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Published online by Cambridge University Press: 23 May 2013
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.
2 Coulton, J. J., ‘Towards understanding Greek temple design: general considerations’, BSA 70 (1975), 93–8Google Scholar.
3 The work of Rottländer, R. C. A. is an exception; see e.g. ‘Studien zur Verwendung des Rasters in der Antike II’ OJh 65 (1996), 1–86Google Scholar.
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), 33–49Google Scholar; ‘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), 20–35CrossRefGoogle Scholar; 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), 44–5Google Scholar; 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), 136Google Scholar.
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), 401Google Scholar, 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), 490–500Google Scholar; Broadbent, S. R., ‘Quantum hypotheses’, Biometrika, 42 (1955), 45–57CrossRefGoogle Scholar; 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), 54–5Google Scholar, 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), 501–6Google Scholar.
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. 358–60Google Scholar. 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), 177Google Scholar 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), 80–4Google Scholar.
27 Efron, B., ‘Bootstrap methods: another look at the jackknife’, Annals of Statistics, 7 (1979), 1–26CrossRefGoogle Scholar; on bootstrap methods in general, see e.g. Efron, B. and Tibshirani, R. J., An Introduction to the Bootstrap (Boca Raton and London, 1993)CrossRefGoogle Scholar and Davison, A. C. and Hinkley, D. V., Bootstrap Methods and Their Application (Cambridge, 1997)CrossRefGoogle Scholar; 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), 7–74CrossRefGoogle Scholar.
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), 397–408Google Scholar; 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), 163–9Google Scholar; 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.
