zbMATH — the first resource for mathematics

Corbelled domes in two and three dimensions: the Treasury of Atreus. (English. French summary) Zbl 1330.62435
Summary: Before the development of the true dome, many ancient cultures used the technique of corbelling to roof spaces. Recently, a series of related statistical models have been proposed in the literature for explaining how corbelled domes might have been constructed. The most sophisticated of these models is based on a piecewise linear structure, with an unknown number of changepoints, to guide the building process. This model is analyzed by the reversible jump Markov Chain Monte Carlo (MCMC) technique. All models considered to date have been two-dimensional, that is, they have taken a single cross section through the dome; even when more extensive data, in the form of measurements on multiple slices through the dome, have been available, these have been averaged together for the purposes of analysis. In this paper, we extend the two-dimensional analysis to a three-dimensional analysis, that takes full advantage of the data collected by the archaeologists and of the rotational symmetries inherent in the structure. We also explore ways of graphically presenting the results from a complex, reversible jump MCMC implementation, in order to check convergence, good mixing, and appropriate exploration of the (high dimensional and varying dimension) parameter space. The model and the graphical techniques are demonstrated on the Treasury of Atreus in Mycenae, Greece, one of the finest extant examples of the corbelling method.
62P99 Applications of statistics
00A67 Mathematics and architecture
Full Text: DOI
[1] http://www.stat.cmu.edu/ nlazar/movie.mpeg A clip from a movie showing how the fitted shape of the tomb changes as the simulation progresses
[2] http://projects.dartmouth.edu/history/bronze_age/ The Prehistoric Archaeology of the Aegean
[3] Buck, Detecting a change in the shape of a prehistoric corbelled tomb., The Statistician 42 pp 483– (1993)
[4] Casella, Explaining the Gibbs sampler., The American Statistician 46 pp 167– (1992)
[5] Cavanagh, The structural mechanics of the Mycenaean tholos tomb., Annual of the British School at Athens 76 pp 109– (1981)
[6] Cavanagh, An application of change-point analysis to the shape of prehistoric corbelled domes; I. The maximum likelihood method., To Pattern the Past 4 pp 191– (1985)
[7] Chib, Marginal likelihood from the Gibbs output., J. Amer. Statist. Assoc. 90 pp 1313– (1995) · Zbl 0868.62027
[8] Chib, Marginal likelihood from the Metropolis-Hastings output., J. Amer. Statist. Assoc. 96 pp 270– (2001) · Zbl 1015.62020
[9] Denison, Automatic Bayesian curve fitting., J. R. Statist. Soc., Series B 60 pp 333– (1998) · Zbl 0907.62031
[10] DiCiccio, Computing Bayes factors by combining simulation and asymptotic approximations., J. Amer. Statist. Assoc. 92 pp 903– (1997) · Zbl 1050.62520
[11] Fan, Bayesian modelling of prehistoric corbelled tombs., The Statistician 49 pp 339– (2000)
[12] Gelman, Simulation normalizing constants: From importance sampling to bridge sampling to path sampling., Statistical Science 13 pp 163– (1998) · Zbl 0966.65004
[13] Green, Reversible jump Markov chain Monte Carlo computation and Bayesian model determination., Biometrika 82 pp 711– (1995) · Zbl 0861.62023
[14] Hinkley, Inference about the intersection of two-phase regression., Biometrika 56 pp 495– (1969) · Zbl 0183.48505
[15] Hinkley, Inference in two-phase regression., J. Amer. Statist. Assoc. 66 pp 736– (1971) · Zbl 0226.62068
[16] Lazar, Movies for the visualization of MCMC output., J. Computational and Graphical Statistics 11 pp 863– (2002)
[17] Meng, Simulating ratios of normalizing constants via a simple identity: A theoretical exploration., Statistica Sinica 6 pp 831– (1996) · Zbl 0857.62017
[18] Raiffa, Applied Statistical Decision Theory (2000)
[19] Richardson, On Bayesian analysis of mixtures with an unknown number of components., J. R. Statist. Soc., Series B 59 pp 731– (1997) · Zbl 0891.62020
[20] Stephens, Bayesian retrospective multiple-changepoint identification., Applied Statistics 43 pp 159– (1994) · Zbl 0825.62412
[21] Tierney, Accurate approximations for posterior moments and marginal densities., J. Amer. Statist. Assoc. 81 pp 82– (1986) · Zbl 0587.62067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.