Event Date Model

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).