Event and accumulation dates are density estimates of the occupation
and duration of an archaeological site (L.
Bellanger, Husi, and Tomassone 2006; L. Bellanger, Tomassone, and Husi
2008; Lise Bellanger and Husi 2012).
Event Date
Event dates are estimated by fitting a Gaussian multiple linear
regression model on the factors resulting from a correspondence analysis
- somewhat similar to the idea introduced by Poblome and Groenen (2003). This model results
from the known dates of a selection of reliable contexts and allows to
predict the event dates of the remaining assemblages.
First, a correspondence analysis (CA) is carried out to summarize the
information in the count matrix \(X\).
The correspondence analysis of \(X\)
provides the coordinates of the \(m\)
rows along the \(q\) factorial
components, denoted \(f_{ik} ~\forall i \in
\left[ 1,m \right], k \in \left[ 1,q \right]\).
Then, assuming that \(n\)
assemblages are reliably dated by another source, a Gaussian multiple
linear regression model is fitted on the factorial components for the
\(n\) dated assemblages:
\[
t^E_i = \beta_{0} + \sum_{k = 1}^{q} \beta_{k} f_{ik} + \epsilon_i
~\forall i \in [1,n]
\] where \(t^E_i\) is the known
date point estimate of the \(i\)th
assemblage, \(\beta_k\) are the
regression coefficients and \(\epsilon_i\) are normally, identically and
independently distributed random variables, \(\epsilon_i \sim
\mathcal{N}(0,\sigma^2)\).
These \(n\) equations are stacked
together and written in matrix notation as
\[
t^E = F \beta + \epsilon
\]
where \(\epsilon \sim
\mathcal{N}_{n}(0,\sigma^2 I_{n})\), \(\beta = \left[ \beta_0 \cdots \beta_q \right]'
\in \mathbb{R}^{q+1}\) and
\[
F =
\begin{bmatrix}
1 & f_{11} & \cdots & f_{1q} \\
1 & f_{21} & \cdots & f_{2q} \\
\vdots & \vdots & \ddots & \vdots \\
1 & f_{n1} & \cdots & f_{nq}
\end{bmatrix}
\]
Assuming that \(F'F\) is
nonsingular, the ordinary least squares estimator of the unknown
parameter vector \(\beta\) is:
\[
\widehat{\beta} = \left( F'F \right)^{-1} F' t^E
\]
Finally, for a given vector of CA coordinates \(f_i\), the predicted event date of an
assemblage \(t^E_i\) is:
\[
\widehat{t^E_i} = f_i \hat{\beta}
\]
The endpoints of the \(100(1 −
\alpha)\)% associated prediction confidence interval are given
as:
\[
\widehat{t^E_i} \pm t_{\alpha/2,n-q-1} \sqrt{\widehat{V}}
\]
where \(\widehat{V_i}\) is an
estimator of the variance of the prediction error:
\[
\widehat{V_i} = \widehat{\sigma}^2 \left( f_i^T \left( F'F
\right)^{-1} f_i + 1 \right)
\]
were \(\widehat{\sigma} =
\frac{\sum_{i=1}^{n} \left( t_i - \widehat{t^E_i} \right)^2}{n - q -
1}\).
The probability density of an event date \(t^E_i\) can be described as a normal
distribution:
\[
t^E_i \sim \mathcal{N}(\widehat{t^E_i},\widehat{V_i})
\]
Accumulation
Date
As row (assemblages) and columns (types) CA coordinates are linked
together through the so-called transition formulae, event dates for each
type \(t^E_j\) can be predicted
following the same procedure as above.
Then, the accumulation date \(t^A_i\) is defined as the weighted mean of
the event date of the ceramic types found in a given assemblage. The
weights are the conditional frequencies of the respective types in the
assemblage (akin to the MCD).
The accumulation date is estimated as: \[
\widehat{t^A_i} = \sum_{j = 1}^{p} \widehat{t^E_j} \times
\frac{x_{ij}}{x_{i \cdot}}
\]
The probability density of an accumulation date \(t^A_i\) can be described as a Gaussian
mixture:
\[
t^A_i \sim \frac{x_{ij}}{x_{i \cdot}}
\mathcal{N}(\widehat{t^E_j},\widehat{V_j}^2)
\]
Interestingly, the integral of the accumulation date offers an
estimates of the cumulative occurrence of archaeological events, which
is close enough to the definition of the tempo plot introduced
by Dye (2016).
