6. Time (or Space) Series Analysis

Transcription

1 ATM 55 otes: Tie Series Analysis - Section 6a Page 8 6. Tie (or Space) Series Analysis In this chapter we will consider soe coon aspects of tie series analysis including autocorrelation, statistical prediction, haronic analysis, power spectru analysis, and cross-spectru analysis. We will also consider space-tie cross spectral analysis, a cobination of tie-fourier and space-fourier analysis, which is often used in eteorology. The techniques of tie series analysis described here are frequently encountered in all of geoscience and in any other fields. We will spend ost of our tie on classical Fourier spectral analysis, but will ention briefly other approaches such as Maxiu Entropy (MEM), Singular Spectru Analysis (SSA) and the Multi-Taper Method (MTM). Although we include a discussion of the historical Lag-correlation spectral analysis ethod, we will focus priarily on the Fast Fourier Transfor (FFT) approach. First a few basics 6. Autocorrelation 6.. The Autocorrelation Function Given a continuous function x(t), defined in the interval t < t < t, the autocovariance function is φ( τ ) = t t τ t τ x' t (6.) t where pries indicate deviations fro the ean value, and we have assued that τ >. In the discrete case where x is defined at equally spaced points, k =,,..,, we can calculate the autocovariance at lag L. φ( L) = L x' k L x' k+ L = x' x' k k+ L ; L =,±,±, ±3,... (6.) k= L The autocovariance is the covariance of a variable with itself (Greek autos = self) at soe other tie, easured by a tie lag (or lead) τ. ote that φ( ) = x', so that the autocovariance at lag zero is just the variance of the variable. The Autocorrelation function is the noralized autocovariance function φ(τ)/φ() = r(τ); - < r(τ) < ; r() = ; if x is not periodic r(τ), as τ. It is norally assued that data sets subjected to tie series analysis are stationary. The ter stationary tie series norally iplies that the true ean of the variable and its higher-order statistical Copyright 4 Dennis L. Hartann /4/4 4:3 PM 8

2 ATM 55 otes: Tie Series Analysis - Section 6a Page 9 oents are independent of the particular tie in question. Therefore it is usually necessary to reove any trends in the tie series before analysis. This also iplies that the autocorrelation function can be assued to be syetric, φ(τ) = φ(-τ). Under the assuption that the statistics of the data set are stationary in tie, it would also be reasonable to extend the suation in (6.) fro k=l to in the case of negative lags, and fro k= to -L in the case of positive lags. Such an assuption of stationarity is inherent in uch of what follows. Visualize the coputation of the autocorrelation fro discrete data Suppose you have data at discrete ties, equally spaced in tie, separated by a tie interval Δt. It would look soething like this: Fig. 6. The tie axis fro past (left) to future(right), sapled at intervals of Δt about soe central, but otherwise arbitrary tie of t i. In Fig. 6. t i represents one of the possible ties at which we have data. If we want to copute the autocovariance at one lag, we use the forula, cov(δt) = x'(t i ) x'(t i + Δt)... or... cov(δt) = i= where x' = x x, and x = i= x i i= x'(t i ) x'(t i Δt) It should be clear that these equations are approxiations to the covariance and the ean, but that they get better as increases, so long as the tie series is stationary. One can copute the autocovariance at any arbitrary lag, nδt, by odifying the equation to read, cov(nδt) = n n x'(t i ) x'(t i + nδt)... or... cov(nδt) = n i= i=n+ x'(t i ) x'(t i nδt) 6.. Red oise: oise with eory We define a red noise tie series as being of the for: x(t) = a x(t Δt) +( a ) / ε(t) (6.3) Copyright 4 Dennis L. Hartann /4/4 4:3 PM 9

3 ATM 55 otes: Tie Series Analysis - Section 6a Page 3 where x is a standardized variable x =, x' =, a is on the interval between zero and one and easures the degree to which eory of previous states is retained ( < a < ), ε is a rando nuber drawn fro a standardized noral distribution, and Δt is the tie interval between data points. This is also called a Markov Process or an Auto- Regressive, or AR- Process, since it reebers only the previous value. Multiply (6.3) by x(t-δt) and average to show that a is the one-lag autocorrelation, or the autocorrelation at one tie step, Δt. x( t Δt) x( t) = ax( t Δt) x( t Δt) + a x( t Δt) x( t) = r( Δt) = a Projecting into the future, we obtain = a + a ( ) / ε x t Δt ( ) / ; i.e., a is the autocorrelation at Δt x( t + Δt) = ax( t) + ( a ) / ε = a x( t Δt ) + a a ( ) ( ) / ε + ( a ) / ε = a x( t Δt) + ( a +) ( a ) / ε ow x(t+δt) = ax(t+δt) + ε. Consistent with (6.), ultiply by x(t) and average using the definitions above. x( t)x( t + Δt) = ax( t + Δt)x( t) + εx( t) r( Δt) = a r( Δt) + or by induction ( ) r( Δt) = r( Δt) r( nδt) = r n ( Δt) So for a red noise tie series, the autocorrelation at a lag of n tie steps is equal to the autocorrelation at one lag, raised to the power n. A function that has this property is the exponential function, e nx = e x function for red noise has an exponential shape. ( ) n, so we ay hypothesize that the autocorrelation r( nδt) = exp{ nδt T}. Copyright 4 Dennis L. Hartann /4/4 4:3 PM 3

4 ATM 55 otes: Tie Series Analysis - Section 6a Page 3 or if τ = nδt, The autocorrelation function for red noise. r( τ ) = exp( τ T) where T = -Δt ln a (6.4) τ/t Figure 6. The autocorrelation function of an AR- or red noise process, is one at lag zero and decays exponentially away to zero with an e-folding tie of T. In suary, if we are given a red noise tie series, or Auto-Regressive - (AR-), process, x( t) = ax( t Δt) + ( a ) / ε(t) (6.5) then its autocorrelation function is, r( τ ) = exp( τ T) (6.6) where the autocorrelation e-folding decay tie is given by, T = -Δt ln a Copyright 4 Dennis L. Hartann /4/4 4:3 PM 3

5 ATM 55 otes: Tie Series Analysis - Section 6a Page 3 Aplitude Aplitude Aplitude Aplitude Aplitude Aplitude - Periodic Tie - Two Periods Tie - White oise Tie Tie Red oise (.8) + Period Tie Red oise - Sawtooth Wave Tie Tie Lag Figure 6.3 Coparison of tie series (left) and autocorrelation functions for various cases. Correlation Correlation Correlation Correlation Correlation Correlation.5 Copyright 4 Dennis L. Hartann /4/4 4:3 PM Periodic Two Periods Tie Lag White oise Tie Lag Red oise (.8) Tie Lag Red oise (.8) + Period Tie Lag Sawtooth Wave Tie Lag

6 ATM 55 otes: Tie Series Analysis - Section 6a Page Statistical Prediction and Red oise Consider a prediction equation of the for x ˆ ( t + Δt) = a' x( t) + b' x( t Δt) (6.7) where x =. a' and b' are chosen to iniize the rs error on dependent data. Recall fro our discussion of ultiple regression that for two predictors x and x used to predict y r( x, y) r( x,y)r( x,x ) In the case where the equality holds, r(x,y) is equal to the iniu useful correlation discussed in Chapter 3 and will not iprove the forecasting skill beyond the level possible by using x alone. In the case of trying to predict future values fro prior ties, r(x,y) = r(δt), and r(x,y) = r(x,x ) = r(δt) so that we ust have r( Δt) > r( Δt) in order to justify using a second predictor at two tie steps in the past. ote that for red noise r( Δt) = r( Δt) so that the value at two lags previous to now always contributes exactly the iniu useful, and nearly autoatic, correlation, and there is no point in using a second predictor if the variable we are trying to predict is red noise. All we can use productively is the present value and the autocorrelation function, x( t + Δt) = x( t ) with an R = a = r( Δt) This is just what is called a persistence forecast, we assue toorrow will be like today White oise In the special case r(δt) = a =, our tie series is a series of rando nubers, uncorrelated in tie so that r(τ) = δ() a delta function. For such a white noise tie series, even the present value is of no help in projecting into the future. The probability density function we use is generally norally distributed about zero ean, and this is generated by the randn function in Matlab. Copyright 4 Dennis L. Hartann /4/4 4:3 PM 33

7 ATM 55 otes: Tie Series Analysis - Section 6a Page Degrees of Freedo/Independent Saples Leith [J. Appl. Meteor., 973, p. 66] has argued that for a tie series of red noise, the nuber of independent saples * is given by * = Δt T = total length of record two ties e - folding tie of autocorrelation (6.9) where is the nuber of data points in the tie series, Δt is the tie interval between data points and T is the tie interval over which the autocorrelation drops to /e. In other words, the nuber of degrees of freedo we have is only half of the nuber of e-folding ties of data we have. The ore autocorrelated our data is in tie, the fewer degrees of freedo we get fro each observation. For Red oise: thus e.g., for τ = Δt ( ) = e τ T ln r( τ ) r τ T = -Δt/ln[r(Δt)], so that T = τ ln( r( τ )) ( ) = τ T * = ln[ r ( Δt ) ] ; * (6.) Table 6. Ratio of degrees of freedo to observations (*/) for a regularly spaced tie series with one-lag autocorrelation of r(δt). r(δt) < */ Leith s forula (6.9) is consistent with Taylor(9) for the case of a red noise process. Taylor said that * = (6.) L Where L is given by, L = r(τ ') dτ ' (6.) Copyright 4 Dennis L. Hartann /4/4 4:3 PM 34

8 ATM 55 otes: Tie Series Analysis - Section 6a Page 35 If we substitute the forula for the autocorrelation function of red noise, (6.4) into 6.), then we get that L=T, and Taylor s forula is the sae as Leith s. You ay see a diensional inconsistency in (6.), but this disappears if you consider that Taylor is using tie in non-diensional units of the tie step, t =t/δt, τ =τ/δt, so that L=T/ Δt. The factor of two coes into the botto of the above expression for * so that the intervening point is not easily predictable fro the ones iediately before and after. If you divide the tie series into units of e-folding tie of the auto-correlation, T, One can show that, for a red noise process, the value at a idpoint, which is separated fro its two adjacent points by the tie period T, can be predicted fro the two adjoining values with cobined correlation coefficient of about e -, or about.5, so about 5% of the variance can be explained at that point, and at all other intervening points ore can be explained. This ay see a bit conservative. Indeed, Bretherton et al, (999) show that, assuing that one is looking at quadratic statistics, such as variance and covariance analysis between two variables x and x, and using Gaussian red noise as a odel then a good approxiation to use is: * ( ) ( ) = r (Δt)r (Δt) + r (Δt)r (Δt) (6.3a) where, of course, if we are covarying a variable with itself, r (Δt)r (Δt) = r(δt). This goes back as far as Bartlett(935). Of course, if the tie or space series is not Gaussian red noise, then the forula is not accurate. But it is still good practice to use it. So you can see that the Bretherton, et al. quadratic forula, which is appropriate for use in covariance probles, is ore generous than Leith s conservative forula, allowing about twice as any degrees of freedo when the autocorrelation at one lag is large. However if one is looking at a first order process, such as the calculation of a ean value, or the coputation of a trend where the exact value of the tie is know, then the forula used should be, * ( ) ( ) = r (Δt) + r (Δt) (6.3b) This looks ore like Leith s forula without the behavior near zero autocorrelation. This for goes back to at least 935. If we copare the functional dependence of */ fro Bretherton et al.(999), forulas (6.3a,b) with that of Leith/Taylor fro forula (6.) we can ake the plot below. Copyright 4 Dennis L. Hartann /4/4 4:3 PM 35

9 ATM 55 otes: Tie Series Analysis - Section 6a Page 36 Figure 6.3 Coparison of */ for Leith and Bretherton et al forulas as a function of r(δt) Degrees of Freedo: And EOFs. Estiates of degrees of freedo discussed here generally rely on a statistical odel. There is a long history that has been suarized by Bretherton et al. (999). Bretherton discusses a spatial data set of diension that is stationary on the tie interval for which it is sapled. Define a quadratic functional of soe vector variable X(t), where the vector is of length. E(t) = [X(t), X(t)] = X i (t) (6.4) The nuber of spatial degrees of freedo * is defined to be the nuber of uncorrelated rando noral variables a k, each having zero ean and the sae population variance a, for which the χ distribution for the specified functional ost closely atches the PDF of the functional of X(t). In order to approxiate this one can require that the distribution atch the observed distributions enseble ean value E and the teporal variance about this ean, χ i= var(e) = E ' = ( E E ) (6.5) Copyright 4 Dennis L. Hartann /4/4 4:3 PM 36

10 ATM 55 otes: Tie Series Analysis - Section 6a Page 37 For the χ distribution E = * a and var(e) = * a. We can then solve for the spatial degrees of freedo that atches the first two oents of the noral distribution of variance. * = E var(e) a = var(e) E (6.6) These estiates can be obtained fro the x covariance atrix of X, C xx., if X(t) is norally distributed and we know C well enough. Suppose we have the eigenvalues λ k and the standardized principle coponents z k (t) of C. We can now calculate * fro the eigenvalues in the following way. and E(t) = λ k z k (t) E = λ k (6.7) k= ( ) k= var(e) = λ k var z k (t) = λ k var z k z k k= k= = λ k z k 4 z k k= ( ) Since we are assuing that the PCs are standardized Gaussian noral variables their variance is one and their kurtosis is 3, and we have that var(e) = λ k z 4 k z k = λ k 3 = λ k (6.8) k= We can now write down an eigenvalue based estiate for the effective nuber of spatial degrees of freedo by substituting (6.7) and (6.8) into (6.6). k= k= * eff = k= k= λ k λ k ( = λ ) (6.9) λ This forula can also be written in ters of the covariance atrix fro which the eigenvalues were derived. Copyright 4 Dennis L. Hartann /4/4 4:3 PM 37

11 ATM 55 otes: Tie Series Analysis - Section 6a Page 38 * eff = i= i, j= C ii C ij = ( trc) tr(c ) (6.) The forula (6.3a) can be obtained by using the correlation function for an AR- red noise process in (6.) and truncating the expansion after one ter. In that way we can see that (6.3) requires both an assuption of Gaussian Red oise and an assuption that the one lag autocorrelation is sall in the sense that r(δt). One can also easily use (6.9) and (6.) to estiate spatial degrees of freedo in a tie series by coputing covariance atrices in tie, or a lagged covariance atrix. In this case the covariance is between the tie series and itself lagged in tie. One has to choose a suitable interval for the axiu lag. This is also called singular spectru analysis and will be discussed later. References: Bartlett, M.S. 935: Soe aspects of the tie-correlation proble in regard to tests of significance. J. Roy. Stat. Soc., 98, Bretherton, C. S., M. Widann, V. P. Dynikov, J. M. Wallace and I. Bladé, 999: The effective nuber of spatial degrees of freedo of a tie-varying field. J. Cliate,, Leith, C. E., 973: The standard error of tie-averaged estiates of cliatic eans. J. Appl. Meteorol.,, Taylor, G. I., 9: Diffusion by continuous oveent. Proc. London Math. Soc.,, 96-. Copyright 4 Dennis L. Hartann /4/4 4:3 PM 38

12 ATM 55 otes: Tie Series Analysis - Section 6a Page Verification of Forecast Models Consider a forecast odel that produces a large nuber of forecasts x f of x. The ean square (s) error is given by error = ( x x f ) (6.) The skill of the odel is related to the ratio of the s error to the variance of x about its cliatological ean. Suppose that the odel is able to reproduce cliatological statistics in the sense that so that If the odel has no skill then x f = x, x' f = x' x' x' f = ( x x f ) = ( x' x' f ) = x' x' x' f + x' f = x' This result ay see soewhat paradoxical at first. Why is it not siply x'?, why twice this? (6.) The average root ean squared difference between two randoly chosen values with the sae ean is larger by ' than that of each of these values about their coon ean. Two tie Series with r()= Tie Figure 6.4 Two rando tie series to illustrate why the standard error of a skill-less prediction is actually twice the variance of the tie series. Once the skill falls below a certain level, it is better to assue cliatology than to use a skill-less prediction. The following figure shows a plot of the rs error versus prediction tie interval τ for a hypothetical forecast odel whose skill deteriorates to zero as τ. Copyright 4 Dennis L. Hartann /4/4 4:3 PM 39

13 ATM 55 otes: Tie Series Analysis - Section 6a Page 4 rs error τ c τ Figure 6.5 RMS error for a siple forecast (solid) and a forecast that optially weights the forecast schee and cliatology. For τ >τ c the odel appears to have no skill relative to cliatology, yet it is clear that it ust still have soe skill in an absolute sense since the error has not yet leveled off. The odel can be ade to produce a forecast superior to cliatology if we use a regression equation of the for. x ˆ f = ax f + ( a)x As an exercise you can show that east-squares regression to iniize ( x x ˆ f ) yields a = x' x' f x' So we should choose a = R(τ). The ultiple correlation factor, R, for the original regression As the skill of the prediction schee approaches zero for large τ the ˆ x f forecast is weighted ore and ore heavily toward cliatology and produces an error growth like the dotted curve in the figure above. Proble: Prove that at the point where the rs error of a siple forecast x f, passes the error of cliatology (the average), where τ = τ c, a =.5, and at that point the rs error of ˆ x f equals.87 ties the rs error of x f. Copyright 4 Dennis L. Hartann /4/4 4:3 PM 4

14 ATM 55 otes: Tie Series Analysis - Section 6a Page 4 6. Haronic Analysis Haronic analysis is the interpretation of a tie or space series as a suation of contributions fro haronic functions, each with a characteristic tie or space scale. Consider that we have a set of values of y(t i ) = y i. Then we can use a least-squares procedure to find the coefficients of the following expansion y ( t ) = A + A cosπk t o k T + B sinπk t k T k = (6.4) where T = the length of the period of record. y(t) is a continuous function of t. orally on a coputer we would have discrete data and y(t) would be specified at a set of ties t = t + iδt. ote that B k = when k=/, since you cannot deterine the phase of the wave with a wavelength of two tie steps. If the data points are not evenly spaced in tie then we ust be careful. The results can be very sensitive to sall changes in y(t i ). One should test for the effects of this sensitivity by iposing sall variations in y i and be particularly careful where there are large gaps. Where the data are unevenly spaced it ay be better to eliinate the higher haronics. In this case one no longer achieves an exact fit, but the behavior ay be uch better between the data points. Copyright 4 Dennis L. Hartann /4/4 4:3 PM 4

15 ATM 55 otes: Tie Series Analysis - Section 6a Page 4 Useful Math Identities: It ay be of use to you to have this reference list of trigonoetric identities: cos(α β) = cosα cosβ + sinα sin β ; tanγ = sinγ cosγ (6.5) You can use the above two relations to show that: C cosθ + S sinθ = ( ) ; where A = C + S ; θ o = Arc tan S C A cos θ θ o (6.6) where you need to note the signs of S and C to get the phase in the correct quadrant. The coplex fors of the trig functions also coe up iportantly here. Also, fro this you can get; e iθ = cosθ + i sinθ where i = (6.7) sinθ = eiθ e iθ i ; cosθ = eiθ + e iθ (6.8) If you need ore of these, check any book of standard atheatical tables. 6.. Evenly Spaced Data Discrete Fourier Transfor On the interval < t < T chosen such that t = and t + = T where is an even nuber. The analytic functions are of the for cos( πk iδt T ), sin( πk iδt T ) (6.9) Δt is the (constant) spacing between the grid points. In the case of evenly spaced data we have: a) a o = the average of y on interval <t<t. b) The functions (predictors) are orthogonal on the interval < t < T so that the covariance atrix is diagonal and the coefficients can be deterined one at a tie (see Section 4.). Hence Copyright 4 Dennis L. Hartann /4/4 4:3 PM 4

16 ATM 55 otes: Tie Series Analysis - Section 6a Page 43 a k = x' k y' x' k (6.3) c) The functions each have a variance x' k = except for A / and B / whose variances are and respectively. These results can also be obtained by analytic integration if the data points define the sine and cosine waves exactly. Hence we derive the rather siple algebraic forulas for the coefficients: A k = y i cosπkiδ t T i= B k = y i sinπkiδt T i k =, A = y i cosπiδ t T i= or ( ) = y + A k cos πk t T y t k = a o = y i b o = + B k sin πk t T + A cos π t T (6.3) (6.3) or, alternatively y( t) = y + C k cos πk T k = ( ) t t k C k = Ak + Bk and tk = + A πt cos T T πk B tan k A k (6.33) Of course, norally these forulas would be obtained analytically using the a priori inforation that equally spaced data on a finite interval can be used to exactly calculate Copyright 4 Dennis L. Hartann /4/4 4:3 PM 43

17 ATM 55 otes: Tie Series Analysis - Section 6a Page 44 the Fourier representation on that interval (assuing cyclic continuation ad infinitu in both directions). The fraction of the variance explained by a particular function is given by r ( y, x k ) = x' k y' x' k y' = A k + B k y' A y' for k =,for k =,,... (6.34) The variance explained by a particular k is C k for k =,,... ; A for k = (6.35) 6.. The Power Spectru The plot of C k vs. k is called the power spectru of y(t) - the frequency spectru if t represents tie and the wavenuber spectru if t represents distance. Strictly speaking C k represents a line spectru since it is defined only for integral values of k, which correspond to particular frequencies or wavenubers. If we are sapling a finite data record fro a larger tie series, then this line spectru has serious drawbacks.. Integral values of k do not have any special significance, but are siply deterined by the length of the data record T, which is usually chosen on the basis of what is available, and is an iportant design paraeter of the analysis. The frequencies that are resolved are a direct result of the length of the tie series chosen for Fourier Transfor. ω k = πk T, k =,,,3,4,... /. The individual spectral lines each contain only about degrees of freedo, since data points were used to deterine a ean, / aplitudes and / - phases (a ean and / variances). Hence, assuing that a reasonable aount of noise is present, a line spectru ay (should) have very poor reproducibility fro one finite sapling interval to another; even if the series is stationary (i.e., its true properties do not change in tie). To obtain reproducible, statistically significant results we need to obtain spectral estiates with any degrees of freedo. Copyright 4 Dennis L. Hartann /4/4 4:3 PM 44

18 ATM 55 otes: Tie Series Analysis - Section 6a Page 45 The nuber of degrees of freedo for each spectral estiate is just twice the nuber of realizations of the spectru that we average together. 3. With the notable exceptions of the annual and diurnal cycles and their higher haronics, ost interesting signals in geophysical data are not truly periodic but only quasi-periodic in character, and are thus better represented by spectral bands of finite width, rather than by spectral lines. Continuous Power Spectru: Φ(k) All of the above considerations suggest the utility of a continuous power spectru which represents the variance of y(t) per unit frequency (or wavenuber) interval such that y' k* = Φ( k) dk (6.36) So that the variance contributed is equal to the area under the curve Φ(k), as shown below. Φ k k k k k* Figure 6.6 Hypothetical continuous power spectru as a function of frequency or wavenuber index, k. k* corresponds to one cycle per Δt, the highest frequency in y(t) that can be resolved with the given spacing of the data points. This k* is called the yquist frequency. If higher frequencies are present in the data set they will be aliased into lower frequencies. k k*. This is a proble when there is a coparatively large aount of variance beyond k *, or at frequencies greater the yquist frequency. Copyright 4 Dennis L. Hartann /4/4 4:3 PM 45

19 ATM 55 otes: Tie Series Analysis - Section 6a Page 46 Δt Δt t true period t = /3 Δt aliased period t =.Δt true wavenuber aliased wavenuber k = 3.k * k =.k * Figure 6.7 A scheatic showing how a wave with a period of /3 Δt, will be aliased into a variance at a period of Δt. Degrees of Freedo Resolution Tradeoff: For a fixed length of record, we ust balance the nuber of degrees of freedo for each spectral estiate against the resolution of our spectru. We increase the degrees of freedo by increasing the bandwidth of our estiates. Soothing the spectru eans that we have fewer independent estiates but greater statistical confidence in the estiate we retain. High resolution Lower resolution High Inforation Sooth/Average Lower Inforation Low Quality High quality in a statistical sense "Always" insist on adequate quality or you could ake a fool of yourself. The nuber of degrees of freedo per spectral estiate is given by /M* where M* is the nuber of independent spectral estiates and is the actual nuber of data points y i (t) regardless of what the autocorrelation is. As long as we use a red-noise fit to the spectru as our null hypothesis, we don t need to reduce the nuber of degrees of freedo to account for autocorrelation, since we are testing whether the spectrue deviates fro a siple red noise spectru which is copletely defined by the autocorrelation at one lag and the total variance of the tie series. Copyright 4 Dennis L. Hartann /4/4 4:3 PM 46

20 ATM 55 otes: Tie Series Analysis - Section 6a Page Methods of Coputing Power Spectra Direct Method: The direct ethod consists of siply perforing a Fourier transfor or regression haronic analysis of y i (t) to obtain C k. This has becoe econoical because of the Fast Fourier Transfor (FFT). Because the transfor assues cyclic continuity, it is desirable to "taper" the ends of the tie series y i (t), as will be discussed in section When we do a Fourier analysis we get estiates of the power spectru at / frequencies, but each spectral estiate has only two degrees of freedo. A spectru with so few degrees of freedo is unlikely to be reproducible, so we want to find ways to increase the reliability of each spectral estiate, which is equivalent to a search for ways to increase the nuber of degrees of freedo of each estiate. How to obtain ore degrees of freedo: a.) Average adjacent spectral estiates together. Suppose we have a 9 day record. If we do a Fourier analysis then the bandwidth will be /9 day -, and each of the 45 spectral estiates will have degrees of freedo. If we averaged each adjacent estiates together, then the bandwidth will be /9 day - and each estiate will have d.o.f. In this case we would replace the value of the power at the central frequency f i, with an average over the band centered on f i. The frequencies represented are separated by the bandwidth of the spectral analysis, Δf. We would replace P( f i ) by P( f i ), defined thusly. n P( f i ) = P( f i + nδf ) n + (6.37) n This spectru, thus soothed, now has (n+) degrees of freedo, rather than degrees of freedo. The bandwidth of this new spectru is Δf (n +), which eans that the effective frequency resolution has been degraded by a factor of (n +). Ideally, we would like the bandwidth to be narrow, with the spectral estiates closely spaced in frequency, but in this case we have soothed the spectru to get ore degrees of freedo. Copyright 4 Dennis L. Hartann /4/4 4:3 PM 47

21 ATM 55 otes: Tie Series Analysis - Section 6a Page 48 b.) Average realizations of the spectra together. Suppose we have tie series of 9 days. If we copute spectra for each of these and then average the individual spectral estiates for each frequency over the saple of spectra, then we can derive a spectru with a bandwidth of /9 days - where each spectral estiate has degrees of freedo. In this case we leave the resolution of the spectru unchanged and averaged together realizations, rather than adjacent frequencies. So if we have a set of spectral estiates P i ( f ), each with bandwidth Δf, and that we have of these. Then we copute the averaged spectru, P( f i ) = P i ( f ) (6.38) i= ow the bandwidth is unchanged, but the averaged spectru is one spectru with degrees of freedo per spectral estiate, rather than spectra, each with degrees of freedo. Get it? So how do we estiate the degrees of freedo in the direct FFT ethod? If we have data points fro which we copute the Fourier transfor and subsequent spectru, then the resulting spectru provides a variance at / frequencies and we have two degrees of freedo per spectral estiate. If we sooth the spectru, then we ust estiate the effect of this soothing on the degrees of freedo, which would be increased. If we average two adjacent spectral estiates together, then we could assue that the nuber of degrees of freedo are doubled. In general, a forula would be d.o.f = M * (6.39) Where is the total nuber of data points used to copute the spectru estiate, and M* is the total nuber of degrees of freedo in the spectru. For exaple, if you used 4 data points to estiate a spectru with 64 independent spectral estiates, then the nuber of degrees of freedo would be 4/64 = 6. Later we will describe a ethod in which we take soe data record of length and break it up into chunks of length M ch. These chunks are chosen to ake the coputation of the bandwidth we want efficient. Then we average together approxiately / M ch of these spectra, giving us about / M ch degrees of freedo. Do you understand where the coes fro? Because we use M ch data points to produce M ch / spectral estiates, each estiate gets two degrees of freedo. We throw away the phase inforation, so each power estiate at each frequency uses two pieces of data. Copyright 4 Dennis L. Hartann /4/4 4:3 PM 48

22 ATM 55 otes: Tie Series Analysis - Section 6a Page 49 Table 6. A table that illustrates the relationship between the chunk length, T, the tie step Δt, the saple size, and the approxiate degrees of freedo for a case with a total saple of 5 days, with tie steps of or ½ days and a chunk length of 8 days. Tie chunk = T 8 days 8 days Tie step = Δt day ½ day Tie Steps in chunk = M ch 8 56 Bandwidth = Δf=/T /8 /8 yquist Frequency = / Δt / / uber of Spectral Estiates = M ch / 64 8 Saples in 5 days = 5 4 Degrees of Freedo ~ / M ch / 8 8 Table 6. illustrates that if you have a finite record of 5 days, which you divide into 8 day segents, the degrees of freedo per spectral estiate does not change when you halve the tie step Δt, and the spacing between frequencies does not change. All that happens is that you resolve double the nuber of frequencies and all of the new ones are at higher frequencies than the original yquist frequency. You double the size of the yquist interval by adding new frequencies at higher frequencies without changing the original set obtained with twice the Δt. Lag Correlation Method: According to a theore by orbert Wiener, that we will illustrate below, the autocovariance (or autocorrelation, if we noralize) and the power spectru are Fourier transfors of each other. So we can obtain the power spectru by perforing haronic analysis on the lag correlation function on the interval -T L τ T L. The resulting spectru can be soothed, or the nuber of lags can be chosen to achieve the desired frequency resolution. The Fourier transfor pair of the continuous spectru and the continuous lag correlation are shown below. Copyright 4 Dennis L. Hartann /4/4 4:3 PM 49

23 ATM 55 otes: Tie Series Analysis - Section 6a Page 5 T L Φ( k) = r( τ) e ikτ dτ T L r( τ ) = π k * k * Φ( k) e ikτ dk (6.47) The axiu nuber of lags L deterines the bandwidth of the spectru and the nuber of degrees of freedo associated with each one. The bandwidth is cycle/t L, and frequencies, cycle/t L, /T L, 3/T L,..., / t. There are Δt T = L Δt T L (6.48) of these estiates. Each with T L Δt = L degrees of freedo. (6.49) The lag correlation ethod is rarely used nowadays, because Fast Fourier Transfor algoriths are ore efficient and widespread. The lag correlation ethod is iportant for intellectual and historical reasons, and because it coes up again if you undertake higher order spectral analysis. Copyright 4 Dennis L. Hartann /4/4 4:3 PM 5