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#+TITLE:   Archaeology in Oceania, Age of O18
#+AUTHOR:    Tom Dye
#+EMAIL:     tsd at tsdye dot com
#+DATE:      2010-07-22 Thu
#+DESCRIPTION: 
#+KEYWORDS: 
#+LANGUAGE:  en
#+OPTIONS:   H:3 num:t toc:nil \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
#+OPTIONS:   TeX:t LaTeX:nil skip:nil d:nil todo:nil pri:nil tags:nil
#+INFOJS_OPT: view:nil toc:nil ltoc:t mouse:underline buttons:0 path:http://orgmode.org/org-info.js
#+EXPORT_SELECT_TAGS: export
#+EXPORT_EXCLUDE_TAGS: noexport
#+LINK_UP:  ../uses.php
#+LINK_HOME: http://orgmode.org/worg/
#+BABEL: :exports code

#+SEQ_TODO: TODO(t) STARTED(s)  | DONE(d) REJECTED(r) REFTEX
  
* LaTeX preamble
#+srcname: latex-preamble
#+begin_src latex :tangle o18_ao.tex
    \documentclass[minion,glossaries]{tsdarticle}
    
    \author{Thomas S. Dye and Jeffrey Pantaleo}
    
    \title{Age of the O18 Site\thanks{Special thanks to Valerie Curtis for
    her commitment to this project and her confidence that it would
    yield interesting results.  Dave Tuggle and Matthew Spriggs
    offered valuable criticism that sharpened the argument
    considerably.  Caitlin Buck offered perceptive advice on Bayesian
    modeling.  Kristin Macak drafted the site map, which was prepared
    for publication by Eric Komori.  Dan Davison and Eric Schulte both
    provided guidance on the use of Org-babel, a software environment
    that integrates thinking, analysis, and writing that greatly
    facilitated production of the paper.}}
    
    \let\alert\textbf
    \let\itemize\compactitem
    
    \begin{document}
    
    \maketitle
    
#+end_src

* Abstract
#+begin_src latex :tangle o18_ao.tex
    \begin{abstract}
      Seven new \rc\ age determinations on short-lived materials yield a
      sound evidential basis for the chronology of the O18 site on O`ahu
      Island, Hawai`i, long thought to be an early settlement site.
      Calibration within a model-based, Bayesian framework indicates
      that the site was established in \textsc{ad} 1040--1219, some 260--459
      years after the current estimate of first settlement, and
      abandoned in the late eighteenth or early nineteenth centuries.
      Previously published age determinations are mostly too old,
      probably due to the ``old wood'' effect.  O18 appears to be the
      oldest site on the Waim\={a}nalo Plain, but earlier sites in
      Waim\={a}nalo likely exist inland of the plain.
  \end{abstract}
#+end_src

* Introduction
#+srcname: latex-intro
#+begin_src latex  :tangle o18_ao.tex

  The age of the O18 site has been an important datum in Hawaiian
  prehistory since the first estimate was published in the pages of
  this journal nearly 40 years ago \citep{pearson71:_bellows}.  Based
  on an internally inconsistent set of \rc\ age determinations, the
  site was interpreted by its excavators as having been established in
  the seventh century \textsc{ad} and abandoned by the twelfth
  century. The estimated date of establishment was subsequently pushed
  back to the fourth century \textsc{ad} by \citet{kirch85}, based
  primarily on volcanic glass hydration dates that are no longer
  believed to be valid \citep{tuggle01:_age_bellow_dune_site_o18}.
  Kirch considered O18 to be one of only two sites representing the
  earliest phase of Polynesian settlement of the Hawaiian Islands.
  This characterization exerted a strong hold on the archaeological
  imagination.  In the early 1980s, it inspired Matthew Spriggs to
  pull additional samples from storage and have them dated. These
  samples yielded a stratigraphically inconsistent set of \rc\ age
  determinations that was interpreted more than a decade later
  with some difficulty by \citet{tuggle01:_age_bellow_dune_site_o18}
  as indicating an occupation span beginning perhaps as early as the
  eighth century \textsc{ad} and ending in the middle of the fifteenth
  century \textsc{ad}.
  
  Here, we present the results of nine new \rc\ age determinations
  from O18, most of them on short-lived materials.  The age
  determinations on short-lived materials are internally consistent
  and provide, for the first time, a sound evidential basis for the
  site's chronology.  The \rc\ age determinations are interpreted
  within a model-based, Bayesian framework.  An estimate of site
  establishment yielded by the model-based analysis, supported by
  the age of an \textit{Aleurites moluccana} nutshell dated by Spriggs,
  indicates that O18 was established several centuries after the
  islands were first settled by Polynesians.  
  
  The O18 chronology yielded by the site-specific Bayesian model is
  extended to include \rc\ age determinations from four other sites in
  the region.  The chronologies of all five sites are broadly similar.
  Like these other sites, O18 was abandoned late in traditional
  Hawaiian times.

#+end_src
* O18 in regional context [2/2]
** DONE Regional context
#+srcname: regional-context
#+begin_src latex  :tangle o18_ao.tex
  \section{The O18 Site}
  \label{sec:O18-site}
  
  Site O18 is located on the Waim\={a}nalo Plain, at the coast
  (fig.~\ref{fig:location}).  It is a small part of a larger traditional
  Hawaiian settlement pattern in which the coastal plain was used on a
  regular basis, primarily for activities associated with fishing and
  shellfishing, by people who kept more established residences inland on
  the volcanic soils that supported their food gardens.  A large portion
  of the coastal plain was developed as a military installation in the
  twentieth century, especially during World War II, and much of the
  traditional Hawaiian deposit was lost during this development.  The
  pattern of sites on the plain today is probably due more to military
  development than it is to patterns of traditional activity in the
  past.
  
  \begin{figure}[htb!]
    \centering
    \includegraphics[width=84mm]{AO_bellows_O18_landscape.png}
    \caption{Traditional Hawaiian sites on a portion of the Waim\={a}nalo Plain.}
    \label{fig:location}
  \end{figure}
  
  Immediately inland of Site O18, and at one time probably coterminous
  with it, is Site 50--80--15--4853, a large expanse of discontinuous
  cultural deposits on the north bank of Puh\={a} Stream that
  represent primarily cooking and eating activities
  \citep{tuggle97:_archaeol_resear_areas_propos_devel,desilets02:bellows}.
  South of Puh\={a} Stream is Site 50--80--15--4851, which is broadly
  similar to Site --4853, but also includes low-lying swamp deposits
  in old stream meanders that were used to cultivate taro
  \citep{tuggle97:_archaeol_resear_areas_propos_devel,dye98}.  On the
  north part of the plain, nearer the foothills of Keolu Hills, are
  Sites 50--80--11--4856 and --4857, which were also likely
  coterminous, and which appear to represent the same range of
  activities as Site --4853.

#+end_src

** DONE Stratigraphy
#+srcname: stratigraphy
#+begin_src latex  :tangle o18_ao.tex
  Excavations for cultural resources management carried out at sites on
  the plain provide data for a model of regional cultural stratigraphy.
  The model groups deposits into one of three horizons:
  \begin{inparaenum}[(i)]
  \item Horizon 1 is the modern surface consisting of secondarily
    deposited sand, historic-era and traditional Hawaiian cultural
    materials, and pockets of volcanic fill material laid down during
    construction of military facilities;
  \item Horizon 2 is the traditional Hawaiian cultural deposit, often
    truncated by heavy machinery during construction of military
    facilities; and
  \item Horizon 3 is the underlying basal sand that was laid down as
    local sea level fell from its mid-Holocene +1.8~m highstand
    \citep{fletcher96} prior to settlement of the islands.
  \end{inparaenum}
  
  The model was developed to capture variability with distance from
  the coast, the source of trade wind-driven sand that represents the
  primary natural mode of deposition since the plain was first
  inhabited, and the degree to which cultural activities included
  excavation of pits primarily for cooking fires, but also for posts
  and trash disposal.  Pit excavation is responsible for moving
  artifacts and other cultural materials down the stratigraphic
  profile and contribute markedly to the thickness of the cultural
  deposit (fig.~\ref{fig:stratigraphy}).
  
  \begin{figure}[htb!]
    \centering
    \includegraphics[width=84mm]{graphics/strat-overview.png}
    \caption{Regional cultural stratigraphy along a hypothetical
    transect running inland from the beach, showing the relative
    effects of ongoing sand deposition and traditional Hawaiian pit
    excavation.}
    \label{fig:stratigraphy}
  \end{figure}
  
  At the inland edge of the plain, illustrated by profile A in
  figure~\ref{fig:stratigraphy}, sand deposition is slight and
  few pits were excavated in traditional Hawaiian times.  The cultural
  deposit here can be characterized as a paleosol whose surface
  includes a low density of cultural material that appears to have
  been discarded upon it in a more-or-less random fashion.  Moving
  toward the coast, through profiles B, C, and D, both the intensity
  of cultural deposition and pit excavation increases, creating a
  thicker cultural deposit beneath which individual pit features can
  be discovered as dark stains in the light-colored basal sands.
  Closer to the coast, represented in the figure by profile E, the
  thickness of the cultural deposit reaches a maximum due to a higher
  intensity of use and a larger volume of aeolian sand deposit from
  the nearby beach.  The frequency of pit excavation here is such that
  it is rarely possible to identify individual features in the
  underlying basal sand.  Instead, the base of the cultural deposit
  consists entirely of the bases of pits excavated atop and through
  one another.  At Site 50--80--15--4856, where the stratigraphy
  corresponded to the model represented by profile E, it was estimated
  that the number and volume of pits excavated in traditional Hawaiian
  times were sufficient to turn over the cultural deposit completely
  three times.  Closer to the beach, the level of cultural activity
  drops somewhat and the influx of aeolian sand increases markedly,
  creating a relatively complex stratigraphy in which cultural
  deposits are interspersed with layers and lenses of beach sand.
  This is the situation encountered during excavations at O18, where
  two primary traditional Hawaiian cultural deposits, Layers II and
  III, along with several smaller sub-layers or lenses were
  identified.
  
  One implication of the model is that the relatively complex
  stratigraphy at O18 in comparison to sites farther inland on the
  plain is not an indication of greater antiquity.  Instead, it is a
  function of the site's proximity to the beach.  In this view, the
  O18 site is the coastal fringe of traditional Hawaiian settlement on
  the plain, where the focus of activity was a short distance inland,
  away from the constant influx of windblown sand and from periodic
  inundation by storm waves.
#+end_src
* Age of the site [3/3]
  - Add interval between first settlement and site establishment
  - 67% hpd 260--459
  - 95% hpd 100--509
** DONE Age of O18

#+srcname: O18-age
#+begin_src latex  :tangle o18_ao.tex 
  \section{Age Determinations and Analysis}
  \label{sec:age-determinations}
  
  The nine new age determinations were processed in two batches
  independently of one another.  Five collections of wood charcoal,
  two made by Lloyd Soehren of Bishop Museum in 1966 and three by the
  University of Hawaii field school in 1967, were submitted by Valerie
  Curtis, then an archaeologist with the U.S. Air Force, to Gail
  Murakami of the International Archaeological Research Institute,
  Inc.\ Wood Identification Laboratory for taxon identification.  The
  identified samples were submitted to Beta-Analytic, Inc.\ for \rc\
  dating by the accelerator mass spectrometry (AMS) method
  (table~\ref{tab:calibration}).
  
  \begin{table}[p]
    \topcaption{Age determinations on mostly short-lived specimens}
    \label{tab:calibration}
    \footnotesize
    \begin{tabularx}{\textwidth}{llYrrllrr}
      \toprule \textbf{Sample} & \textbf{Unit} & \textbf{Material} &
      $\mathbf{\delta^{13}}$\textbf{C} &
      \multicolumn{1}{c}{\textbf{CRA}}& \textbf{Age (\textsc{ad})}\fn{1} &
      $\mathbf{j}$ & $\mathbf{P_{j1}}$ & $\mathbf{P_{j2}}$ \\
      \midrule
      \multicolumn{7}{l}{Layer II} \\
      Beta-248821 & B-20 & Pearl shell & -1.6 & 620 $\pm$ 40 &
      1670--1859 & $\theta_1$  & 0.14 &  0.05 \\
      Beta-231223 & A-3 & \alert{\textit{Nestegis sandwicensis}} & -23.5
      & \alert{710 $\pm$ 40} & \multicolumn{1}{c}{---} & $\theta_2$ &
      \alert{0.98}  & \multicolumn{1}{c}{---} \\
      \addlinespace
      \multicolumn{7}{l}{Layer III} \\
      Beta-231220 & EE-15 & \textit{Dodonaea viscosa} & -24.6 & 870
      $\pm$ 40  & 1060--1279 & $\theta_3$ & 0.10 & 0.09 \\
      Beta-231221 & EE-15 & \textit{Diospyros sandwicensis} & -26.2 &
      \alert{680 $\pm$ 40} & 1260--1399 & $\theta_4$ & 0.11 & 0.11 \\
      Beta-231222 & C-5 & \textit{Canthium odoratum} & -26.5 &
      \alert{490 $\pm$ 40} & 1310--1499 & $\theta_5$ & 0.14 & 0.15 \\ 
      Beta-248818 & C-6 & Pearl shell & +0.5 &  820 $\pm$ 40 &
      1430--1689 & $\theta_6$ & 0.12 & 0.08  \\ 
      Beta-248819 & C-6 & Pearl shell & +2.3 &  840 $\pm$ 40 &
      1420--1679 & $\theta_7$ & 0.11 & 0.08 \\ 
      Beta-248820 & A-6 & Pearl shell & +1.5 &  790 $\pm$ 40 &
      1440--1699 & $\theta_8$ & 0.15 & 0.09 \\ 
      \addlinespace
      \multicolumn{7}{l}{Layer not identified} \\
      Beta-231224 & A-3 & \textit{Canthium odoratum} & -24.0 & 690 $\pm$
      40 & \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---} &
      \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---}\\
      \bottomrule
      \addlinespace
      \multicolumn{9}{l}{\fn{1}95\% highest posterior density region.}\\
    \end{tabularx}
  \end{table}
    
  A second set of four age determinations on pearl shell manufacturing
  waste was selected from the O18 collections held by the U.S. Air
  Force and submitted by \tsdye* to Beta-Analytic, Inc.\ for AMS
  dating (table~\ref{tab:calibration}).  Pearl shell, produced by
  mollusks in the genus \textit{Pinctada}, was a favored material for
  fishhook manufacture in traditional Hawai`i.  The cross-laminar
  structure of the shell gives it exceptional strength for
  applications like fishhooks that generate high levels of stress at
  the bend.  \textit{Pinctada} shell is a suitable dating material
  because the animal is a sessile filter-feeder that takes up its
  carbon from the general ocean water around it, and not from an old
  limestone substrate \citep{dye94b}.  The current best estimate of
  the apparent age of the ocean water around Hawai`i yields a
  reservoir correction factor of 110 $\pm$ 80.  The large standard
  deviation of this estimate is likely due to regional patterns of
  variability in the apparent age of surface waters around Hawai`i
  that are not yet understood completely.  Additional information on
  this variability might make it possible in the future to apply a
  more precise estimate in the calibration of these samples.  This
  might yield slightly different calibrated ages for the samples, one
  from Layer II and three from Layer III, but will not alter the fact
  that these samples returned \rc\ age estimates that were internally
  consistent, a first in the long history of \rc\ dating at O18.
  
  % \begin{figure}[htb!]
  %   \includegraphics[width=\textwidth]{graphics/pearl-shell}
  %   \caption{Pearl shell manufacturing waste submitted for \rc\ dating:
  %     \textit{a}, Beta-248818; \textit{b}, Beta-248819; \textit{c},
  %     Beta-248820; \textit{d}, Beta-248821. The scale bar is
  %     1~cm.}  \label{fig:pearl-shell}
  % \end{figure}
  
  Notable features of Table~\ref{tab:calibration} have been set off in
  boldface.  One of the samples, Beta-231224, could not be assigned to
  either Layer II or Layer III and is not considered further here.
  The single sample from Layer II is wood charcoal from a tree known
  in Hawai`i as \gls{olopua}.  Although the life span of
  \gls{olopua} is not known, the fact that it is a tree indicates the
  possibility that the sample has in-built age.  In fact, the age
  determination returned by the laboratory is stratigraphically
  inverted with two of the Layer III samples.  Beta-231220, the age
  estimate for charcoal from a shrub known in Hawai`i as
  \gls{`a`ali`i}, does not suffer the effects of in-built age and is
  the most reliable estimate for the antiquity of settlement at O18.
#+end_src
  
** DONE Calibration
#+srcname: calibration
#+begin_src latex  :tangle o18_ao.tex
  A Bayesian model of O18 stratigraphy relates each of the dated samples
  to the calendric ages represented by the two primary cultural
  deposits.  The symbols $\theta_{2-5}$ represent the calendar ages of
  the archaeological events associated with burning the four dated wood
  charcoal pieces and $\theta_1$ and $\theta_{6-8}$ represent calendar
  ages of manufacturing events, presumably of pearl shell fishhooks
  (table~\ref{tab:calibration}, column \textbf{j}).  These are related
  to the calendar ages of the start and end of deposition of the two
  primary cultural deposits; $\alpha_3$ and $\beta_3$ represent the
  start and end of deposition, respectively, of Layer III, and
  $\alpha_2$ and $\beta_2$ represent the start and end of deposition,
  respectively, of Layer II.  The known stratigraphic relations of
  $\theta_{2-8}$ to the layer boundaries are set out in (\ref{eq:1}),
  where $>$ means ``is older than'' and $\geq$ means ``is older than or
  the same age as.''
    
  \begin{equation}
    \label{eq:1}
    \phi_2 \geq \alpha_3 \geq \theta_{3-8} \geq \beta_3 > \alpha_2
    \geq \theta_{1, 2} \geq \beta_2 \geq \phi_1
  \end{equation}

  
  For the sake of brevity, (\ref{eq:1}) groups the $\theta$ from each
  layer in an unconventional way; the $\theta$ are understood to be
  unordered so there are no stratigraphic relations among them.  
    
  The salient points of (\ref{eq:1}) are:
  \begin{itemize}
  \item the onset of Layer III deposition, $\alpha_3$, began either at,
    or sometime after, the time Hawai`i was colonized by Polynesians,
    which is modeled here as normally distributed, $\phi_2$ =
    \textsc{ad} 800 $\pm$ 50
    \citep{athens02:_avifaun_extin_veget_chang_and};
  \item the calendar ages of three dated burning events, $\theta_{3-5}$,
    and three dated manufacturing events, $\theta_{6-8}$, fall within
    the period of time represented by the deposition of Layer III;
  \item the calendar ages of the burning and manufacturing events,
    $\theta_{3-8}$, are unordered, i.e., there is no stratigraphic
    information on their ages relative to one another;
  \item the calendar ages of a burning event, $\theta_{2}$, and a
    manufacturing event, $\theta_{1}$, fall within the period of time
    represented by the deposition of Layer II;
  \item the calendar ages of the burning and manufacturing events,
    $\theta_{2}$ and $\theta_{1}$, are unordered, i.e., there is no
    stratigraphic information on their ages relative to one another;
  \item there is a hiatus between the end of deposition of Layer III,
    $\beta_3$, and the start of deposition of Layer II, $\alpha_2$, as
    indicated by the $>$ symbol; and
  \item the end of layer II deposition, $\beta_2$, was either before or
    during the time cattle ranching was established on the Waim\={a}nalo
    Plain, which is modeled here as normally distributed, $\phi_1$ =
    \textsc{ad} 1830 $\pm$ 20.
  \end{itemize}
    
  This model was implemented with the BCal software package
  \citep{bcal} using the most recent atmospheric and marine
  calibration curves \citep{reimer09:_intcal_marin_radioc_age_calib}.
  In an effort to identify outliers among the age determinations, each
  one was assigned an uninformative outlier prior probability of 0.1,
  following a procedure set out by
  \citet{christen94:_summar_set_of_radioc_deter}.  The intial run of
  the software clearly identified Beta-231223 as an outlier; the value
  of 0.98 in the column, $\mathbf{P_{j1}}$ stands out from the rest of
  the values in the column, which differ little from their initial
  values.  Beta-231223 was omitted from the analysis and a subsequent
  run of the software failed to detect outliers, as shown in the
  column, $\mathbf{P_{j2}}$, where values are all close to their
  initial values.  The seven age determinations for O18 used in
  subsequent analyses are one more than the six potentially useful age
  determinations available previously.
    
  Age estimates returned by the software for parameters of the model
  establish a chronology for the O18 site and its constituent layers.
  The 67\% highest posterior density region, equivalent to a one
  standard deviation estimate, for initial settlement of the site,
  $\alpha_3$, is \textsc{ad} 1040--1219 (fig.~\ref{fig:a3},
  \textit{bottom left}).  This initial period of deposition at the
  site, identified by archaeologists as Layer III, came to an end in
  \textsc{ad} 1580--1699 (fig.~\ref{fig:a3}, \textit{bottom right}).
  After a hiatus marked stratigraphically by a layer of beach sand,
  cultural deposition of Layer II began in \textsc{ad} 1670--1789
  (fig.~\ref{fig:a3}, \textit{top left}) and continued until
  \textsc{ad} 1770--1859 (fig.~\ref{fig:a3}, \textit{top right}).
  There is little evidence that the site was abandoned in traditional
  Hawaiian times.  For example, the probability that $\beta_2$ is
  older than \textsc{ad} 1778, the year Cook sailed to Hawai`i, is
  0.31.  Thus, given the present dating evidence and the stratigraphic
  model of the O18 site, it is more than twice as likely that the site
  was abandoned sometime after Cook.
  
  \begin{figure}[htb!]
  \centering
  \includegraphics[width=84mm]{o18_layers.png}
    \caption{Estimated ages of Layers II and III at O18: \textit{top
    left}, early boundary of Layer II; \textit{top right}, late
    boundary of Layer II; \textit{bottom left}, early boundary of Layer
    III; \textit{bottom right}, late boundary of Layer III.}
    \label{fig:a3}
  \end{figure}
    
  An advantage of a model-based Bayesian calibration is that it is
  possible to derive estimates for time intervals of interest.  The
  O18 site has figured in interpretations of initial Polynesian
  settlement of Hawai`i \citep{kirch85}; it is interesting to estimate
  the interval between settlement and establishment of the site.  The
  67\% highest posterior density region for the time interval between
  $\phi_2$ and $\alpha_3$ is 260--459 years
  (fig.~\ref{fig:duration-3}, \textit{top left}).  The initial period
  of cultural deposition at the site, represented by Layer III, was
  quite long.  The 67\% highest posterior density region for the time
  interval between $\alpha_3$ and $\beta_3$ is 400--629 years
  (fig.~\ref{fig:duration-3}, \textit{top right}).  In contrast, the
  hiatus between Layers III and II appears to have been relatively
  short.  The estimated duration of this hiatus, which is represented
  stratigraphically by a layer of light-colored beach sand, has a 67\%
  highest posterior density region of 10--109 years
  (fig.~\ref{fig:duration-3}, \textit{bottom left}).  The duration of
  Layer II was short compared to Layer III.  The 67\% highest
  posterior density region for the time interval between $\alpha_2$
  and $\beta_2$ is 10--80 years.
    
  \begin{figure}[htb!]
    \centering
    \includegraphics[width=84mm]{o18_intervals.png}
    \caption{Time intervals at O18: \textit{top left}, the interval
      between Polynesian settlement of Hawai`i and establishment of
      O18; \textit{top right}, duration of Layer III; \textit{bottom
        left}, duration of hiatus between Layers II and III;
      \textit{bottom right}, duration of Layer II.}
    \label{fig:duration-3}
  \end{figure}
#+end_src latex

** DONE Age summary
#+srcname: age-summary
#+begin_src latex  :tangle no
  In summary, the chronology of Site O18 as estimated by \rc\ dates on
  short-lived materials within a Bayesian model appears to begin
  sometime in the eleventh to thirteenth centuries \textsc{ad} and to
  have continued, with interruptions, through to the end of the
  traditional Hawaiian period.  These interruptions, indicated
  statigraphically by deposits of light-colored sand, appear to have
  been relatively brief.  The hiatus between Layers III and II was
  probably less than a century and could have been as short as a
  decade.  These results fit well with the regional model of cultural
  stratigraphy, which places O18 at the coastal fringe of widespread
  traditional Hawaiian use of the Waim\={a}nalo Plain.
#+end_src
* Regional picture [2/2]

** DONE Regional data

#+srcname: regional-data
#+begin_src latex  :tangle o18_ao.tex
  \section{O18 in Regional Perspective}
  \label{sec:regional-perspective}
  
  The Bayesian model can be extended to include other sites on the
  Waim\={a}nalo Plain.  Cultural resources management excavations at
  sites 50--80--15--4851 and --4853 and 50--80--11--4856 and --4857
  have yielded 37 \rc\ age determinations, 35 on charcoal from
  identified short-lived taxa and two on pearl shell manufacturing
  waste (table~\ref{tab:ages}).  Each of the sites consists of the
  remnants of a single cultural deposit that typically lacks internal
  stratification.  Because no stratigraphic relationships between the
  deposits of these sites and the layers of O18 have been established,
  they are each modeled as single phases independent of one another
  and of Layers II and III at O18.  Using the short-hand described
  earlier, the model can be extended with the addition of the
  following inequalities:
  
  \begin{equation}
    \label{eq:4851}
    \alpha_{4851} \geq \theta_{9-11} \geq \beta_{4851}
  \end{equation}
  
  \begin{equation}
    \label{eq:4853}
    \alpha_{4853} \geq \theta_{12-27} \geq \beta_{4853}  
  \end{equation}
  
  \begin{equation}
    \label{eq:4856}
    \alpha_{4856} \geq \theta_{28-41} \geq \beta_{4856}  
  \end{equation}
  
  \begin{equation}
    \label{eq:4857}
    \alpha_{4857} \geq \theta_{42-45} \geq \beta_{4857}  
  \end{equation}
  
  \begin{table}[htb!]
    \centering
    \scriptsize
    \topcaption{\rc\ ages of short-lived materials from other sites on the
      Waim\={a}nalo Plain}
    \label{tab:ages}
    \begin{tabularx}{\textwidth}{llXrrl}
      \toprule Laboratory & Fire-pit & Material & $\delta^{13}$C &
      CRA\fn{1} & $\mathbf{j}$ \\
      \midrule
      \multicolumn{5}{l}{Site 50--80--15--4851} \\
      Beta-111023\fn{2} & Feature 3 & cf.\ \latin{Rauvolfia
        sandwicensis} &
      -26.9 & 310$\pm$40  & $\theta_9$ \\
      Beta-111024\fn{2} & Feature 2 & \latin{Sida} cf.\ \latin{fallax} &
      -26.8 & 140$\pm$60 & $\theta_{10}$ \\
      Beta-111025\fn{2} & Feature 1 & \latin{Sida} cf.\ \latin{fallax} &
      -24.2 & 540$\pm$50 &  $\theta_{11}$ \\
      \addlinespace
      \multicolumn{5}{l}{Site 50--80--15--4853} \\
      Beta-101869\fn{2} & Feature 6 & \latin{Chamaesyce} sp. & -12.9 &
      230$\pm$60 & $\theta_{12}$ \\
      Beta-101871\fn{2} & Feature 9 & cf.\ \latin{Osteomeles
        anthyllidifolia} & -25.3 & 720$\pm$40 & $\theta_{13}$ \\
      Beta-101872\fn{2} & Feature 10 & cf.\ \latin{Osteomeles
        anthyllidifolia} & -24.7 & 680$\pm$40 & $\theta_{14}$ \\
      Beta-111022\fn{2} & Feature 1 & \latin{Sida} cf.\ \latin{fallax}
      & -27.5 & 150$\pm$40 & $\theta_{15}$ \\
      Beta-120317\fn{2} & Feature 1 & \latin{Sida} cf.\ \latin{fallax}
      & -21.3 & 140$\pm$50 & $\theta_{16}$\\
      Beta-120318\fn{2} & Feature 5 & \latin{Sida} cf.\ \latin{fallax}
      & -26.1 & 150$\pm$50 & $\theta_{17}$ \\
      Beta-120319\fn{2} & Feature 9 & \latin{Aleurites molucanna}
      nutshell, \latin{Chenopodium oahuense}, \latin{Sida} cf.\
      \latin{fallax}
      & -25.9 & 350$\pm$80 & $\theta_{18}$ \\
      Beta-120320\fn{2} & Feature 13 & \latin{Aleurites molucanna}
      nutshell & -25.6 & 230$\pm$50 & $\theta_{19}$ \\
      Beta-120321\fn{2} & Feature 15 & \latin{Aleurites molucanna}
      nutshell & -25.0 & 110$\pm$70 & $\theta_{20}$ \\
      Beta-120322\fn{2} & Feature 16 & \latin{Chamaesyce} sp.
      & -16.8 & 310$\pm$60 & $\theta_{21}$ \\
      Beta-120323\fn{2} & Feature 17 & \latin{Aleurites molucanna}
      nutshell, \latin{Chenopodium oahuense}, \latin{Sida} cf.\
      \latin{fallax} & -27.5 & 170$\pm$60 & $\theta_{22}$ \\
      Beta-120324\fn{2} & Feature 18 & \latin{Aleurites molucanna}
      nutshell & -25.2 & 250$\pm$50 & $\theta_{23}$ \\
      Beta-120325\fn{2} & Feature 19 & \latin{Aleurites molucanna}
      nutshell & -25.2 & 270$\pm$70 & $\theta_{24}$ \\
      Beta-120326\fn{2} & Feature 20 & \latin{Aleurites molucanna}
      nutshell, \latin{Chenopodium oahuense}, \latin{Sida} cf.\
      \latin{fallax} & -14.0 & 330$\pm$60 & $\theta_{25}$ \\
      Beta-120327\fn{2} & Feature 24 & \latin{Aleurites molucanna}
      nutshell & -23.0 & 400$\pm$70 & $\theta_{26}$ \\
      Beta-120328\fn{2} & Feature 25 & \latin{Sida} cf.\ \latin{fallax}
      & -25.5 & 220$\pm$50 & $\theta_{27}$ \\
      
      \addlinespace
      \multicolumn{5}{l}{Site 50--80--11--4856}\\
      Beta-208589\fn{3} & & \latin{Chenopodium oahuense} wood charcoal & -26.6 & 140$\pm$40 & $\theta_{28}$\\
      Beta-208590\fn{3} & & \latin{Sida} cf.\ \latin{fallax} wood
      charcoal & -24.9  &  90$\pm$40 & $\theta_{29}$\\
      Beta-208591\fn{3} & & \latin{Aleurites moluccana} nutshell & -25.7
      &  140$\pm$40 & $\theta_{30}$\\
      Beta-246786\fn{4} & Feature 4 & \latin{Sida} cf.\ \latin{fallax}
      wood charcoal & -25.4 & 380$\pm$40 & $\theta_{31}$\\
      Beta-251245\fn{4} & Feature 5 & \latin{Chenopodium oahuense} wood
      charcoal & -24.5 & 260$\pm$40 & $\theta_{32}$ \\
      Beta-251243\fn{4} & Feature 9 & \latin{Aleurites moluccana}
      nutshell charcoal & -24.9 & 350$\pm$40 & $\theta_{33}$ \\
      Beta-251244\fn{4} & Feature 10 & \latin{Sida} cf.\ \latin{fallax}
      wood charcoal & -24 & 250$\pm$40 & $\theta_{34}$ \\
      Beta-251242\fn{4} & Feature 12 & \latin{Sida} cf.\ \latin{fallax}
      wood charcoal & -24.4 & 200$\pm$40 & $\theta_{35}$ \\
      Beta-251246\fn{4} & Feature 17 & \latin{Chenopodium oahuense} wood
      charcoal & -21.9 & 240$\pm$40 & $\theta_{36}$ \\
      Beta-251247\fn{4} & Feature 22 & \latin{Cordyline fruticosa} wood
      charcoal & -22.6 & 450$\pm$40  & $\theta_{37}$ \\
      Beta-251248\fn{4} & Feature 23 & \latin{Aleurites moluccana}
      nutshell
      charcoal & -25.6 & 390$\pm$40  & $\theta_{38}$ \\
      Beta-200230\fn{5} & Feature 22 & \latin{Chamaesyce sp.} wood charcoal
      & -11.3 & 550$\pm$40 & $\theta_{39}$ \\
      Beta-208588\fn{3} & & Pearl shell & -0.1 & 630$\pm$40 & $\theta_{40}$ \\
      Beta-208587\fn{3} & & Pearl shell & -2.7 & 630$\pm$40 &
      $\theta_{41}$ \\
  
      \addlinespace
      \multicolumn{5}{l}{Site 50--80--11--4857}\\
      Beta-200229\fn{5} & Feature 11 & \latin{Sida} cf.\ \latin{fallax}
      wood charcoal & -25.6  &  170$\pm$40 & $\theta_{42}$ \\
      Beta-200228\fn{5} & Feature 12 & \latin{Chamaesyce} sp.\ wood
      charcoal & -25.7  &  200$\pm$40 & $\theta_{43}$ \\
      Beta-260904\fn{6} & Context 12 & cf.\ \latin{Chamaesyce} sp.\ wood
      charcoal  & -23.4  &  580$\pm$40 & $\theta_{44}$ \\
      Beta-260905\fn{6} & Context 13 & \latin{Sida} cf.\ \latin{fallax}
      wood charcoal & -26.4  &  400$\pm$40 & $\theta_{45}$\\
  
  
      \bottomrule \multicolumn{5}{l}{\fn{1} Conventional \rc\ age
        \citep{stuiver-polach77}.} \\
      \multicolumn{5}{l}{\fn{2} \citet{dye00:_effec}.} \\
      \multicolumn{5}{l}{\fn{3}
        \citet*{mcelroy06:_archaeol_monit_and_inves_durin}.} \\
      \multicolumn{5}{l}{\fn{4}
        \citet*{zzz_lebo09:_pre_const_archaeol_survey_for}.}\\
      \multicolumn{5}{l}{\fn{5}
        \citet*{putzi05:_archaeol_monit_repor_for_replac}.}\\
      \multicolumn{5}{l}{\fn{6} \citet*{dye09:_pre_archaeol_resour_survey_new}.}\\
    \end{tabularx}
    
  \end{table}
#+end_src
    

** DONE Regional analysis
   - When other sites were established
   - All sites presumably abandoned in early historic period
   - Intervals between establishment of O18 and other sites
   - 4856, 150--359 years after O18
   - 4851, -20--349 years after O18
   - 4857, 0--309 years after O18
   - 4853, 70--279 years after O18
   - p 4851 > Layer III = 0.21
   - alpha 3 at O18, 67%, 1040--1219
   - alpha 4853, 67% 710-571 BP, 1240--1379 AD
   - alpha 4857, 67% 760-551 BP, 1190--1409 AD
   - alpha 4851, 67% 770-521 BP, 1160--1429 AD
   - alpha 4856, 67% 620-521 BP, 1360--1429 AD


#+srcname: regional-analysis-1
#+begin_src latex  :tangle o18_ao.tex
  Based on the current dating evidence, sites 50--80--15--4851 and
  --4853 and 50--80--11--4856 and --4857 were all established after O18.
  Site 50--80--15--4851, located on the opposite bank of Puh\={a} Stream
  from O18, is likely to be the oldest among the four.  It was
  established \textsc{ad} 1160--1429, based on the 67\% highest
  posterior density region (fig.~\ref{fig:initial-use}, \textit{top left}).
  Penecontemporaneously, Site 50--80--11--4857, located inland and
  north of O18, was established in \textsc{ad} 1190--1409
  (fig.~\ref{fig:initial-use}, \textit{bottom right}).  Site 50--80--15--4853,
  immediately inland of site O18, has been extensively dated and appears
  to have been established at a later time.  The 67\% highest posterior
  density region for the site's establishment is \textsc{ad} 1240--1379
  (fig.~\ref{fig:initial-use}, \textit{top right}).  Finally, site
  50--80--11--4856, located on the coast north of O18, was established
  in \textsc{ad} 1360--1429 (fig.~\ref{fig:initial-use}, \textit{bottom left}),
  apparently later than Site 50--80--11--4857 located immediately
  inland.  The probability that 50--80--11--4857 was established earlier
  than 50--80--11--4856 is 0.88.
  
  \begin{figure}[htb!]
    \centering
    \includegraphics[width=84mm]{site-establishment.png}
    \caption{Initial site use on the Waim\={a}nalo Plain: \textit{top
        left}, 50--80--15--4851; \textit{top right}, 50--80--15--4853;
      \textit{bottom left}, 50--80--11--4856; \textit{bottom right},
      50--80--11--4857.}
    \label{fig:initial-use}
  \end{figure}
#+end_src

#+results: regional-analysis-1
#+BEGIN_LaTeX
Based on the current dating evidence, sites 50--80--15--4851 and
--4853 and 50--80--11--4856 and --4857 were all established after O18.
Site 50--80--15--4851, located on the opposite bank of Puh\={a} Stream
from O18, is likely to be the oldest among the four.  It was
established \textsc{ad} 1160--1429, based on the 67\% highest
posterior density region (fig.~\ref{fig:initial-use}, \textit{top left}).
Pene-contemporaneously, Site 50--80--11--4857, located inland and
north of O18, was established in \textsc{ad} 1190--1409
(fig.~\ref{fig:initial-use}, \textit{bottom right}).  Site 50--80--15--4853,
immediately inland of site O18, has been extensively dated and appears
to have been established at a later time.  The 67\% highest posterior
density region for the site's establishment is \textsc{ad} 1240--1379
(fig.~\ref{fig:initial-use}, \textit{top right}).  Finally, site
50--80--11--4856, located on the coast north of O18, was established
in \textsc{ad} 1360--1429 (fig.~\ref{fig:initial-use}, \textit{bottom left}),
apparently later than Site 50--80--11--4857 located immediately
inland.  The probability that 50--80--11--4857 was established earlier
than 50--80--11--4856 is 0.88.

\begin{figure}[htb!]
  \centering
  \includegraphics[width=84mm]{site-establishment.png}
  \caption{Initial site use on the Waim\={a}nalo Plain: \textit{top
      left}, 50--80--15--4851; \textit{top right}, 50--80--15--4853;
    \textit{bottom left}, 50--80--11--4856; \textit{bottom right},
    50--80--11--4857.}
  \label{fig:initial-use}
\end{figure}
#+END_LaTeX
#+srcname: regional-analysis-2
#+begin_src latex :tangle o18_ao.tex     
  Another way to look at the site establishment estimates is relative to
  the establishment of O18.  All of the
  posterior probability distributions have left tails that extend past
  zero and thus each site retains some probability of having been
  established before O18.  These probabilities are all rather slim,
  however.  The site with the greatest probability of having been
  established before O18, 50--80--11--4851, has a probability of 0.2.
  Using 67\% highest posterior density regions: Site 50--80--15--4851
  was settled 10 years earlier than to 349 years after O18
  (fig.~\ref{fig:site-establishment}, \textit{top left}); site
  50--80--11--4857 was settled at the same time as O18 to 319 years
  later (fig.~\ref{fig:site-establishment}, \textit{bottom right}); site
  50--80--11--4853 was settled 60--279 years after O18
  (fig.~\ref{fig:site-establishment}, \textit{top right}); and site
  50--80--11--4856 was settled 160--359 years after O18
  (fig.~\ref{fig:site-establishment}, \textit{bottom left}).
  
       \begin{figure}[htb!]  \centering
         \includegraphics[width=84mm]{after-o18.png} \caption[Sequence of
         site establishment]{Sequence of site establishment---the
           interval between establishment of O18 and other sites:
           \textit{top left}, Site 50--80--15--4851; \textit{top right},
           Site 50--80--15--4853; \textit{bottom left}, Site
           50--80--11--4856; \textit{bottom right}, Site
           50--80--11--4857.  Note that there is a small probability
           that each of the sites was established before O18.}
         \label{fig:site-establishment} \end{figure}
#+end_src
* Conclusion [4/4]
#+srcname: conclusion
#+begin_src latex  :tangle o18_ao.tex 
  \section{Summary and Conclusion}
  \label{sec:conclusion}
#+end_src

** DONE Regional prehistory [5/5]
    - [X] O18 not an example of an early site, it was settled at least
      240 years after Polynesian colonization and perhaps as much as
      449 years after
    - [X] O18 is likely the earliest site on the plain
    - [X] Other sites settled over the next approximately 3-4 centuries,
      in what appears to be a piece-meal fashion, and not a radiation
      out from O18.
    - [X] Discuss the early date on kukui nutshell.  How it got into the
      O18 deposit is a mystery, but if it is not spuriously old due
      to laboratory error, then it might indicate early establishment
      of the tree on the Waimanalo plain.  The calibrated age shown
      in Table 2 accepts the material as primarily deposited in Layer
      II.  If this constraint is relaxed, and the date is calibrated
      as constrained only by phi1 and phi2, then at 67% AD 850--1160.
      The probability it was growing before O18 was established is
      XX%
    - [X] This carries with it the possibility that there were
        settlements in Waimanalo that pre-dated O18
    - [ ] Results differ by timing and by the fact that events of
      interest are estimated directly.
#+srcname: prehistory
#+begin_src latex  :tangle o18_ao.tex 
  Seven new \rc\ age determinations on short-lived materials yield a
  chronology for O18 that differs from previous estimates.  The
  results clearly indicate that O18 was settled later than previously
  estimated.  The 67\% highest posterior density region for the true age of $\alpha_3$ is
  \textsc{ad} 1040--1219, which is 4--9 centuries younger than
  previous estimates.  The hypothesis that O18 was occupied during an
  early phase of Polynesian settlement is, on present evidence, false.
  The best estimate, based on present evidence, places initial site
  use 260--459 years after the archipelago was discovered and
  colonized.  With this new, ``late'' chronology, O18 joins site H1 on
  Hawai`i Island \citep{dye92} and the H\={a}lawa Dune site on
  Moloka`i \citep{kirch07:_recon_hawaiian_cultur_sequen} in a growing
  group of relatively late sites once believed to have been examples
  of early Hawaiian settlement.
      
  
  
      % Contrast this with
      % the previous situation, where estimates were based on \textit{ad
      % hoc} procedures.  The initial set of five GaK dates supported two
      % estimates that differed by three centuries. The addition of the
      % Beta-Analytic dates did little to change the shape of the
      % data,\footnote{The two ANU dates are clearly anomalous.} in
      % fact it is remarkable how closely the new dates reprised the old
      % ones.  Yet, this similar but augmented data set supported a 500 year
      % revision of the estimate.  Clearly, these \textit{ad hoc} estimates
      % were responding to information adjunct to the dates
      % themselves---initially to now-discredited old ``dates'' on volcanic
      % glass from O18 and subsequently to a widely-accepted
      % paleoenvironmental estimate for first settlement in the eighth or
      % ninth centuries \textsc{ad}.  In both cases these adjunct data are
      % not explicitly modeled in the calibration, but are instead applied
      % idiosyncratically.  The Bayesian model is sufficiently robust to
      % accommodate both the addition of new data and a revised estimate of
      % when the islands were initially colonized.
      
      The situation is similar with respect to when O18 was abandoned.
      The new dates on short-lived materials, calibrated and
      interepreted within a Bayesian framework, indicate that the site was
      abandoned at the end of traditional Hawaiian times in the late
      eighteenth or early nineteenth centures, some 3--6 centuries later
      than earlier estimates.  The estimate brings the
      abandonment of O18 in line with abandonment date estimates for other sites
      on the Waim\={a}nalo Plain.
  
      One reason that previous estimates of O18 chronology were too
      old by centuries was a failure to control for the potential
      effects of old wood during the dating process, but errors
      assigning the dated samples to their correct archaeological
      contexts in a field school situation, and statistical and other
      errors in the dating laboratory probably had effects, too.  It
      is worthwhile to emphasize the ill effects of old wood; cultural
      resources management archaeologists working in Hawaii routinely
      date unidentified wood charcoal.  There is no reason to believe
      that their age determinations on unidentified wood charcoal will
      perform any better than those from O18, which proved to be poor
      estimators of site chronology.  They are essentially worthless for
      establishing archaeological chronologies.
    
      In most cases, the old dates that do a poor job of estimating
      the age of O18 provide no other useful information.  An
      exception to this is Beta-20852b on \textit{A. moluccana}
      nutshell.  This age determination does a poor job of estimating
      the age of its archaeological context in Layer II, but because
      the identified material derived from a tree introduced to the
      islands by Polynesians the age estimate itself is of interest.
      If the calendar age, $\theta_{46}$, of this age determination is
      associated with the archaeological event of planting
      \textit{kukui} trees in Waim\={a}nalo and calibrated in the
      context of a model that specifies only that this event dates to
      traditional Hawaiian times (\ref{eq:kukui}), then the 67\%
      highest posterior density region for $\theta_{46}$ is
      \textsc{ad} 840--1159, an estimate that has a 70\% probability
      of dating an event older than the establishment of O18.  Thus,
      it is likely that the \textit{A. moluccana} tree was planted by
      Hawaiians who lived at some other site in Waim\={a}nalo prior to
      settlement at O18.  Because dates from nearby sites indicate
      that O18 was established before them, this putative earlier
      settlement is likely to be located somewhere inland, probably on
      the volcanic soils that supported gardens in traditional
      Hawaiian times.  Whether cultural deposits associated with this
      putative early settlement still exist is a question for future
      research.
      
      \begin{equation}
        \label{eq:kukui}
        \phi_2 \geq \theta_{46} \geq \phi_1
      \end{equation}
  
      Finally, development of an explicit chronological model relating
      regional archaeological events to one another and set out in
      inequalities (\ref{eq:1}--\ref{eq:kukui}) means that anyone can
      replicate the estimate and explore how different parameters of
      the model affect it.  It is not possible to do this in a precise
      way with an approach that is not strictly model-based.  Changes
      in chronological estimates for sites on the Waim\={a}nalo Plain
      will most likely result from new dates on short-lived materials
      from secure stratigraphic contexts both on the Waim\={a}nalo
      Plain and beyond.  Excavation of deposits at the coastal fringe
      of Site 50--80--11--4856, for instance, might help clarify the
      processes responsible for deposition of charcoal in this active
      and variable environment at the fringe of traditional Hawaiian
      settlement on the Waim\={a}nalo Plain.  And certainly, any
      change in the estimated settlement date of the Hawaiian Islands
      would have a direct effect on the estimate of the interval
      between this event and establishment of O18.  If the change
      in the estimated settlement date were sufficiently large, it
      might even have an effect on the estimate of when O18 was
      established.
#+end_src

** REJECTED The weaknesses of the present estimates
    - Settlement dates heavily dependent on dated samples, sample size
      potentially important in interpretation
    - Calibration of shell dates, might change a bit, but not much
    - Layer II represented by a single sample, if shell calibrates a
      bit older, then the boundary between Layers II and III will change.
    - The Layer III shell dates are all relatively young and a change
      in calibration likely won't affect the estimated establishment
      date
#+srcname: evaluation
#+begin_src latex 
  The incorporation into the Bayesian calibration of a stratigraphic
  model of traditional Hawaiian sites on the Waim\={a}nalo Plain,
  expressed in inequalities (\ref{eq:1}) and (\ref{eq:4851}) through
  (\ref{eq:4857}), makes it possible to answer interpretive questions
  with probabilistic estimates, rather than with guesses or ad hoc
  arguments.  Among the probabilistic estimates
  are: \begin{inparaenum}[(i)] \item the ages of archaeological events
  of interest that could not be dated directly, including first
  settlement and abandonment of the five site, as well as the upper
  boundary of Layer III and the lower boundary of Layer II at O18;
  \item the elapsed time between events,
  including \begin{inparaenum}[(a)] \item the duration of Layers II,
  III, and the hiatus between them, \item the lag between settlement
  of O18 and other sites on the Waim\={a}nalo Plain, and \item the
  time between first settlement of Hawai`i by Polynesians and
  establishment of O18; and \end{inparaenum} \item the relative timing
  of events of interest, such as the first planting of
  \textit{A. moluccana} in Waim\={a}nalo relative to the establishment
  of O18.  \end{inparaenum} The ability of Bayesian calibration to
  yield direct answers to interpretive questions is one of its great
  strengths.
  
  Another strength of the model-based, Bayesian approach to
  calibration is that new information can be used to augment and
  refine the results.  For example, a new age determination that is
  older than others from the same site will push the estimate of site
  establishment back in time; the Bayesian calibration will indicate
  directly with probabilistic estimates how much effect the new date
  has on the regional chronology.  Similarly, refinement of $\Delta R$
  for Hawai`i might change the calibrated ages of the pearl shell
  samples by some decades.  Here, the Bayesian calibration will
  calculate the effect this has on the age of the transition from
  Layer III to Layer II at O18.  In this way, work on regional
  chronology can be collaborative, rather than based on arguments over
  how the ad hoc methods of chronologic hygiene might be applied in a
  particular instance.
#+end_src

** REJECTED Chronometric hygiene is non-scientific.
    - Although Gak dates are often dismissed, they did no worse than
      other laboratories did.
    - It is possible to use dates with large standard deviations: they
      don't mean much, but there is no intrinsic reason to discard what
      little information they might hold.
#+srcname: hygiene
#+begin_src latex 
      
#+end_src

** REJECTED There are no known old sites
     - Graph of time interval between settlement and site
       establishment for O18
     - H1 is late, cite Dye NZJA
     - Halawa Dune is late, too.
#+srcname: hawaii
#+begin_src latex 
       
#+end_src
* LaTeX postamble
#+srcname: latex-ending
#+begin_src latex :tangle o18_ao.tex
% Comment or uncomment as needed
% style=altlist another possibility
%\printglossary[type=main, style=tsdlist]
%\printglossary[type=hawaiian, style=tsdlist]
% \printglossary[type=polynesian, style=tsdlist]
% \printglossary[type=gazetteer, style=tsdlist]
% \printglossary[type=acronym, style=tsdlist]
% \printglossary[type=oldenglish, style=tsdlist]
% \printglossary[type=bio, style=tsdlist]

\addcontentsline{toc}{section}{Bibliography}
\bibliographystyle{chicago}

% Comment or uncomment as needed
% \bibliography{tsd}
\bibliography{tsd,local}

\end{document}

#+end_src
* R code for graphics

** Dated events
   - A kludge
   - Put all the csv file names in a table, then edit the table for
     input to the R function
#+srcname: thetas()
#+begin_src shell
cd r && ls *.csv
#+End_src

#+results: thetas
| alpha-2.csv                |
| beta-2.csv                 |
| alpha-3.csv                |
| beta-3.csv                 |
  
- thetas is a list inside the function

- run lapply, use string substitution to make variable names, file
  names

- need to add cbind() an identifier that can be used to label the plot
  and to segregate the plots

#+srcname: dated-events(files = thetas)
#+begin_src R :session :file output_dated_events.png
  library(ggplot2)  
  make.plot.file <- function(x, y)
    {
      r <- y
      afile <- paste("r/",x,sep="")
      anobject <- strsplit(x,".",fixed=TRUE)[[1]][1]
      z <- read.csv(file = afile)
      z <- cbind(z,label=rep(anobject,dim(z)[1]))
      r <- rbind(r, z)
      r
    }
  res <- data.frame(cal.BP=numeric(0),Posterior.probability=numeric(0),label=character(0))
  for (f in files[,1]) res <- make.plot.file(f, res)
  g <-  ggplot(res, aes(x=1950 + cal.BP, y=Posterior.probability))
  png(file="output_dated_events.png",width=168,height=100,unit='mm',res=600)
  g + geom_bar(stat='identity') + xlab("Year AD") +
  ylab("Probability") + facet_wrap(~ label)
  dev.off()
#+end_src

#+results: dated-events
[[file:output_dated_events.png]]

#+srcname: single-date(x = "alpha-4856.csv")
#+begin_src R :session
  afile <- paste("r/",x,sep="")
  anobject <- strsplit(x,".",fixed=TRUE)[[1]][1]
  ofname <- paste(anobject,".pdf",sep="")
  z <- read.csv(file = afile)
  g <-  ggplot(z, aes(x=1950 + cal.BP, y=Posterior.probability))
  g + geom_bar(stat='identity') + xlab("Year AD") +
    ylab("Probability")
  ggsave(ofname)
  ofname
#+end_src

#+results: single-date
: theta-7.pdf
** Intervals
   - This is a kludge that works for a one-off situation
   - The shell source shows all the csv files
   - Edit the results table to select the files to use as input to the
     plotting routine
#+srcname: intervals 
#+begin_src sh
cd r && ls *.csv
#+end_src

#+results: intervals
| alpha-4851-and-alpha-3.csv |
| alpha-4853-and-alpha-3.csv |
| alpha-4856-and-alpha-3.csv |
| alpha-4857-and-alpha-3.csv |

#+srcname: interval-estimates(files = intervals)
#+begin_src R :session :file output_intervals.png
  library(ggplot2)
  
  make.plot.file <- function(x, y)
    {
      r <- y
      afile <- paste("r/",x,sep="")
      anobject <- strsplit(x,".",fixed=TRUE)[[1]][1]
      z <- read.csv(file = afile)
      z <- cbind(z,label=rep(anobject,dim(z)[1]))
      r <- rbind(r, z)
      r
    }
  res <- data.frame(cal.BP=numeric(0),Posterior.probability=numeric(0),label=character(0))
  for (f in files[,1]) res <- make.plot.file(f, res)
  g <-  ggplot(res, aes(x=cal.BP, y=Posterior.probability))
  png(file="output_intervals.png",width=168,height=100,unit='mm',res=600)
  g + geom_bar(stat='identity') + xlab("Time Interval (Years)") +
  ylab("Probability") + facet_wrap(~ label, scales = "fixed")
  dev.off()
#+end_src

#+results: interval-estimates
[[file:output_intervals.png]]


** Regional date graph
   - Hard code a complex graphic
#+srcname: r-regional-data
#+begin_src R :session
  library(ggplot2)
  load(".RData")
  a3_gg <- cbind(a3, rep("O18",dim(a3)[1]))
  names(a3_gg)[3] <- "name"
  a4851 <- read.csv("alpha-4851.csv")
  a4851_gg <- cbind(a4851, rep("4851", dim(a4851)[1]))
  names(a4851_gg)[3] <- "name"
  a4853 <- read.csv("alpha-4853.csv")
  a4853_gg <- cbind(a4853, rep("4853", dim(a4853)[1]))
  names(a4853_gg)[3] <- "name"
  a4856 <- read.csv("alpha-4856.csv")
  a4856_gg <- cbind(a4856, rep("4856", dim(a4856)[1]))
  names(a4856_gg)[3] <- "name"
  a4857 <- read.csv("alpha-4857.csv")
  a4857_gg <- cbind(a4857, rep("4857", dim(a4857)[1]))
  names(a4857_gg)[3] <- "name"
  alpha_gg <- rbind(a3_gg, a4851_gg, a4853_gg, a4856_gg, a4857_gg)
#+end_src

#+results: r-regional-data
| ggplot2   |
| reshape   |
| plyr      |
| grid      |
| proto     |
| stats     |
| graphics  |
| grDevices |
| utils     |
| datasets  |
| methods   |
| base      |

#+srcname: r-regional-plot
#+begin_src R :session
  alpha_plot <- ggplot(alpha_gg, aes(x=1950 + cal.BP, y=Posterior.probability))
  pdf(file="alpha-regional.pdf", height=3.75, width=7.5)
  alpha_plot + geom_bar(stat='identity') + xlab("Year AD") +
  ylab("Probability") + facet_wrap(~ name) 
#+end_src

* REJECTED LaTeX article makefile
#+begin_src latex  
  <<latex-preamble>>
  <<latex-intro>>
  <<regional-context>>

  <<stratigraphy>>
  <<O18-age>>
  
  <<calibration>>

  <<regional-data>>
  <<regional-analysis-1>>
  <<regional-analysis-2>>

  <<conclusion>>
  <<prehistory>>

  <<latex-ending>>
#+end_src
* REJECTED Post-mortem on earlier data [2/2]
** DONE Post mortem data
#+srcname: post-mortem-data
#+begin_src latex
  \section{Post-mortem on the Early Dates}
  \label{sec:post-mortem}
    
  The Bayesian model described in the previous section can be extended
  to incorporate the early age determinations reported by
  \citet{pearson71:_bellows} and
  \citet{tuggle01:_age_bellow_dune_site_o18}.  The purpose of extending
  the model in this way is to compare the chronology produced by the
  early age determinations with the one produced by the new age
  determinations on short-lived materials.  Thus, it is important that
  the model be extended in a way that keeps the two chronologies
  separate.  This is accomplished in the BCal software by creating a
  separate set of phases for Layers II and III, as if they belonged to a
  separate site, and assigning the early age determinations to this
  separate set of phases.  This extension to the model can be
  represented by a second inequality, (\ref{eq:2}), which establishes
  new layer boundaries, indicated by an \textit{o} appended to the subscript,
  and includes six of the early age determinations, $\theta_{2, 9-13}$
  (table~\ref{tab:early-dates}).  GaK-1819 is clearly an outlier in
  Layer II and has been excluded from the analysis.
  
  \begin{equation}
    \label{eq:2}
    \phi_2 \geq \alpha_{3o} \geq \theta_{12-13} \geq \beta_{3o} > \alpha_{2o}
    \geq \theta_{2, 9-11} \geq \beta_{2o} \geq \phi_1
  \end{equation}
  
  \begin{table}[htb!]
    \topcaption{Early O18 age determinations}
    \label{tab:early-dates}
    \footnotesize
    \begin{tabular}{llrrllrr}
      \toprule
      \textbf{Sample} & \textbf{Material} &
      $\mathbf{\delta^{13}}$\textbf{C} &
      \multicolumn{1}{c}{\textbf{CRA}} &\multicolumn{1}{c}{\textbf{Age
          (\textsc{ad})}} & \multicolumn{1}{c}{\textbf{j}} &
      \multicolumn{1}{c}{$\mathbf{P_j}$} &
      \multicolumn{1}{c}{$\mathbf{P_{dep}}$} \\
      \midrule
      \multicolumn{4}{l}{Layer II} \\
      GaK-1818 & \alert{unidentified} & -25.0 & 1126 $\pm$ 124 &
      1070--1229 &
      $\theta_2$  & 0.19 & 0.00002 \\
      GaK-1819 & \alert{unidentified} & -25.0 & \alert{1616 $\pm$ 96} & \multicolumn{1}{c}{---} &
      \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---} \\
      Beta-20852a & \alert{unidentified} & -24.1 & 720 $\pm$ 130 & 1120--1309 &
      $\theta_9$ & 0.08 & 0.0002\\
      Beta-20852b & \textit{A. moluccana} nutshell & -26.4 & 1330 $\pm$
      230 & 1080--1259
      & $\theta_{10}$ & 0.20 & 0.0001\\
      GaK-1816 & \alert{unidentified} & -25.0 & 716 $\pm$ 129 & 1120--1309 &
      $\theta_{11}$ & 0.09 & 0.00008\\
      \addlinespace
      \multicolumn{4}{l}{Layer III} \\
      GaK-1817 & \alert{unidentified} & -25.0 & 1046 $\pm$ 115 & 870--1019 & 
      $\theta_{12}$ & 0.06 & 0.12 \\
      GaK-1820 & \alert{unidentified} & -25.0  & \alert{modern} & \multicolumn{1}{c}{---} &
      \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---} \\
      Beta-20853 & \alert{unidentified} & -25.0 & 1070 $\pm$ 370 &
      850--1019 &
      $\theta_{13}$ & 0.04 & 0.12 \\
      ANU-6179 & \alert{unidentified} & -24.8 & \alert{modern} & \multicolumn{1}{c}{---} &
      \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---} \\
      ANU-7027 & \textit{Cocos nucifera} & -23.1 & \alert{120 $\pm$ 132}
      & \multicolumn{1}{c}{---} 
      & \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---} & \multicolumn{1}{c}{---} \\
      \bottomrule
    \end{tabular} 
    \end{table}
#+end_src

** DONE Post mortem results
#+srcname: post-mortem-results
#+begin_src latex
  The Bayesian calibration of the early age determinations yields layer
  boundaries that correspond fairly closely to those proposed by
  \citet{tuggle01:_age_bellow_dune_site_o18}.  According to this
  analysis: the site was established sometime shortly after initial
  settlement of the islands, \textsc{ad} 800--949, a range that is
  influenced strongly by the constraints imposed by $\phi_2$; the hiatus
  between Layers III and II began in \textsc{ad} 900--1069 and ended
  when Layer II began to be deposited in \textsc{ad} 990--1169; and the
  site was abandoned in \textsc{ad} 1160--1399.%   In this case, it
  % appears that the ad hoc methods of ``chronological hygiene'' yielded
  % relatively reasonable results.
  
  A potentially more interesting question has to do with the
  probabilities that the early age determinations date events that
  took place within the period of time represented by the
  stratigraphic layers from which they were collected, as these are
  estimated by the Bayesian calibration of short-lived materials as
  represented by inequality (\ref{eq:1}).  These probabilities are
  shown in the column, $P_{dep}$ of table~\ref{tab:early-dates} where
  it can be seen that all four of the Layer II age determinations have
  vanishingly small probabilities of representing events that took
  place while that layer was being deposited.  The two Layer III age
  determinations have somewhat larger, but equally low probabilities
  of belonging to that layer.  In all cases, the age determinations
  are too old.
  
  In the cases of two Layer III dates on unidentified wood, GaK-1817 and
  Beta-20853, the most likely culprit is in-built age.  Both of these
  dates are about 200 years older than the early date on
  \textit{D. viscosa} (see table~\ref{tab:calibration}).
  \citet{dye00:_effec} has shown that controlling for in-built age
  yields age estimates that are, on average, about 100 years younger
  than age estimates on samples that don't control for in-built age.
  Discrepancies of 200 years are well within the range of potential
  in-built ages.  The three other Layer III dates, GaK-1820, ANU-6179,
  and ANU-7027 do not differ statistically from the modern standard.
  Deposition of Layer III ended sometime in the late sixteenth or
  seventeenth centuries, so these three dates are all more than 250
  years too young for their stratigraphic positions.  It seems unlikely
  that modern material could have made its way down through the fairly
  complex site stratigraphy to the base of the section, so problems of
  context are unlikely \textit{a priori}.  In practical terms, this
  leaves some type of laboratory error as the likely culprit, where
  laboratory error is interpreted broadly to include both the
  archaeological and dating laboratories.
  
  The Layer II age determinations are all older than the period during
  which the layer was deposited.  Two of the dates on unidentified
  wood charcoal, Beta-20852a and GaK-1816, are about 200 years too old
  and, like the Layer III samples, might be reasonably interpreted as
  old wood.  The other three age estimates, two on unidentified wood
  charcoal and a third on \textit{A. moluccana} nutshell, are very old
  for their stratigraphic positions.  Although the two estimates on
  unidentified wood charcoal might represent rather extreme examples
  of in-built age, perhaps because they are pieces of driftwood, the
  old age estimate on the short-lived nutshell sample suggests that
  other factors might be active.  It is tempting to suggest that all
  of these samples derived from Layer III, but they are each quite a
  bit older than the early Layer III date on \textit{D. viscosa}, so
  this possibility seems unlikely.  One is thrown back on laboratory
  error or the possibility that remnants of a deposit older than Layer
  III exposed material that was somehow redeposited in Layer II.
  Although this scenario might seem rather far-fetched, it is
  certainly within the realm of depositional processes at this coastal
  location on the fringe of traditional Hawaiian settlement.
#+end_src
* Spell check
 LocalWords:  LaTeX src ao tex noweb srcname documentclass tsdarticle Pantaleo
 LocalWords:  maketitle uncomment altlist printglossary tsdlist hawaiian toc rc
 LocalWords:  polynesian oldenglish addcontentsline bibliographystyle chicago
 LocalWords:  tsd citep pearson citet tuggle kirch Spriggs hoc Hawai Waim nalo
 LocalWords:  htb includegraphics Puh archaeol resear propos devel desilets gg
 LocalWords:  Keolu strat mortem
* Notes
** DONE Get reference for 2009 calibration curves
** The current calibration
   - copy of bellows region aug
   - note that theta numbers do not match with paper
** DONE Correct footnotes in regional 14C table
** DONE Check all date range estimates
** DONE Regenerate graphs
** DONE Add pearl shells from project 40 to the BCal analysis
   - 380 208588 Pteriidae shell <0.1 630􏰂40 -0.1 ad 1682–1846 
   - 368 208587 Pteriidae shell <0.1 630􏰂40 -2.7 ad 1682–1846 

** DONE Get 95% and 67% hpd regions from BCal, put them in text
** DONE Calculate 67% hpd region for interval between alpha 2 and beta 2
** DONE Check date list for 4856
** DONE Make argument for pearl shell being a suitable dating material.
** DONE Full information for Beta-200230
[2010-01-31 Sun]

** DONE Reference for Kirch and McCoy's re-dating of Halawa Dune site

** DONE Spriggs and Anderson argument against Gak dates
** Renumber \theta [28/28]
   - [X] 45 -> 2
   - [X] 15 -> 9
   - [X] 14 -> 10
   - [X] 46 -> 11
   - [X] 31 -> 12
   - [X] 30 -> 13
   - [X] 29 -> 14
   - [X] 28 -> 15
   - [X] 47 -> 28
   - [X] 48 -> 29
   - [X] 49 -> 30
   - [X] 32 -> 31
   - [X] 33 -> 32
   - [X] 34 -> 33
   - [X] 35 -> 34
   - [X] 36 -> 35
   - [X] 37 -> 36
   - [X] 38 -> 37
   - [X] 39 -> 38
   - [X] 40 -> 39
   - [X] 41 -> 40
   - [X] 44 -> 41
   - [X] 43 -> 44
   - [X] 44 -> 43, then added to pearl shell dates from project 40
   - [X] 40 -> 44
   - [X] 41 -> 43
   - [X] 42 -> 44
   - [X] 43 -> 45
* Changes to Page Proofs
** Page 113, column a, note
   - change "jeffrey.pantaleo.ctr@hickam.af.mil" to "jpanta4149@aol.com"
** Page 115, Table 1
   - Column *j*, all numbers following thetas should be subscripts,
     \theta_1, \theta_2, etc.
   - Column head *Pj1*, the j1 should be subscript to the P, *P_{j1}*
   - Column head *Pj2*, the j2 should be subscript to the P, *P_{j2}*
** Page 115, column b, line 2
   - Change "The single sample from Layer II is wood charcoal" to "The
     single wood charcoal sample from Layer II is"
** Page 117, column a, inequalities (2), (3), (4), and (5)
   - Need subscripts for all the greek letters,
     e.g. \alpha 4851\geq\theta 9-11\geq\beta 4851 should be \alpha_{4851}\geq\theta_{9-11}\geq\beta_{4851}
** Page 119, column a, inequality (6)
   - Need subscripts, like so, \phi_2 \geq \theta_{46} \geq \phi_1