The permutation test as a non-parametric method for testing the statistical signi cance of power spectrum estimation in cyclostratigraphic research

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1 Earth and Planetary Science Letters 181 (2000) 175^189 The permutation test as a non-parametric method for testing the statistical signi cance of power spectrum estimation in cyclostratigraphic research Eulogio Pardo-Igüzquiza a, Francisco J. Rodr guez-tovar b; * b a Department of Mining and Mineral Engineering, University of Leeds, Leeds LS2 9JT, UK Departamento de Estratigraf a y Paleontolog a, Facultad de Ciencias, Universidad de Granada, Granada, Spain Received 25 February 2000; accepted 13 June 2000 Abstract A computer-intensive significance test for estimated power spectra of cyclic sedimentary successions is presented. This simple method requires no more than a few minutes in computer time for a PC-486, and does not require distributional assumptions. It is suitable for all the spectral analysis approaches used in practice. Moreover, good performance is achieved with relatively short stratigraphical series. The method is similar to a permutation test that has been successfully applied to other statistical problems. In the proposed application of the permutation test to the spectral analysis of time series, the data of a stratigraphic sequence are ordered at random (random permutation) and the power spectrum is estimated by the given approach. The process is repeated many times (e.g times) and thus it is possible to assess the statistical significance of the power spectrum of the original sequence for each frequency. Simulation results and the application to real data are shown in order to discuss the performance of the method. ß 2000 Elsevier Science B.V. All rights reserved. Keywords: cyclostratigraphy; statistical analysis; time-series analysis 1. Introduction In the last two decades, numerous studies have focused on the analysis of rhythmic sedimentary successions, in an e ort to recognise the possible cyclic in uence of astronomical parameters on the Earth's climate [1^7]. From the rst papers, a wide variety of spectral analysis techniques have been used for the detection of cyclical components * Corresponding author. fjrtovar@goliat.ugr.es in stratigraphic sequences [8^10]. However, one of the most common handicaps in the spectral analysis of rhythmic stratigraphic successions is to discern signi cant periodicities from a noisy background. Usually, statistical tests are used with this approach. However, statistical tests of the estimated power spectrum are not easily available for all the estimators, and, when available, they depend on distributional assumptions or on asymptotic results which are not very well suited to short stratigraphical series. Thus, the aim of this paper is to present a computer-intensive signi cance test of general use in estimating the power spectrum of cyclic sedimentary successions X / 00 / $ ^ see front matter ß 2000 Elsevier Science B.V. All rights reserved. PII: S X(00)

2 176 E. Pardo-Igüzquiza, F.J. Rodr guez-tovar / Earth and Planetary Science Letters 181 (2000) 175^189 A simulation case study and the application of this method to a series of calcareous nannofossil, using the periodogram, and the maximum entropy estimator are presented. 2. Standard statistical tests A complete explanation of standard tests for the power spectrum is found in Brockwell and Davis [11], while such methods as the multitaper estimator [12,13] have their own statistical test. We include here only a very brief discussion of a method which is often used in cyclostratigraphy [5,14,15]. In cyclostratigraphy, the successions studied are comparatively short and have normally been a ected by di erent processes such as tectonics, diagenesis, hiatus, variable sedimentation rates, etc., which can make it di cult to calibrate the absolute time represented along the sequence. This contrasts with other disciplines, such as electrical engineering or geophysics, where long time series with very precise measurements are often the norm. However, even with the limitations of the stratigraphic sequences, the signature of a cyclical process is frequently detected by the spectral analysis of the succession. Furthermore, together with the recognition of peaks at given frequencies in the estimated power spectrum, in cyclostratigraphic research, it is of special interest to assess the statistical signi cance of these peaks. Statistical theory provides estimators of the power spectrum P(f k ) and con dence intervals for the estimators. For example, it is well known that the periodogram I(f k ) [16] is an unbiased estimate of the power spectrum [17]: E I f k Š ˆ P f k and with variance: var I f k Š ˆ P 2 f k where f k denotes the k-th frequency. Also, if the data are Gaussian: 2WI f k P f k WM where M n 2 is a M-squared random variable with n degrees of freedom and V denotes that it is distributed asymptotically. Then, the 100(13K)% con dence intervals for I(f k ) are given by: 2WI f k M 2 9P f k 9 2WI f k 2;K =2 M 2 2;13K =2 4 where K is the con dence level. Con dence intervals are closely related to hypothesis testing. For example the null hypothesis: H 0 : fthere are no significant cyclical components in the seriesg may be tested by the method proposed by Schwarzacher [18]: in a logarithmic plot of the estimated power spectrum, a smooth line (i.e. without any peak) is tted as the mean power spectrum (null hypothesis H 0 ), which is the assumed form for P(f k ). Then, applying Eq. 4, it is possible to draw the 100(13K)% con dence bands and to check visually which frequencies have a signi cant value of the power spectrum estimated by I(f k ). However, the main drawback of this hypothesis test is that the result depends heavily on which estimate is adopted for representing P(f k ), the null hypothesis [14]. Because of that disadvantage, a non-parametric computer-intensive test may be used as a complement. The signi cance level K is also known as the probability of the type I error (i.e. the probability of rejecting the null hypothesis when the null hypothesis is true, or, in other words, the probability of accepting a peak as signi cant when it in fact is not). If K is increased, we increase the probability of assigning signi cance to spurious peaks; and if K is reduced, we increase the probability of assigning non-signi cance to peaks that are really cyclical components. This fact is inherent with any test for a statistical hypothesis. In practice the signi cance level is usually xed at 0.05 or In testing the signi cance of the power spectrum, we perform (n) simultaneous hypothesis

3 E. Pardo-Igüzquiza, F.J. Rodr guez-tovar / Earth and Planetary Science Letters 181 (2000) 175^ tests where (n) is the number of frequencies where the power spectrum has been estimated, and thus is it more appropriate to choose 0.01 as the signi cance level, while for single hypothesis testing the 0.05 signi cance level is generally used. 3. Method: the computer-intensive permutation test The permutation test, suggested by R.A. Fisher in the 1930s [19], gained applicability from the wide availability of powerful computers. The basic idea is that if the series contains some cyclic components, a reordering at random (random permutation) of the elements in the sequence will destroy the cyclic pattern of the sequence. If the random permutation is repeated a large number of times, it is possible to evaluate the chances of nding the power spectrum of the original series at random. After this, for each frequency, a range of possible values is established and the probability of observing a value larger than the power spectrum of the original ordering is calculated. The smaller the value of that probability, the stronger the evidence becomes against the null hypothesis, which is that the value of the peak is not signi cant Procedure and perspectives The steps required by the algorithm are: 1. Given the original sequence: fz t ;tˆ0; T; N31g the power spectrum is estimated for example by the periodogram: fi f k ;kˆ 0; T; N=2g 2. From the original sequence a random permutation is obtained: fz t ;tˆ0; T; N31g 3. The power spectrum is estimated for the new sequence: fi f k ;kˆ 0; T; N=2g 4. Steps (2) and (3) are repeated a large number of times M (e.g. 1000). 5. For each frequency k, there are M values {I*(f k )} and the signi cance of the estimate {I(f k )} of the original sequence may be assessed by the achieved signi cance level (ASL) of the test [19]. The ASL of a given estimate {I(f k )} is de ned as the probability of nding at least as large a value when the null hypothesis is true (i.e. there is no signi cant contribution at that frequency that gives a signi cant cyclic component): ASL f k ˆProbfI f k v I f k g 5 The smaller the value of the ASL, the stronger the evidence against the null hypothesis. It should be noted that in Eq. 5, the value I(f k ) is xed (for a given frequency), while I*(f k )isa random variable with values provided by the power spectra estimated from the series given by random permutation. 6. A small value of the signi cance level K is chosen (like 0.01) and the peaks for which the ASL is less than K are considered signi cant with a con dence of 100(13K)%. The ASL is estimated in practice by: ASL f k ˆfnumber of timesw I f k vi f k Šg=M 6 One important characteristic of permutation testing is its accuracy, because, if the null hypothesis is true, then [19]: ProbfASL6K gˆk 7 for any value of K between 0 and 1. It may be noted that each random permutation is not a bootstrap sample because the resampling is not done with replacement.

4 178 E. Pardo-Igüzquiza, F.J. Rodr guez-tovar / Earth and Planetary Science Letters 181 (2000) 175^189 Interpretations on the existence of any possible cyclicity in rhythmic successions have been conventionally approached by spectral analysis of stratigraphic series. However, until now, to analyse the signi cance of these power spectra, very dependent methods have been used. The application of the methodology explained here provides a new tool that can be of great use to test the signi cance of peaks recognised in the power spectrum. Some advantages of the permutation test for testing the signi cance of the power spectrum are: b b b it may be applied with any power spectrum estimator: periodogram, Blackman and Tukey approach, maximum entropy, autoregressive spectral estimator, etc.; it does not require any distributional assumption; it may be applied with short as well as long time series. The permutation test could be useful in at least three di erent circumstances: b b b Many examples of orbitally induced cyclicity are so clear that spectral analysis only serves as additional proof. In such cases the permutation test is useful to eliminate any doubt. In cases in which orbital cyclicity is not obvious, a misuse of various spectral techniques and ltering may provide a curve with an accidental peak that is more or less similar to an `expected' astronomical technique. The permutation test will check if a real periodicity is present. Testing the presence and stability of the periodicities. The permutation test could be applied to subspectra or to the series after the removal of the rst or last 10% or 25% of the data. In case of a stable periodicity throughout the succession, spectra should be fairly similar. The permutation test should provide evidence of the signi cance of the peaks in these cases too. The implicit model assumed by the method is white noise plus sinusoidal cyclic components. If a trend is suspected, some detrending method should be applied rst. If red noise is suspected, a moving window spectral analysis could be performed. 4. Examples of application 4.1. Simulation case study The rst example is the application of the permutation test to the estimated periodogram of a simulated time series with a white noise stochastic Table 1 Number of signi cant values in the permutation test for simulated white noise for di erent sample sizes and number of permutations White noise model Length of the series Number of random permutations Number of signi cant values K = 0.05 Number of signi cant values K = 0.01 Gaussian (3) 0 (1) Gaussian (6) 0 (1) Gaussian (13) 3 (3) Gaussian (3) 0 (1) Gaussian (6) 0 (1) Gaussian (13) 2 (3) Uniform (3) 1 (1) Uniform (6) 0 (1) Uniform (13) 1 (3) Uniform (3) 1 (1) Uniform (6) 0 (1) Uniform (13) 3 (3) In the last two columns, the number in parentheses indicates the expected number of signi cant values that is equal to (N/2)K where, N is the length of the series (the number of data of the series).

5 E. Pardo-Igüzquiza, F.J. Rodr guez-tovar / Earth and Planetary Science Letters 181 (2000) 175^ Fig. 1. (A) Periodogram of the uniform white noise time series (256 data), and (B) results of the permutation test in the form SL = (1.03ASL(f k ))100%; for example, in the gure, the values crossing SL = 95% (dashed line) are signi cant with a signi cance level of K = model. Two kinds of white noise have been used: uniform white noise and Gaussian white noise, with three di erent numbers of data for the series: 128, 256 and 512. The permutation test has been repeated with 1000 and 5000 random permutations. Given a times series of length N, the periodogram is estimated at N/2+1 frequencies, but, because the series are zero mean or transformed to be zero mean, the periodogram for frequency zero is always zero I(0) = 0.0, and the signi cance of this frequency is not considered. In this case, a signi cance test of the estimated power spectrum implies performing the N/2 independent test for N/2 periodogram estimates simultaneously. Thus, accepting a signi cance level of 0.05, the number of signi cant values explained by randomness is

6 180 E. Pardo-Igüzquiza, F.J. Rodr guez-tovar / Earth and Planetary Science Letters 181 (2000) 175^189 Fig. 2. (A) Periodogram of the Gaussian white noise time series (256 data), and (B) results of the permutation test in the form SL = (1.03ASL(f k ))100%; for example, in the gure, the values crossing SL = 95% (dashed line) are signi cant with a signi cance level of K = approximately three, six and 13 for a sample size of 128, 256 and 512, or one, one and three accepting a signi cance level of The results for the simulation example are given in Table 1. The table shows that the number of signi cant values agrees quite well with the expected number, especially for Gaussian white noise and for the three sample sizes, as well as with uniform white noise for sample sizes of 256 and 512. With the short time series (128 values) of uniform white noise and with signi cance level of 0.05, six signi cant values are found, whereas three were expected. With 1000 random permutations results are good, while increasing the number of random per-

7 E. Pardo-Igüzquiza, F.J. Rodr guez-tovar / Earth and Planetary Science Letters 181 (2000) 175^ Fig. 3. (A) Periodogram of the two sinusoidal components and Gaussian white noise (with variance equal to 1.0) time series (256 data), and (B) results of the permutation test in the form SL = (1.03ASL(f k ))100%; for example, in the gure, the values crossing SL = 95% (dashed line) are signi cant with a signi cance level of K = mutations to 5000 o ers little improvement. In any case, at least 1000 permutations should be used and if the computer is e cient enough this number should be increased to Figs. 1 and 2 show the periodogram and the signi cance level of the di erent periodogram values for the uniform white noise and Gaussian white noise, respectively. The results of the permutation test are expressed in the form SL = (1.03ASL(f k ))100%, where ASL(f k ) is the ASL given by Eq. 6. Thus, for example, signi cant values with signi cance level of 0.05 are the values with SL s 95%, as shown in Figs. 1B and 2B. The second simulation example consists of three time series with 256 data each, which were generated as two sinusoidal components plus

8 182 E. Pardo-Igüzquiza, F.J. Rodr guez-tovar / Earth and Planetary Science Letters 181 (2000) 175^189 Fig. 4. (A) Periodogram of the two sinusoidal components and Gaussian white noise (with variance equal to 3.0) time series (256 data), and (B) results of the permutation test in the form SL = (1.03ASL(f k ))100%; for example, in the gure, the values crossing SL = 95% (dashed line) are signi cant with a signi cance level of K = white noise: x t ˆw t X2 kˆ1 A k Wcos 2ZWf k Wt P k Š 8 with amplitudes A 1 =1, A 2 = 0.75, phases P 1 =0, P 2 = Z and frequencies f 1 = , f 2 = 0.20 cycles per unit of distance, which correspond to wavelengths of 16 units and 5 units of distance, respectively. The di erence between the three series is the level of the noise given by the variance of the Gaussian white noise w(t) added to the sinusoidal components, which are 1.0, 3.0 and 8.0, respectively. The periodogram and the results of the permutation test are expressed in the form SL =

9 E. Pardo-Igüzquiza, F.J. Rodr guez-tovar / Earth and Planetary Science Letters 181 (2000) 175^ Fig. 5. (A) Periodogram of the two sinusoidal components and Gaussian white noise (with variance equal to 8.0) time series (256 data), and (B) results of the permutation test in the form SL = (1.03ASL(f k ))100%; for example, in the gure, the values crossing SL = 95% (dashed line) are signi cant with a signi cance level of K = (1.03ASL(f k ))100%, and given in Figs. 3, 4 and 5 for the three di erent series, respectively. For the periodogram (Figs. 3A, 4A and 5A), it is clear that, as the level of noise increases, the peaks from the background noise (spurious peaks) become more relevant in relation to the peaks for the cyclic components. The number of signi cant peaks with a signi cance level of 0.05 is four, seven and seven for the three time series, respectively, and with 1000 random permutations in all the cases (four, six, seven signi cant peaks for the three series, respectively, using 5000 random permutations). For the periodogram, without any kind of smoothing, a signi cance level of 95% is not high enough because we are simultaneously performing a number of hypothesis tests equal

10 184 E. Pardo-Igüzquiza, F.J. Rodr guez-tovar / Earth and Planetary Science Letters 181 (2000) 175^189 to the number of frequencies for which the periodogram is evaluated. With a signi cance level of 0.01 (99% con dence in the right decision), the number of signi cant peaks is two, three, two for the three previous time series and for both cases, using 1000 and 5000 random permutations. For the rst time series, with the highest signal to noise ratio, the two estimated components are the true ( and 0.2) with an ASL of 0.0 and 0.0, respectively. For the second time series, the three frequencies are , and 0.2 with an ASL of 0.0, and 0.0, respectively. For the third time series, the two signi cant components with a signi cance level of 0.01 appear at frequencies of and with an ASL of and 0.002, respectively. The component at frequency 0.2 is not signi cant at 99% signi cance level, but it is signi cant at 95% signi cance level, and, from all of the components which are signi cant at 95% signi cance level, but not at 99%, the 0.2 component has the lowest ASL (equal to 0.016). The results are very similar using 1000 or 5000 random permutations Application to real data The experimental data used are the percentages of the nannofossil Watznaueria barnesae, measured on samples from a core from the Albian Gault Clay Formation in southern England (original data in table 1 from Erba et al. [15]). This species is regarded as a non-fertility index, and cyclic variations of their percentage are related to Milankovitch cycles [15]. Thus, the spectral analysis ( gure 7C in [15]) shows peaks at periodicities of yr and yr at a 95% con- dence level. Other spectral peaks over the 95% con dence level were not reported by Erba et al. [15] because their periods are not coincident with Milankovitch cycles. The result of the signi cance test also depends on the estimation of the average underlying spectrum (solid line in gure 7C of [15]). The same problem with other geological series has been reported in Berger et al. [14]. The spectral analysis is revisited and the permutation test is applied to the power spectra estimated by the periodogram and the maximum entropy methods. The original succession used is composed of 135 data, most of them on a regular spacing of 0.05 m, measured on a core 6.25 m long between and 18.5 m depth [15]. In order to have a succession with a constant sampling interval of 0.05 m, interpolating splines have been applied. Thus, 126 Fig. 6. Series of percentages of the calcareous nannofossil W. barnesae [15] at a sampling interval of 0.05 m along a 6.25 m long core.

11 E. Pardo-Igüzquiza, F.J. Rodr guez-tovar / Earth and Planetary Science Letters 181 (2000) 175^ Fig. 7. (A) Experimental periodogram of the succession showed in Fig. 6, and (B) results of the permutation test in the form SL = (1.03ASL(f k ))100%; only SL s 70% has been represented, and the SL of 95 and 99% have been highlighted with slashed lines. measurements from to 18.5 at a sampling rate of 0.05 m were obtained; 118 were identical to the original ones and eight were obtained by the interpolating splines. The sequence with a constant sampling interval (Fig. 6) showed no apparent trend, and therefore only the substraction of the mean was performed before applying the spectral analysis estimators Periodogram The estimated periodogram is shown in Fig. 7A. The ASL from the permutation test shows

12 186 E. Pardo-Igüzquiza, F.J. Rodr guez-tovar / Earth and Planetary Science Letters 181 (2000) 175^189 that frequencies signi cant at a 99% con dence level are only and , corresponding to periods of yr and yr (Fig. 7B). At a signi cance level of 90%, only one additional frequency appeared at , which corresponds to a period of yr (Fig. 7B). The periods were calculated using the sedimentation rate of 5 cm every yr, as used in Erba et al. [15] Maximum entropy estimator Even though for the maximum entropy spectral estimator [20] it is not easy to derive the standard hypothesis test [14], the permutation test can still Fig. 8. (A) Estimated power spectrum of the series showed in Fig. 6 by the method of maximum entropy with L = 30, and (B) results of the permutation test in the form SL = (1.03ASL(f k ))100%; only SL s 70% has been represented, and the SL of 95 and 99% have been highlighted with slashed lines.

13 E. Pardo-Igüzquiza, F.J. Rodr guez-tovar / Earth and Planetary Science Letters 181 (2000) 175^ be applied. The maximum entropy estimates are a function of a parameter L [20] that must be speci- ed when the estimator is applied. A low L value decreases the variance but increases the bias of the estimator, and the reverse happens if L is increased. With a small L value, few periodicities are detected, with a high L value, a large number of spectral peaks appear, most of them spurious. The power spectra estimated by maximum entropy with L = 30 and L = 50 are shown in Figs. 8A and 9A, respectively. With L = 30, the frequencies ( yr) and ( yr) are signi cant at 99% and 95% con dence levels, respectively (Fig. 8B). No other frequencies are sig- Fig. 9. (A) Estimated power spectrum of the series showed in Fig. 6 by the method of maximum entropy with L = 50 (B), and (B) results of the permutation test in the form SL = (1.03ASL(f k ))100%; only SL s 70% has been represented, and the SL of 95 and 99% have been highlighted with slashed lines.

14 188 E. Pardo-Igüzquiza, F.J. Rodr guez-tovar / Earth and Planetary Science Letters 181 (2000) 175^189 ni cant at a 90% con dence level (Fig. 8B). With L = 50, the frequency ( yr) is signi cant at 99% con dence level and the frequency ( yr) is signi cant at 95% con dence level (Fig. 9B). At a 90% con dence level, only one additional frequency at ( yr) appears as signi cant (Fig. 9B). This illustrates how the permutation test gives basically the same results for the periodogram and the maximum entropy methods, even for di erent values of the parameter L in the maximum entropy estimator [10]. 5. Conclusions The permutation test is revealed as a non-parametric method for testing the statistical signi cance of the estimated power spectrum that can be used complementarily with other methodologies. The test directly assesses the ASL of the estimate for each frequency, and while it does not require statistical assumptions, it is computer-intensive. Implicit in the method is the model of white noise plus hidden sinusoidal signals. However, with 1000 random permutations, good results are provided for relatively short series (between 128 and 512 data), requiring no more than a few minutes in computer time for a PC-486. Nevertheless, if time allows, it is advisable to increase the number of permutations to Although only the periodogram without smoothing together with the maximum entropy estimator have been used in this paper, the method may be applied generally to other spectral estimators. In cyclostratigraphic studies, the importance of the spectral peaks that have been shown to be statistically signi cant by the permutation test should be corroborated by other information, such as the structure of the correlogram and the physical meaning of the periodicities. Acknowledgements We would like to thank the reviewers by providing constructive criticism. The work of P.-I. was supported by a European Community Training Fellowship in the Department of Meteorology (Reading University, UK) as part of the Training and Mobility of Researches (TMR) program. The research of R.-T. was done under the program and nancial support of the EMMI Group (RNM-178 Junta de Andaluc a, Spain) and project PB (DGICYT).[AC] References [1] G. Einsele and A. Seilacher, Cyclic and Event Strati cation, Springer-Verlag, Berlin, 1982, 536 pp. [2] A. Berger, J. Imbrie, J. Hays, G. Kukla and B. Saltzman, Milankovitch and Climate, NATO ASI Ser. 126, Reidel Publ. Company, Dordrecht, 1984, 895 pp. [3] G. Einsele, W. Ricken and A. Seilacher, Cycles and Events in Stratigraphy, Springer-Verlag, Berlin, 1991, 955 pp. [4] A.G. Fischer, D.J. Bottjer, Orbital forcing and sedimentary sequences, J. Sediment. Petrol. Spec. Iss. 61 (7) (1991) 1063^1270. [5] W. Schwarzacher, Cyclostratigraphy and the Milankovitch Theory, Developments in Sedimentology 52, Elsevier, Amsterdam, 1993, 225 pp. [6] P.L. de Boer and D.G. Smith, Orbital Forcing and Cyclic Sequences, IAS Spec. Publ. 19, Blackwell Scienti c Publications, Oxford, 1994, 559 pp. [7] M.R. House and A.S. Gale, Orbital Forcing Timescales and Cyclostratigraphy, Geol. Soc. Spec. Publ. 85, The Geological Society, London, 1995, 559 pp. [8] G.P. Weedon, The spectral analysis of stratigraphic time series, in: G. Einsele, W. Ricken and A. Seilacher (Eds.), Cycles and Events in Stratigraphy, Springer-Verlag, Berlin, 1991, pp. 840^854. [9] F.J. Rodr guez-tovar, Evoluciön sedimentaria y ecoestratigrä ca en plataformas epicontinentales del margen Sudibërico durante el Kimmeridgiense inferior, Ph.D. Thesis, Univ. Granada, 1993, 344 pp. [10] E. Pardo-Igüzquiza, M. Chica, F.J. Rodr guez-tovar, CYSTRATI: a computer program for spectral analysis of stratigraphic successions, Comput. Geosci. 20 (1994) 511^584. [11] P.J. Brockwell and R.A. Davis, Time Series: Theory and Methods, Springer-Verlag, New York, 1991, 577 pp. [12] D.J. Thompson, Spectrum estimation and harmonic analysis, IEEE Proc. 70 (1982) 1055^1096. [13] J.M. Lees, J. Park, Multiple-Taper spectral analysis: A stand-alone C-subroutine, Comput. Geosci. 21 (1995) 199^236. [14] A. Berger, J.L. Melice, L.A. Hinnov, A strategy for frequency spectra of Quaternary climate records, Clim. Dyn. 5 (1991) 240^277. [15] E. Erba, D. Castradori, G. Guasti, M. Ripepe, Calcareous nannofossils and Milankovitch cycles: the example of the

15 E. Pardo-Igüzquiza, F.J. Rodr guez-tovar / Earth and Planetary Science Letters 181 (2000) 175^ Albian Gault Clay Formation (southern England), Palaeogeogr. Palaeoclimatol. Palaeoecol. 93 (1992) 47^69. [16] M.B. Priestley, Spectral Analysis and Time Series, Acad. Press, London, 1981, 890 pp. [17] C. Chat eld, The Analysis of Time Series, Chapman and Hall, London, 1991, 241 pp. [18] W. Schwarzacher, Sedimentation Models and Quantitative Stratigraphy, Developments in Sedimentology 19, Elsevier, Amsterdam, 1975, 382 pp. [19] B. Efron and R.J. Tibshirani, An Introduction to the Bootstrap, Chapman and Hall, New York, 1993, 436 pp. [20] A. Papoulis, Probability, Random Variables and Stochastic Processes, McGraw-Hill, Singapore, 1984, 576 pp.