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The Toumba building at Lefkandi: a statistical method for detecting a design-unit1

  • Jari Pakkanen (a1)

Abstract

This methodological paper uses the measurements of the Early Iron Age Toumba building at Lefkandi to study whether a single design-unit can be detected in the data set. Cosine quantogram analysis is used in the initial analysis of the building dimensions and, in the second phase, the relevance of the obtained results is calculated by using Monte Carlo computer simulations. A statistically significant unit of c. 49 cm can be isolated, but because of the very limited number of precise dimensions, this result should only be accepted with caution. The case study demonstrates how the complex statistical problem of deriving the lengths of possibly used design-units in ancient architecture can be approached; metrological analyses can only gain from employing appropriate quantitative methods.

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2 Coulton, J. J., ‘Towards understanding Greek temple design: general considerations’, BSA 70 (1975), 93–8.

3 The work of Rottländer, R. C. A. is an exception; see e.g. ‘Studien zur Verwendung des Rasters in der Antike IIOJh 65 (1996), 186.

4 See ‘Layout’; the analysis presented in this paper supersedes the brief metrological study in ‘Methodological reflections’, 242-5, where de Waele's view is shown to be problematic.

5 For the measurements, see Popham, M. R., Calligas, P. G., and Sackett, L. H. (eds) with Coulton, J. and Catling, H. W., The Protogeometric Building at Toumba, Part 2: The Excavation, Architecture and Finds (BSA Supp. 23; London, 1992), 3349; ‘Layout’, 380-3.

6 The Bayesian approach assumes that a quantum exists, an assumption I do not think we can make in this case; cf. Freeman, P. R., ‘A Bayesian analysis of the Megalithic yard’, Journal of the Royal Statistical Society A 139 (1976), 2035; Freeman's posterior distributions are actually very closely connected to Kendall's method used in this paper and presented in his article ‘Hunting quanta’, as has been demonstrated by Silverman, B. W. in ‘Discussion of Dr Freeman's paper’, Journal of the Royal Statistical Society, A 139 (1976), 44–5; see also Statistics in Archaeology, 233-4. For an overview of, and further references on, posterior distributions, see ibid., 176-8.

7 Sonntagbauer, W., ‘Zum Grundriß des Parthenon’, ÖJh 67 (1998), 136.

8 Unfortunately, I was unaware of its existence at the time of writing my contribution to ‘Methodological reflections’.

9 Kendall did detect a ‘real quantum’ in the data, but he demonstrated that it could equally well be a result of laying out the relevant dimensions by pacing (‘Hunting quanta’, 258); a synopsis of the discussion is presented in Renfrew, C. and Bahn, P., Archaeology: Theories, Methods, and Practice (3rd edn., London, 2000), 401, and Baxter sums up the argument in Statistics in Archaeology, 235: ‘Although the megalithic yard may be dead, the methodology that some regard as having buried it lives on.’ Several more relevant examples of quantal problems in archaeology are discussed in ‘Archaeostatistics’, 282-6.

10 von Mises, R., ‘Über die “Ganzzahligkeit” der Atomgewichte und verwandte Fragen’, Physikalische Zeitschrift, 19 (1918), 490500; Broadbent, S. R., ‘Quantum hypotheses’, Biometrika, 42 (1955), 4557; id., ‘Examination of quantum hypothesis based on a single set of data’, Biometrika, 43 (1956), 32-44; ‘Hunting quanta’, 234. A review of Kendall's method with a larger number of executed simulations is presented in Statistics in Archaeology, 228-35. on the use of Monte Carlo methods in archaeology, see Pakkanen, J., The Temple of Athena Alea at Tegea: A Reconstruction of the Peristyle Column (Publications of the Department of Art History at the University of Helsinki, 18; Helsinki, 1998), 54–5, esp. n. 18; Statistics in Archaeology, 147-58.

11 ‘Hunting quanta’, 233-4.

12 Ibid., 235-9; ‘Archaeostatistics’, 282; Statistics in Archaeology, 231.

13 ‘Hunting quanta’, 241-9; ‘Archaeostatistics’, 282-3; Statistics in Archaeology, 232-3. I have implemented the computer programs used in the cosine quantogram analyses, Monte Carlo simulations, and kernel density estimations on the basis of statistical program Survo MM.

14 ‘Layout’, 380–3; ‘Methodological reflections’, 244–5. The question of measurement accuracy and significant digits is discussed in ‘Methodological reflections’, 243.

15 The issue of the small number of observations is also addressed in Pakkanen, J., ‘Deriving ancient foot-units from building dimensions: a statistical approach employing cosine quantogram analysis’, in Burenhult, G. and Arvidsson, J. (eds), Archaeological Informatics: Pushing the Envelope. CAA 2001 (BAR S1016; Oxford, 2002), 501–6.

16 ‘Hunting quanta’, 249, 254–7; the altitude of the peak varies as the square-root of N.

17 Ibid., 252–3.

18 One foot is two thirds of the length of a cubit, so 489.71 mm × ⅔ 326.5 mm. The classical general account on the length of the ‘Doric’ foot, though without any detailed analyses, is Dinsmoor, W. B., ‘The basis of Greek temple design: Asia Minor, Greece, Italy’, Atti del settimo congresso internazionale di archeologia classica (Rome, 1961), i. 358–60. The unit is also called ‘Pheidonian’ after the legendary king of Argos, who, according to Hdt. vi. 127, introduced a new system of weights and measures in his kingdom; see Dörpfeld, W., ‘Metrologische Beiträge’, AM 15 (1890), 177 for connecting the name of Pheidon with the foot-unit.

19 ‘Hunting quanta’, 241.

20 Measurement Xi is replaced by Xi + mU, where m is 5 for accurate measurements and 50 for all the others, U a random number between -1 and 1 selected from a uniform distribution, and i = 1, 2, 3, …, N.

21 For this length of the unit, see e.g. Dinsmoor (n. 18), 357–8.

22 On measurement selection and metrology, see ‘Archaeostatistics’, 285–6.

23 ‘Hunting quanta’, 245.

24 Ibid., 245–6.

25 Freeman (n. 6), 23.

26 On the number of Monte Carlo simulations, see Manly, B. F. J., Randomization, Bootstrap and Monte Carlo Methods in Biology (2nd edn., London and Weinheim, 1997), 80–4.

27 Efron, B., ‘Bootstrap methods: another look at the jackknife’, Annals of Statistics, 7 (1979), 126; on bootstrap methods in general, see e.g. Efron, B. and Tibshirani, R. J., An Introduction to the Bootstrap (Boca Raton and London, 1993) and Davison, A. C. and Hinkley, D. V., Bootstrap Methods and Their Application (Cambridge, 1997); a summary of bootstrap in archaeological context is presented in Statistics in Archaeology, 148–53.

28 Manly (n. 26), 34.

29 S = sup{ϕ(q): 200 ≤ q ≤ 600}.

30 In formula (2) the cosine value would be 1 for all measurements, and the maximum peak height could be calculated as .

31 All the used continuous probability distributions satisfy the condition

where [a, b] = [600, 520000]; cf. FIGS. 6–7.

32 The simulated measurements were created using the formula X = 6583 ∣Z∣ + 600; 6583 mm is the mean of the 21 measurements in col. 2 of TABLE 1, and the distribution is moved to the right by 600 mm; cf. ‘Hunting quanta’, 245–6.

33 The formula used was X = αχ2 + 600, where the chisquared distribution had 2 degrees of freedom and the multiplier a three different values of 2600, 2800 and 3000 in the simulation runs (see lines -–11 of TABLE 1). The χ2 distribution in FIG. 6 was produced with a = 2800.

34 The measurements were created from formula X = 2600F + 580, where the degrees of freedom were v, = 4 and v 2, = 4.

35 On univariate kernel density estimation in general, see Silverman, B. W., Density Estimation for Statistics and Data Analysis (London and New York, 1986), 774.

36 Baxter, M. J. and Beardah, C. C., ‘Beyond the histogram: improved approaches to simple data display in archaeology using kernel density estimates’, Archeologia e calcolatori, 7 (1996), 397408; for a recent bibliography on the use of kernel density estimates in archaeology, see Beardah, C. C. and Baxter, M.J., ‘Three-dimensional data display using kernel density estimates’, in Barceló, J. A. et al. (eds), New Techniques for Old Times: Computer Applications and Quantitative Methods in Archaeology. Proceedings of the 26th Conference, Barcelona, March 1998 (BAR S757; Oxford, 1999), 163–9; see also Statistics in Archaeology, 29–37.

37 The parallel with bootstrapping techniques is evident, but the problems with bootstrap demonstrated in the beginning of this section can be avoided; cf. Manly (n. 26), 34.

38 For evaluating the band-widths I have employed the MATLAB routines programmed by C. C. Beardah; see Baxter and Beardah (n. 36), 405–8.

39 In order to keep FIG. 7 intelligible, I have not illustrated KDE with h = 1800 in the plot: it continues the trend and is still smoother than the KDE with h = 1500.

40 The band-width h calculated using Solve-The-Equation method (STE) is 945.3, one-, two- and three stage Direct-Plug-In (DPI) methods 2049.1, 1488.2 and 1132.5 respectively, and Smooth-Cross-Validation (SCV) 1383.1. For the methods, see Baxter and Beardah (n. 36), 397–408.

41 ‘Hunting quanta’, 260.

42 Ibid., 253.

1 I wish to dedicate this article to the memory of my father Ahti Pakkanen for all his support over the years; especially, I should have liked to have had the opportunity to thank him also in print for locating and sending a copy of von Mises's article published in 1918, the first study employing the sum of cosine values for determining the size of an unknown unit in a data set (for the full reference, see n. 10). I am grateful to Michael Baxter for pointing me towards D. G. Kendall's work on unit detection, and I also wish to thank the anonymous referee for the comments I received on the manuscript of the paper.

The following short titles are used in this article:

‘Archaeostatistics’ = N. R. J. Fieller, ‘Archaeostatistics: old statistics in ancient contexts’, Statistician, 42 (1993), 279-95.

‘Hunting quanta’ = D. G. Kendall, ‘Hunting quanta’, Philosophical Transactions of the Royal Society of London. Mathematical and Physical Sciences, A 276 (1974), 231-66.

‘Layout’ = J. A. K. E. de Waele, ‘The layout of the Lefkandi “Heroon”’, BSA 93 (1998), 379-84.

‘Methodological reflections’ = J. and P. Pakkanen, ‘The Toumba building at Lefkandi: some methodological reflections on its plan and function’, BSA 95 (2000), 239-52.

Statistics in Archaeology = M. Baxter, Statistics in Archaeology (London, 2003).

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The Toumba building at Lefkandi: a statistical method for detecting a design-unit1

  • Jari Pakkanen (a1)

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