical properties of the spectral estimates or to speed up the computations In spectral analysis a time series is regarded as a nite sample of an infmitely long series and the objective is to infer the properties of the in nitely long series If the observed time series is viewed as repeating itself an infmite number of times the sample can be considered as resulting from applying a data window to the in nite series The data window is a series of weights equal to 1 for the N observations of the time series and zero elsewhere This data window is rectangular in appearance The effect of the rectangular data window on spectral estimation is to distort the estimated spectrum of the unknown in nitelength series by introducing leakage Leakage refers to the phenomenon by which variance at an important frequency say a frequency of a strong periodicity leaks into other frequencies in the estimated spectrum The net effect is to produce misleading peaks in the estimated spectrum The objective of tapering is to reduce leakage Tapering consists of altering the ends of the meanadjusted time series so that they taper gradually down to zero Before tapering the mean is subtracted so that the series has mean zero A mathematical taper is then applied A frequently used taper function is the split cosine bell given by 17cos27rtp 0 tltp2 p2 tltlip2 23 17cos27r17tp lip2gtgl J WFU where p is the proportion of data desired to be tapered t is the time index and wp t are the taper weights A suggested proportion is 10 or p 010 which means that 5 is tapered on each end Bloom eld 2000 p 69 Padding The Fast Fourier Transform FFT introduced by Cooley and Tukey 1965 is a computational algorithm that can greatly speed up computation of the Fourier transform and spectral analysis The FFT is most effective if the length of time series n has small prime numbers One way of achieving this is to pad the time series with zeros until the length of the series is a power of 2 before computing the Fourier transform The padded data are de ned as Jet 0 g t lt n 24 0 n tltri39 where x is the original time series after subtracting the mean It can be shown Bloom eld 2000 p 61 that the discrete Fourier transform of the padded series differs trivially from that of the original series As a side effect of padding the grid of frequencies on which the transform is calculated is changed to a ner spacing This change suggests that padding with zeros can also be used to alter the Fourier frequencies such that some period of apriori interest falls near a Fourier frequency This is an acceptable procedure eg Mitchell et al 1966 The ner spacing of Fourier frequencies for a given span of Daniell lter gives a spectral estimate with a narrower bandwidth see 3 under Resolution above but the increase in resolution comes at the expense of a decrease in stability of the spectral estimate see eqn 26 below Notes76 GEOS 585A Spring 2009 15 Effect of padding and tapering on stability Tapering and padding both have the effect of increasing the variance of the spectral estimate If the time series is tapered by the split cosine bell taper and the total proportion of the series tapered is p the variance of the spectral estimate see eqn 20 is increased by a factor of C 7128793 25 2 8 e 5P If the time series is padded from an initial length of N to a padded length of N 39 the variance is increased by a factor of N op 7 26 If a time series has been padded and tapered an equation of form 22 can still be used for the con dence interval for the spectrum except with an effective degrees of freedom de ned as v 22 27 g where g3 0751ng 28 A simple example will serve to illustrate the computation of a con dence interval when the series has been padded and tapered before computation of the spectrum Say the original time series has a length 300 years a total of 20 of the series has been tapered and that the tapered series has then been padded to length 512 by appending zeros Equations 25 and 26 give variance in ation factors 6 7 128793 7 1287932 7 2 2 11163 29 2875p 28752 and 512 17067 30 0P 300 If the periodogram is smoothed by a 5weight Daniell lter 125 25 25 25 125 the quantity g3 is given by g3 0701ng 11163170672188 04169 31 equivalent degrees of freedom are 2 2 480 m 5 32 g3 04169 and the 95 confidence interval is 53f 53f g s g 1283 f 8312 33 039 f g s f g 601 f Notes76 GEOS 585A Spring 2009 16 67 References Anderson OD 1976 Time series analysis and forecasting the BoxJenkins approach Butterworths London 182 p Blackman RB and Tukey JW 1959 The measurement of power spectra from the point of View of communications engineering New York Dover Bloom eld P 2000 Fourier analysis of time series an introduction second edition New York John Wiley amp Sons Inc 261 p Chat eld C 2004 The analysis of time series an introduction sixth edition New York Chapman amp HallCRC Cooley JW and Tukey JW 1965 An algorithm for the machine computation of complex Fourier series Math Comput V 19 p 297301 Daniell PJ 1946 Discussion on the symposium on autocorrelation in time series J Roy Statist Soc Suppl V 8 p 8890 Einstein A 1914 Arch Sci Phys Natur V 437 p 254256 Hamming RW and Tukey JW 1949 Measuring noise color Bell Telephone Laboratories Memorandum Hayes MH 1996 Statistical digital signal processing and modeling New York John Wiley amp Sons Inc Kay SM 1988 Modern spectral estimation Engelwood Cliffs NJ Prentice Hall Lagrange 1873 Recherches sur la maniere de former des tables des planetes d apres les seules observations in Oevres de Lagrange V V1 p 507627 Mitchell JM Jr Dzerdzeevskii B Flohn H Hofmeyr WL Lamb HH Rao KN and Wallen CC 1966 Climatic change Technicall Note No 79 report of a working group of the Commission for Climatology WMO No 195 TP 100 Geneva Switzerland World Meteorological Organizaton 81 p Percival DB and Walden AT 1993 Spectral analysis for physical applications Cambridge University Press Schuster A 1897 On lunar and solar periodicities of earthquakes Proc Roy Soc p 455465 The MathWorks 1 1996 MATLAB signal processing toolbox Natick MA The MathWorks Inc Thomson W 1876 On an instrument for calculating the integral of te product of two given functions Proc Roy Soc V 24 p 26668 Vautard R and Ghil M 1989 Singular spectrum analysis in nonlinear dynamics with applications to paleoclimatic time series Physica D 35 395424 Welch PD 1967 The use of fast Fourier transform for the estimation of power spectra A method based on time averaging over short modified periodograms IEEE Trans Audio Electroacoust V AU15 p 7073 Wilks Daniel S 1995 Statistical methods in the atmospheric sciences Academic Press New York 467p Notes76 GEOS 585A Spring 2009 17

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