Flexible Bayesian Methods for Archaeological Dating Angela Jane Karlsberg Thesis submitted to ...

217 Pages·2013·18.27 MB·English

Flexible Bayesian Methods for Archaeological Dating
Angela Jane Karlsberg
Thesis submitted to the University of Sheffield
for the degree of Doctor of Philosophy
Department of Probability and Statistics
School of Mathematics and Statistics
University of Sheffield
Sheffield, U.K.
July 2006
Acknowledgements
There are lots of people that I would like to thank for their support and encouragement
and without whom completion of this thesis would not have been possible.
Firstly I would like to acknowledge the enthusiastic supervision of both Dr. Caitlin Buck
and Prof. Paul Blackwell who provided advice, criticism and friendly encouragement
whenever I needed it throughout the project.
Secondly, I would like to thank both EPSRC and English Heritage for providing resources
and funding for my research. lowe particular gratitude to Alex Bayliss of English
Heritage for her archaeological supervision and willingness for me to get involved in
some practical archaeology.
I would also like to thank my office mate Lynsey McColl (for all those serious discussions
and all those lunches) as well as all the other PhD students, the rest of the academic
staff, research assistants and secretaries that made my time so enjoyable at Sheffield.
Finally I would like to thank my husband, Simon, and my parents, who have been great
over the years and never raised an eyebrow when I claimed that my thesis would be
"finished in the next month" for nearly a year ...
Abstract
Statistical models for the calibration of both independent and related groups of
radiocarbon determinations are now well established and there exists a number of
software packages such as BCal, OxCal and CALIB that can perform the necessary
calculations to implement them. When devising new statistical models it is important
to understand the motivations and needs of the archaeologists. When researchers select
samples for radiocarbon dating, they are often not interested in when a specific plant or
animal died. Instead, they want to use the radiocarbon evidence to help them to learn
about the dates of other events, which cannot be dated directly but which are of greater
historical or archaeological significance (e.g. the founding of a site).
Our initial research focuses on formulating prior distributions that reliably represent
a priori information relating to the rate of deposition of dateable material within an
archaeological time period or phase. In archaeology, a phase is defined to be a collection
of excavated material (context or layers) bounded early and late by events that are of
archaeological importance. Current software for estimating boundary dates only allows
for one possible type of a priori distribution, which assumes that material suitable for
dating was deposited at a uniform rate between the start and end points of the phase.
Although this model has been useful for many real problems, researchers have become
increasingly aware of its limitations. We therefore propose a family of alternative prior
models (with properties tailored to particular problems within archaeological research)
which includes the uniform as a special case and allows for more realistic and robust
modelling of the deposition process. We illustrate, via two case studies, the difference in
archaeological conclusions drawn from the data when implementing both uniform and
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non-uniform prior deposition models.
The second area of research, that we take the first steps towards tackling, is spatio
temporal modelling of archaeological calibration problems. This area of research is
of particular interest to those studying the response of plants and animals, including
humans, to climate change. In archaeological problems our temporal information
typically arises from radiocarbon dating, which leads to estimated rather than exactly
known calendar dates. Many of these problems have some form of spatial structure yet
it is very rare that the spatial structure is formally accounted for. The combination of
temporal uncertainty and spatial structure means that we cannot use standard models
to tackle archaeological problems of this kind. Alongside this, our knowledge of past
landscapes is generally very poor as they were often very different from modern ones;
this limits the amount of spatial detail that can be included in the modelling.
In this thesis we aim to make reliable inferences in spatio-temporal problems by carefully
devising a model that takes account of the temporal uncertainty as well as incorporating
spatial structure, to provide probabilistic solutions to the questions posed. We illustrate
the properties of both the conventional models and the spatio-temporal models using a
case study relating to the radiocarbon evidence for the Late glacial reoccupation of NW
Europe.
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Contents
1 Introduction 1
1.1 Background 3
1.1.1 Radiocarbon dating 3
1.1.2 Assumptions of the radiocarbon dating method . 6
1.1.3 Measurement of radiocarbon .......... . 11
1.1.4 Modelling the relationship between calendar and radiocarbon years 13
1.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20
2 Statistics in radiocarbon dating 22
2.1 Introduction.......... . ..... . ...... . ........... 22
2.2 Bayesian inference ........ . ....... . ... 23
2.2.1 Prior probability distributions. . . . . . . . . . . . . . . . . . . .. 24
2.3 Interpreting radiocarbon data. . . . . . . . . . . . . . . . . . . . . . . .. 25
2.3.1 Basic model and calibration . ........ . ...... ..... 26
2.3.2 Interpreting groups of related radiocarbon determinations . . . .. 28
2.3.3 The first use of Bayesian statistics . . . . . . . . . . . . . . . . ., 36
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2.4 Some case studies and simple extensions . . . . . . . . . . . . . . . . . .. 39
2.5 The MCMC revolution . . . . . . . . . . . . . . .. 41
2.5.1 Prior information about time elapsed between deposits. . . . . .. 43
2.5.2 Prior information about the rate of deposition ..... . ..... 46
2.5.3 Outlier detection . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47
2.5.4 Remodelling the calibration curve .................. 49
2.6 Alternative prior specification . . . . . . . . . . . . . . . . . . . . . . . .. 50
2.7 Model comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55
2.8 Implementation of methods discussed within the chapter . 58
3 Modelling the deposition process 60
3.1 Introd uction. . . . . . . . . . . . . . . . 60
3.2 Methods for general Bayesian inference ................... , 60
3.2.1 Numerical integration 61
3.2.2 Simulation methods . 63
3.2.3 Practical considerations in MCMC . . . . . . . . . . . . . . . . .. 66
3.3 Devising alternative deposition models . . 70
3.4 Set-up of the trapezium and sigmoidal priors for the deposition rate of
datable material within an archaeological phase . . . . . . . . . . . . . .. 73
3.4.1 Trapezium prior 74
3.4.2 Sigmoidal prior . 74
3.4.3 Single phase of activity ........................ 76
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3.4.4 Multiple phases of activity. . . . . . . . . . . . . . . . . . . . . .. 78
3.5 Implementing the trapezium and sigmoidal prior distributions using a
Metropolis-Hastings algorithm ..... . . . . . . . . . . . . . . . . . .. 80
3.6 Coding the MCMC: problems encountered. . ..... 83
3.6.1 Computing aspects of the case studies within Chapter 4 . . . . .. 85
4 Case studies for the uniform, trapezium and sigmoidal models 86
4.1 Introduction............................... 86
4.2 Case study: Radiocarbon dating and art historic dating of Roman and
Coptic textiles ................................. 87
4.2.1 Introduction ....................... 87
4.2.2 A simple temporal model .................... , 90
4.2.3 Prior beliefs .............................. , 91
4.2.4 The trapezium model ................... 92
4.2.5 Dating Coptic textiles ..................... 93
4.3 Case study: Human reoccupation of NW Europe after the last Ice Age. 96
4.3.1 Introduction ................. . . ... . . 96
4.3.2 Prior beliefs. ............ 98
4.3.3 IntCal98 versus IntCal04: Will it make a difference to the
archaeological interpretations? .... ............. 99
4.3.4 Setting up the multiple phase problem 101
4.3.5 The human reoccupation of NW Europe . 102
4.3.6 Does the reoccupation process overlap in the eight regions? 108
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4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5 First steps towards fully spatio-temporal modelling 112
5.1 Introduction. . . . . . . . . . 112
5.2 Types of a priori information 113
5.3 Defining joint (bivariate) prior distributions 116
5.4 Generalizing to higher dimensions ..... . 122
5.4.1 Rewriting the two-dimensional case. 122
5.4.2 Alternative formalization 123
5.4.3 Priors in ]R3 ..... . . . 125
5.4.4 Asymmetric priors in ]R3 . 127
5.5 Summary . . . . . . . . . . . . . ............... . ...... . 130
6 Spatio-temporal modelling 131
6.1 Introduction . . . .... . 131
6.2 Example of an archaeological spatia-temporal problem 132
6.3 Incorporating spatia-temporal information ...... . 134
6.3.1 The prior distributions when no spatial dependence is incorporated 136
6.3.2 The prior distributions when incorporating spatial dependence
between regions. . . . . . . . . . . . . . . . . . . . . . . . . . .. 138
6.4 Alternative spatia-temporal priors 142
6.4.1 First idea - constrain aj - 1j $ c years . 143
6.4.2 Second idea - define the prior differently . 143
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6.5 Fully spatia-temporal modelling. . ... 147
6.5.1 Uniform spatio-temporal model. 147
6.5.2 Trapezium model . 151
6.6 Summary . . . . . . . . . 155
7 Spatia-temporal case study: human reoccupation of NW Europe 156
7.1 Introduction................................. 156
7.2 Recapping the inferences obtained from the human reoccupation case
study in Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.3 Spatial information arising from the reoccupation case study 158
7.4 Setting up the spatia-temporal models ............ . 160
7.5 Results: Conventional uniform model versus the uniform spatio-temporal
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 163
7.6 Results: Trapezium model versus the trapezium spatio-temporal model 168
7.6.1 Possible routes of migration through NW Europe . 172
7.7 Summary .......................... . 175
8 Conclusions and further work 176
8.1 Conclusions . 176
8.2 Further work 182
8.2.1 Model choice 182
8.2.2 Extending Spatio-temporal models 184
8.2.3 Outlier detection . . . . . . . . . . 185
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A C code 181
B Archaeological data 189
C Spatia-temporal modelling: results 193
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