the permutation test as a non-parametric method …hera.ugr.es › doi › 1500000x.pdfthe...

The permutation test as a non-parametric method for testingthe statistical signi¢cance of power spectrum estimation in

cyclostratigraphic research

Eulogio Pardo-Iguzquiza a, Francisco J. Rodr|guez-Tovar b;*a Department of Mining and Mineral Engineering, University of Leeds, Leeds LS2 9JT, UK

b Departamento de Estratigraf|a y Paleontolog|a, Facultad de Ciencias, Universidad de Granada, 18002 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 requiredistributional assumptions. It is suitable for all the spectral analysis approaches used in practice. Moreover, goodperformance is achieved with relatively short stratigraphical series. The method is similar to a permutation test that hasbeen successfully applied to other statistical problems. In the proposed application of the permutation test to thespectral analysis of time series, the data of a stratigraphic sequence are ordered at random (random permutation) andthe power spectrum is estimated by the given approach. The process is repeated many times (e.g. 1000 times) and thus itis 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 themethod. 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 havefocused on the analysis of rhythmic sedimentarysuccessions, in an e¡ort to recognise the possiblecyclic in£uence of astronomical parameters on theEarth's climate [1^7]. From the ¢rst papers, awide variety of spectral analysis techniques havebeen used for the detection of cyclical components

in stratigraphic sequences [8^10]. However, one ofthe most common handicaps in the spectral anal-ysis of rhythmic stratigraphic successions is todiscern signi¢cant periodicities from a noisy back-ground. Usually, statistical tests are used with thisapproach. However, statistical tests of the esti-mated power spectrum are not easily availablefor all the estimators, and, when available, theydepend on distributional assumptions or onasymptotic results which are not very well suitedto short stratigraphical series. Thus, the aim ofthis paper is to present a computer-intensive sig-ni¢cance test of general use in estimating thepower spectrum of cyclic sedimentary successions.

0012-821X / 00 / $ ^ see front matter ß 2000 Elsevier Science B.V. All rights reserved.PII: S 0 0 1 2 - 8 2 1 X ( 0 0 ) 0 0 1 9 1 - 6

* Corresponding author. E-mail: [email protected]

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Earth and Planetary Science Letters 181 (2000) 175^189

www.elsevier.com/locate/epsl

A simulation case study and the application ofthis method to a series of calcareous nannofossil,using the periodogram, and the maximum entropyestimator are presented.

2. Standard statistical tests

A complete explanation of standard tests forthe power spectrum is found in Brockwell andDavis [11], while such methods as the multitaperestimator [12,13] have their own statistical test.We include here only a very brief discussion ofa method which is often used in cyclostratigraphy[5,14,15]. In cyclostratigraphy, the successionsstudied are comparatively short and have nor-mally been a¡ected by di¡erent processes suchas tectonics, diagenesis, hiatus, variable sedimen-tation rates, etc., which can make it di¤cult tocalibrate the absolute time represented along thesequence. This contrasts with other disciplines,such as electrical engineering or geophysics, wherelong time series with very precise measurementsare often the norm. However, even with the lim-itations of the stratigraphic sequences, the signa-ture of a cyclical process is frequently detected bythe spectral analysis of the succession. Further-more, together with the recognition of peaks atgiven frequencies in the estimated power spec-trum, in cyclostratigraphic research, it is of specialinterest to assess the statistical signi¢cance ofthese peaks.

Statistical theory provides estimators of thepower spectrum P(fk) and con¢dence intervalsfor the estimators. For example, it is well knownthat the periodogram I(fk) [16] is an unbiased es-timate of the power spectrum [17] :

E�I�f k�� P�f k� �1�

and with variance:

var�I�f k�� P2�f k� �2�

where fk denotes the k-th frequency.Also, if the data are Gaussian:

2WI�f k�P�f k�WM 2

2 �3�

where Mn2 is a M-squared random variable with n

degrees of freedom and V denotes that it is dis-tributed asymptotically.

Then, the 100(13K)% con¢dence intervals forI(fk) are given by:

2WI�f k�M 2

2;K=2

9P�f k�9 2WI�f k�M 2

2;13K=2

�4�

where K is the con¢dence level.Con¢dence intervals are closely related to hy-

pothesis testing. For example the null hypothesis :

H0 : fthere are no significant cyclical

components in the seriesg

may be tested by the method proposed bySchwarzacher [18] : in a logarithmic plot of theestimated power spectrum, a smooth line (i.e.without any peak) is ¢tted as the mean powerspectrum (null hypothesis H0), which is the as-sumed form for P(fk). Then, applying Eq. 4, itis possible to draw the 100(13K)% con¢dencebands and to check visually which frequencieshave a signi¢cant value of the power spectrumestimated by I(fk).

However, the main drawback of this hypothesistest is that the result depends heavily on whichestimate is adopted for representing P(fk), thenull hypothesis [14]. Because of that disadvantage,a non-parametric computer-intensive test may beused as a complement.

The signi¢cance level K is also known as theprobability of the type I error (i.e. the probabilityof rejecting the null hypothesis when the null hy-pothesis is true, or, in other words, the probabilityof accepting a peak as signi¢cant when it in fact isnot). If K is increased, we increase the probabilityof assigning signi¢cance to spurious peaks; and ifK is reduced, we increase the probability of as-signing non-signi¢cance to peaks that are reallycyclical components. This fact is inherent withany test for a statistical hypothesis. In practicethe signi¢cance level is usually ¢xed at 0.05 or0.01.

In testing the signi¢cance of the power spec-trum, we perform (n) simultaneous hypothesis

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E. Pardo-Iguzquiza, F.J. Rodr|guez-Tovar / Earth and Planetary Science Letters 181 (2000) 175^189176

tests where (n) is the number of frequencies wherethe power spectrum has been estimated, and thusis it more appropriate to choose 0.01 as the sig-ni¢cance level, while for single hypothesis testingthe 0.05 signi¢cance level is generally used.

3. Method: the computer-intensive permutationtest

The permutation test, suggested by R.A. Fisherin the 1930s [19], gained applicability from thewide availability of powerful computers. The ba-sic idea is that if the series contains some cycliccomponents, a reordering at random (randompermutation) of the elements in the sequencewill destroy the cyclic pattern of the sequence. Ifthe random permutation is repeated a large num-ber of times, it is possible to evaluate the chancesof ¢nding the power spectrum of the original ser-ies at random. After this, for each frequency, arange of possible values is established and theprobability of observing a value larger than thepower spectrum of the original ordering is calcu-lated. The smaller the value of that probability,the stronger the evidence becomes against the nullhypothesis, which is that the value of the peak isnot signi¢cant.

3.1. Procedure and perspectives

The steps required by the algorithm are:

1. Given the original sequence:

fzt; t � 0; T; N31g

the power spectrum is estimated for exampleby the periodogram:

fI�f k�; k � 0; T; N=2g

2. From the original sequence a random permu-tation is obtained:

fz�t ; t � 0; T; N31g

3. The power spectrum is estimated for the new

sequence:

fI��f k�; k � 0; T; N=2g

4. Steps (2) and (3) are repeated a large numberof times M (e.g. 1000).

5. For each frequency k, there are M values{I*(fk)} and the signi¢cance of the estimate{I(fk)} of the original sequence may be as-sessed by the achieved signi¢cance level(ASL) of the test [19]. The ASL of a givenestimate {I(fk)} is de¢ned as the probabilityof ¢nding at least as large a value when thenull hypothesis is true (i.e. there is no signi¢-cant contribution at that frequency that gives asigni¢cant cyclic component):

ASL�f k� � ProbfI��f k� v I�f k�g �5�

The smaller the value of the ASL, the strongerthe evidence against the null hypothesis. Itshould be noted that in Eq. 5, the value I(fk)is ¢xed (for a given frequency), while I*(fk) is arandom variable with values provided by thepower spectra estimated from the series givenby random permutation.

6. A small value of the signi¢cance level K ischosen (like 0.01) and the peaks for whichthe 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� � fnumber of timesW�I��f k�vI�f k��g=M

�6�

One important characteristic of permutationtesting is its accuracy, because, if the null hypoth-esis is true, then [19] :

ProbfASL6Kg � K �7�

for any value of K between 0 and 1.

It may be noted that each random permutationis not a bootstrap sample because the resamplingis not done with replacement.

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E. Pardo-Iguzquiza, F.J. Rodr|guez-Tovar / Earth and Planetary Science Letters 181 (2000) 175^189 177

Interpretations on the existence of any possiblecyclicity in rhythmic successions have been con-ventionally approached by spectral analysis ofstratigraphic series. However, until now, to ana-lyse the signi¢cance of these power spectra, verydependent methods have been used. The applica-tion of the methodology explained here provides anew tool that can be of great use to test the sig-ni¢cance of peaks recognised in the power spec-trum. Some advantages of the permutation testfor testing the signi¢cance of the power spectrumare:

b it may be applied with any power spectrumestimator: periodogram, Blackman and Tukeyapproach, maximum entropy, autoregressivespectral estimator, etc. ;

b it does not require any distributional assump-tion;

b it may be applied with short as well as longtime series.

The permutation test could be useful in at leastthree di¡erent circumstances:

b Many examples of orbitally induced cyclicityare so clear that spectral analysis only servesas additional proof. In such cases the permuta-tion test is useful to eliminate any doubt.

b In cases in which orbital cyclicity is not ob-

vious, a misuse of various spectral techniquesand ¢ltering may provide a curve with an acci-dental peak that is more or less similar to an`expected' astronomical technique. The permu-tation test will check if a real periodicity ispresent.

b Testing the presence and stability of the perio-dicities. The permutation test could be appliedto subspectra or to the series after the removalof the ¢rst or last 10% or 25% of the data. Incase of a stable periodicity throughout the suc-cession, spectra should be fairly similar. Thepermutation test should provide evidence ofthe signi¢cance of the peaks in these cases too.

The implicit model assumed by the method iswhite noise plus sinusoidal cyclic components. If atrend is suspected, some detrending methodshould be applied ¢rst. If red noise is suspected,a moving window spectral analysis could be per-formed.

4. Examples of application

4.1. Simulation case study

The ¢rst example is the application of the per-mutation test to the estimated periodogram of asimulated time series with a white noise stochastic

Table 1Number of signi¢cant values in the permutation test for simulated white noise for di¡erent sample sizes and number of permuta-tions

White noisemodel

Length ofthe series

Number of randompermutations

Number of signi¢cant valuesK= 0.05

Number of signi¢cant valuesK= 0.01

Gaussian 128 1000 3 (3) 0 (1)Gaussian 256 1000 7 (6) 0 (1)Gaussian 512 1000 15 (13) 3 (3)Gaussian 128 5000 3 (3) 0 (1)Gaussian 256 5000 7 (6) 0 (1)Gaussian 512 5000 12 (13) 2 (3)Uniform 128 1000 6 (3) 1 (1)Uniform 256 1000 4 (6) 0 (1)Uniform 512 1000 13 (13) 1 (3)Uniform 128 5000 6 (3) 1 (1)Uniform 256 5000 4 (6) 0 (1)Uniform 512 5000 15 (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)Kwhere, N is the length of the series (the number of data of the series).

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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 ser-ies : 128, 256 and 512. The permutation test hasbeen repeated with 1000 and 5000 random permu-tations. Given a times series of length N, the pe-riodogram is estimated at N/2+1 frequencies, but,because the series are zero mean or transformed

to be zero mean, the periodogram for frequencyzero is always zero I(0) = 0.0, and the signi¢canceof this frequency is not considered. In this case, asigni¢cance test of the estimated power spectrumimplies performing the N/2 independent test forN/2 periodogram estimates simultaneously. Thus,accepting a signi¢cance level of 0.05, the numberof signi¢cant values explained by randomness is

Fig. 1. (A) Periodogram of the uniform white noise time series (256 data), and (B) results of the permutation test in the formSL = (1.03ASL(fk))100%; for example, in the ¢gure, the values crossing SL = 95% (dashed line) are signi¢cant with a signi¢cancelevel of K= 0.05.

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approximately three, six and 13 for a sample sizeof 128, 256 and 512, or one, one and three accept-ing a signi¢cance level of 0.01. The results for thesimulation example are given in Table 1. The ta-ble shows that the number of signi¢cant valuesagrees quite well with the expected number, espe-cially for Gaussian white noise and for the three

sample sizes, as well as with uniform white noisefor sample sizes of 256 and 512. With the shorttime series (128 values) of uniform white noiseand with signi¢cance level of 0.05, six signi¢cantvalues are found, whereas three were expected.

With 1000 random permutations results aregood, while increasing the number of random per-

Fig. 2. (A) Periodogram of the Gaussian white noise time series (256 data), and (B) results of the permutation test in the formSL = (1.03ASL(fk))100%; for example, in the ¢gure, the values crossing SL = 95% (dashed line) are signi¢cant with a signi¢cancelevel of K= 0.05.

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mutations to 5000 o¡ers little improvement. Inany case, at least 1000 permutations should beused and if the computer is e¤cient enough thisnumber should be increased to 5000. Figs. 1 and 2show the periodogram and the signi¢cance levelof the di¡erent periodogram values for the uni-form white noise and Gaussian white noise,respectively. The results of the permutation

test are expressed in the form SL =(1.03ASL(fk))100%, where ASL(fk) is the ASLgiven by Eq. 6. Thus, for example, signi¢cant val-ues with signi¢cance level of 0.05 are the valueswith SLs 95%, as shown in Figs. 1B and 2B.

The second simulation example consists ofthree time series with 256 data each, which weregenerated as two sinusoidal components plus

Fig. 3. (A) Periodogram of the two sinusoidal components and Gaussian white noise (with variance equal to 1.0) time series (256data), and (B) results of the permutation test in the form SL = (1.03ASL(fk))100%; for example, in the ¢gure, the values crossingSL = 95% (dashed line) are signi¢cant with a signi¢cance level of K= 0.05.

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white noise:

x�t� � w�t� �X2

k�1

AkWcos��2ZWf kWt� � P k� �8�

with amplitudes A1 = 1, A2 = 0.75, phases P1 = 0,P2 = Z and frequencies f1 = 0.0625, f2 = 0.20 cyclesper unit of distance, which correspond to wave-

lengths of 16 units and 5 units of distance, respec-tively. The di¡erence between the three series isthe level of the noise given by the variance of theGaussian white noise w(t) added to the sinusoidalcomponents, which are 1.0, 3.0 and 8.0, respec-tively.

The periodogram and the results of the permu-tation test are expressed in the form SL =


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(1.03ASL(fk))100%, and given in Figs. 3, 4 and 5for the three di¡erent series, respectively. For theperiodogram (Figs. 3A, 4A and 5A), it is clearthat, as the level of noise increases, the peaksfrom the background noise (spurious peaks) be-come more relevant in relation to the peaks forthe cyclic components. The number of signi¢cantpeaks with a signi¢cance level of 0.05 is four,

seven and seven for the three time series, respec-tively, and with 1000 random permutations in allthe cases (four, six, seven signi¢cant peaks for thethree series, respectively, using 5000 random per-mutations). For the periodogram, without anykind of smoothing, a signi¢cance level of 95% isnot high enough because we are simultaneouslyperforming a number of hypothesis tests equal


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to the number of frequencies for which the perio-dogram is evaluated. With a signi¢cance level of0.01 (99% con¢dence in the right decision), thenumber of signi¢cant peaks is two, three, twofor the three previous time series and for bothcases, using 1000 and 5000 random permutations.For the ¢rst time series, with the highest signal tonoise ratio, the two estimated components are thetrue (0.0625 and 0.2) with an ASL of 0.0 and 0.0,respectively. For the second time series, the threefrequencies are 0.0625, 0.10938 and 0.2 with anASL of 0.0, 0.003 and 0.0, respectively. For thethird time series, the two signi¢cant componentswith a signi¢cance level of 0.01 appear at frequen-cies of 0.0625 and 0.10938 with an ASL of 0.001and 0.002, respectively. The component at fre-quency 0.2 is not signi¢cant at 99% signi¢cancelevel, 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%, the0.2 component has the lowest ASL (equal to0.016). The results are very similar using 1000 or5000 random permutations.

4.2. Application to real data

The experimental data used are the percentages

of the nannofossil Watznaueria barnesae, mea-sured on samples from a core from the AlbianGault Clay Formation in southern England (orig-inal data in table 1 from Erba et al. [15]). Thisspecies is regarded as a non-fertility index, andcyclic variations of their percentage are relatedto Milankovitch cycles [15]. Thus, the spectralanalysis (¢gure 7C in [15]) shows peaks at perio-dicities of 100 000 yr and 38 000 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 withMilankovitch cycles. The result of the signi¢cancetest also depends on the estimation of the averageunderlying spectrum (solid line in ¢gure 7C of[15]). The same problem with other geological ser-ies has been reported in Berger et al. [14].

The spectral analysis is revisited and the permu-tation test is applied to the power spectra esti-mated by the periodogram and the maximum en-tropy methods.

The original succession used is composed of 135data, most of them on a regular spacing of 0.05m, measured on a core 6.25 m long between 12.25and 18.5 m depth [15]. In order to have a succes-sion 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 longcore.

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measurements from 12.25 to 18.5 at a samplingrate of 0.05 m were obtained; 118 were identicalto the original ones and eight were obtained bythe interpolating splines. The sequence with aconstant sampling interval (Fig. 6) showed no ap-parent trend, and therefore only the substraction

of the mean was performed before applying thespectral analysis estimators.

4.2.1. PeriodogramThe estimated periodogram is shown in Fig.

7A. The ASL from the permutation test shows

Fig. 7. (A) Experimental periodogram of the succession showed in Fig. 6, and (B) results of the permutation test in the formSL = (1.03ASL(fk))100%; only SLs 70% has been represented, and the SL of 95 and 99% have been highlighted with slashedlines.

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that frequencies signi¢cant at a 99% con¢dencelevel are only 0.02344 and 0.08594, correspondingto periods of 426 000 yr and 116 000 yr (Fig. 7B).At a signi¢cance level of 90%, only one additionalfrequency appeared at 0.21875, which corre-sponds to a period of 45 000 yr (Fig. 7B). Theperiods were calculated using the sedimentation

rate of 5 cm every 10 000 yr, as used in Erba etal. [15].

4.2.2. Maximum entropy estimatorEven though for the maximum entropy spectral

estimator [20] it is not easy to derive the standardhypothesis 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) re-sults of the permutation test in the form SL = (1.03ASL(fk))100%; only SLs 70% has been represented, and the SL of 95 and99% have been highlighted with slashed lines.

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be applied. The maximum entropy estimates are afunction of a parameter L [20] that must be speci-¢ed when the estimator is applied. A low L valuedecreases the variance but increases the bias of theestimator, and the reverse happens if L is in-creased. With a small L value, few periodicitiesare detected, with a high L value, a large number

of spectral peaks appear, most of them spurious.The power spectra estimated by maximum entro-py with L = 30 and L = 50 are shown in Figs. 8Aand 9A, respectively. With L = 30, the frequencies0.02344 (426 000 yr) and 0.08594 (116 000 yr) aresigni¢cant at 99% and 95% con¢dence levels, re-spectively (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(fk))100%; only SLs 70% has been represented, and the SL of 95and 99% have been highlighted with slashed lines.

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ni¢cant at a 90% con¢dence level (Fig. 8B). WithL = 50, the frequency 0.02344 (426 000 yr) is sig-ni¢cant at 99% con¢dence level and the frequency0.09375 (106 000 yr) is signi¢cant at 95% con¢-dence level (Fig. 9B). At a 90% con¢dence level,only one additional frequency at 0.48438 (21 000yr) appears as signi¢cant (Fig. 9B).

This illustrates how the permutation test givesbasically the same results for the periodogram andthe maximum entropy methods, even for di¡erentvalues of the parameter L in the maximum entro-py estimator [10].

5. Conclusions

The permutation test is revealed as a non-para-metric method for testing the statistical signi¢-cance of the estimated power spectrum that canbe used complementarily with other methodolo-gies. The test directly assesses the ASL of theestimate for each frequency, and while it doesnot require statistical assumptions, it is com-puter-intensive. Implicit in the method is the mod-el of white noise plus hidden sinusoidal signals.However, with 1000 random permutations, goodresults are provided for relatively short series (be-tween 128 and 512 data), requiring no more thana few minutes in computer time for a PC-486.Nevertheless, if time allows, it is advisable to in-crease the number of permutations to 5000.Although only the periodogram without smooth-ing together with the maximum entropy estimatorhave been used in this paper, the method may beapplied generally to other spectral estimators. Incyclostratigraphic studies, the importance of thespectral peaks that have been shown to be statis-tically signi¢cant by the permutation test shouldbe corroborated by other information, such as thestructure of the correlogram and the physicalmeaning of the periodicities.

Acknowledgements

We would like to thank the reviewers by pro-viding constructive criticism. The work of P.-I.was supported by a European Community Train-

ing Fellowship in the Department of Meteorology(Reading University, UK) as part of the Trainingand Mobility of Researches (TMR) program. Theresearch of R.-T. was done under the programand ¢nancial support of the EMMI Group(RNM-178 Junta de Andaluc|a, Spain) and proj-ect PB97-0803 (DGICYT).[AC]

References

[1] G. Einsele and A. Seilacher, Cyclic and Event Strati¢ca-tion, 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, ReidelPubl. Company, Dordrecht, 1984, 895 pp.

[3] G. Einsele, W. Ricken and A. Seilacher, Cycles andEvents in Stratigraphy, Springer-Verlag, Berlin, 1991,955 pp.

[4] A.G. Fischer, D.J. Bottjer, Orbital forcing and sedimen-tary sequences, J. Sediment. Petrol. Spec. Iss. 61 (7) (1991)1063^1270.

[5] W. Schwarzacher, Cyclostratigraphy and the Milanko-vitch Theory, Developments in Sedimentology 52, Else-vier, Amsterdam, 1993, 225 pp.

[6] P.L. de Boer and D.G. Smith, Orbital Forcing and CyclicSequences, IAS Spec. Publ. 19, Blackwell Scienti¢c Pub-lications, Oxford, 1994, 559 pp.

[7] M.R. House and A.S. Gale, Orbital Forcing Timescalesand Cyclostratigraphy, Geol. Soc. Spec. Publ. 85, TheGeological Society, London, 1995, 559 pp.

[8] G.P. Weedon, The spectral analysis of stratigraphic timeseries, in: G. Einsele, W. Ricken and A. Seilacher (Eds.),Cycles and Events in Stratigraphy, Springer-Verlag, Ber-lin, 1991, pp. 840^854.

[9] F.J. Rodr|guez-Tovar, Evolucion sedimentaria y ecoestra-tigra¢ca en plataformas epicontinentales del margen Su-diberico durante el Kimmeridgiense inferior, Ph.D. The-sis, Univ. Granada, 1993, 344 pp.

[10] E. Pardo-Iguzquiza, M. Chica, F.J. Rodr|guez-Tovar,CYSTRATI: a computer program for spectral analysisof stratigraphic successions, Comput. Geosci. 20 (1994)511^584.

[11] P.J. Brockwell and R.A. Davis, Time Series: Theory andMethods, Springer-Verlag, New York, 1991, 577 pp.

[12] D.J. Thompson, Spectrum estimation and harmonic anal-ysis, IEEE Proc. 70 (1982) 1055^1096.

[13] J.M. Lees, J. Park, Multiple-Taper spectral analysis: Astand-alone C-subroutine, Comput. Geosci. 21 (1995)199^236.

[14] A. Berger, J.L. Melice, L.A. Hinnov, A strategy for fre-quency spectra of Quaternary climate records, Clim. Dyn.5 (1991) 240^277.

[15] E. Erba, D. Castradori, G. Guasti, M. Ripepe, Calcareousnannofossils and Milankovitch cycles: the example of the

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Albian Gault Clay Formation (southern England), Palae-ogeogr. 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 andHall, London, 1991, 241 pp.

[18] W. Schwarzacher, Sedimentation Models and Quantita-

tive Stratigraphy, Developments in Sedimentology 19,Elsevier, Amsterdam, 1975, 382 pp.

[19] B. Efron and R.J. Tibshirani, An Introduction to theBootstrap, Chapman and Hall, New York, 1993, 436 pp.

[20] A. Papoulis, Probability, Random Variables and Stochas-tic Processes, McGraw-Hill, Singapore, 1984, 576 pp.

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