A STATISTICAL EXAMINATION OF THE VARIATION IN THE CHLOROPHYLL DEGRADATION PRODUCT: EPIPHASIC CAROTENOID RATIO IN THE ESTHWAITE DEPOSITS

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1 Pigments in lake deposits 139 APPENDIX A STATISTICAL EXAMINATION OF THE VARIATION IN THE CHLOROPHYLL DEGRADATION PRODUCT: EPIPHASIC CAROTENOID RATIO IN THE ESTHWAITE DEPOSITS BY D. E. BARTON Department of Statistics, University College, London Tbe statistical analysis of tbese data presents several points of interest and since a part of tbe metbod of analysis is of fairly recent origin and not yet in tbe elementary texts it is thougbt to be wortb considering in some detail. In tbe absence of a bypotbesis as to bow tbe variations bave arisen, tbe data bave been analysed so as to bring out tbe structure of tbe oscillatory part of tbeir bebaviour as evidently as possible. On tbis basis a simple model bas been assigned to enable constants to be estimated. Tbe qualitative analysis falls into two parts, tbe computation of tbe 'correlogram' and of tbe 'estimated spectrum'. The correlogram analysis. Any periodicity in a series of measures, x\, x-2... Xn, will reflect itself in a correlation between every/)"^ element (if/) is tbe lengtb of tbe period) and a complex of periods will reflect itself in a set of sucb correlations. For a given number, s, tbe correlation coeflicient, r,,, is computed between (.v'l, x\+s), (A'2, A'2+S),... [xn-s, Xn). Explicitly, if n s and if Cs n s n s tben rs = CsjCo (wbere Co is tbe overall sum of squares about tbe mean). The 'serial correlation coefficient, as Ts is called, is tben grapbed against s. Periods of lengtb,/), zp, 3/),... etc., arising out of any periodicity of lengtb p, are tbus revealed. A grapb of tbis kind for an 'autoregressive' model will sbow evidence of 'damping', reflecting tbe fact that such a model may be regarded as periodic but witb wandering phase. On the whole, the correlogram for the Esthwaite data (Fig. 6) would seem to show rather more of a sustained periodicity than a damped oscillation. The estimated spectrum. If we suppose, provisionally, a purely periodic (harmonic) effect we may write, in general, Xj = { Al sin (WITT; + ^1) + A-2 sin (w-ynj + ^.3) +...} + random error

2 140 G. E. FOGG AND J. H. BELCHER where: pi = 2lwi,p2 = 2/OT2,,.. etc., are the periods, w, <ZV2<---, Al, A2, etc., are the amplitudes of the respective periodic components, and ^i, (f>2, etc, are the phase constants. The phase constants refiect the fact that the periodic components have their peaks and troughs at different points and these have to be determined from the data. Since a period pi implies a repetition once in every pi observations then ilpi = roi/2 is the frequency. The graph of the amplitudes against their corresponding frequencies is called the spectrum. (Strictly one should plot this, oscro < 2, but since frequencies w, z w are indistinguishable we take o<ro^i). The phase constants are not reflected in this graph. If the cumulative sums ^i = Ai, S2 = A1+A2, S3 = A1+A2+A3, etc., are formed and plotted against w the cumulative spectrum is obtained. Fig, 6. Correlogram of the chlorophyll degradation product: epiphasic carotenoid ratio data. Several ways of estimating the spectrum exist (see, for example, the review by Jenkins and Priestley, 1957), none of them entirely satisfactory, but all giving similar graphs, smooth, but roughly rising in 'steps', for the estimated cumulative spectrum. We will use the integral form of Jenkins and Priestley's equation 10, m = n, viz. 34 F(w) =- (w-\-2 / Ts ) TT ' ^ ^ s=l The best method of dealing with the present data would seem to be the subjective one of picking out the steeper bits of such a graph and finding the corresponding values of wi, W2, etc. The values of Ai, A2, etc., are estimated by the increments of S over the steep portions of the curve and the rest of the change in S is assigned to random error. The actual estimated spectrum {alias the estimated spectral density function or periodogram) would give a graph in which it is virtually impossible to decide which peaks are real and which adventitious. A compromise, as between the cumulative spectrum and periodogram approaches, is to divide the range of w into equal portions ('frequency bands') and plot the increments of estimated S (say A 5) as a histogram. From this histogram the intervals of w wherein any real frequencies corresponding to large amplitudes lie, may be

3 Pigments in lake deposits 141 picked out visually. The graph of the estimated cumulati^'e spectrum for the Esthwaite data is given in Fig. 7 and the corresponding histogram in Fig. 8. The estimated spectral histogram divides up the overall variance into components belonging to the different frequency bands so that it is reasonable to say from Fig. 8 that about 35''o of the overall variance is due to a periodicity of frequency in the (0.35, 0.40) band with the bare possibility of another 5% due to one in the (0.80, 0.85) band. T 0-2 O O O-45 Fig. 7. Estimated cumulative spectrum for the chlorophyll degradation product: epiphasic carotenoid data. (B) shows detail of (A). Thus, a qualitative conclusion is that one-third of the variability is due to one single frequency in the band (0.35, 0.40). To give a number to this frequency we return to Fig. 7B (detail) which shows a fairly steady slope in the range 0.36 to 0.39 suggesting a value of = 3/^ so that the period is estimated as 2 X 8/3 = 5.33 (i.e cm). This is entirely consistent with the fact that the original sequence shows six cycles in thirty-two observations. Having chosen this value, w = 0.375, the mean value of Xj is then A sin (wjtx-\-<f>) and the constants A and <!> are estimated by least squares, viz minimizing with respect to A and 0.

4 142 G. E, FOGG AND J. H. BELCHER Finally we have, in terms of depth, d, in cm, mean x at depth d estimated as 1, ,258 sin {0,0375(^ 100)+0,024} This cycle accounts for 33.6 per cent of the variance. Further computation would give a precise test of the hypothesis that this exhausts the periodic component but it is visually evident from Figs. 7 and 8 that the rest of the variation may be attributed to random variation, i.e. there is no significant departure from simple periodicity. On the assumption of only one periodicity, the troughs (the 'basic value', see p. 134) have an estimated true level of = The mean of the values of the ratio in the first 90 cm is with an estimated standard error of 0,034. Since the standard error of the estimated trough level is roughly these values differ significantly at about the 0,10 probability level. Further, had a second periodicity been assumed the estimated basic value would have been lower and it does not appear that there is any important difference between the trough level and that in the first 90 cm. 5-O i As Fig, 8, Estimated spectral histogram for the chlorophyll degradation product: epiphasic carotenoid data. The dotted line shows the effect of fine division of zo (the periodogram). It should be emphasized that the approximate and 'visual inspection' nature of our analysis is due to the very short series (thirty-two observations) at our disposal and to the absence of a satisfactory 'small sample' statistical theory. For comparison it is worth looking at Whittle's (1954) spectral analysis of a series twenty times this length for which a more thoroughgoing treatment has been possible. 0-5 REFERENCES JENKINS, G, M, & PRIESTLEY, M, B, (1957), The spectral analysis of time series, J', roy. Statist. Soc, B, 19, 1-12, WHITTLE, P, (1954), The statistical analysis of a seiche record, J*, mar. Res., 13,

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