```
# Load packages
library(folio) # Datasets
library(kairos)
```

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).

The event date is an estimation of the *terminus post-quem* of
an archaeological assemblage. The accumulation date represents the
“chronological profile” of the assemblage. According to Lise Bellanger and Husi (2012), accumulation
date can be interpreted “at best […] as a formation process reflecting
the duration or succession of events on the scale of archaeological
time, and at worst, as imprecise dating due to contamination of the
context by residual or intrusive material.” In other words, accumulation
dates estimate occurrence of archaeological events and rhythms of the
long term.

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:

\[ t^E_i \sim \mathcal{N}(\widehat{t^E_i},\widehat{V_i}) \]

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).

Event and accumulation dates estimation relies on the same conditions and assumptions as the matrix seriation problem. Dunnell (1970) summarizes these conditions and assumptions as follows.

The homogeneity conditions state that all the groups included in a seriation must:

- Be of comparable duration.
- Belong to the same cultural tradition.
- Come from the same local area.

The mathematical assumptions state that the distribution of any historical or temporal class:

- Is continuous through time.
- Exhibits the form of a unimodal curve.

Theses assumptions create a distributional model and ordering is accomplished by arranging the matrix so that the class distributions approximate the required pattern. The resulting order is inferred to be chronological.

Predicted dates have to be interpreted with care: these dates are highly dependent on the range of the known dates and the fit of the regression.

This package provides an implementation of the chronological modeling
method developed by Lise Bellanger and Husi
(2012). This method is slightly modified here and allows the
construction of different probability density curves of archaeological
assemblage dates (*event*, *activity* and
*tempo*).

```
## Bellanger et al. did not publish the data supporting their demonstration:
## no replication of their results is possible.
## Here is a pseudo-reproduction using the zuni dataset
## Assume that some assemblages are reliably dated (this is NOT a real example)
## The names of the vector entries must match the names of the assemblages
<- c(
zuni_dates LZ0569 = 1097, LZ0279 = 1119, CS16 = 1328, LZ0066 = 1111,
LZ0852 = 1216, LZ1209 = 1251, CS144 = 1262, LZ0563 = 1206,
LZ0329 = 1076, LZ0005Q = 859, LZ0322 = 1109, LZ0067 = 863,
LZ0578 = 1180, LZ0227 = 1104, LZ0610 = 1074
)
## Model the event and accumulation date for each assemblage
<- event(zuni, dates = zuni_dates, cutoff = 90)
model #> Warning: Argument 'cutoff' is defunct; please use 'rank' instead.
summary(get_model(model))
#>
#> Call:
#> stats::lm(formula = date ~ ., data = data)
#>
#> Residuals:
#> LZ0569 LZ0279 CS16 LZ0066 LZ0852 LZ1209 CS144 LZ0563 LZ0329 LZ0005Q
#> 0.4466 -3.9079 -0.2887 0.7279 -1.3127 0.5037 2.7819 -4.4530 -0.7960 0.1254
#> LZ0322 LZ0067 LZ0578 LZ0227 LZ0610
#> 2.6011 -0.1375 3.5140 -0.7672 0.9624
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 1164.6501 2.9154 399.481 2.36e-10 ***
#> F1 -158.2879 1.6497 -95.948 7.07e-08 ***
#> F2 25.7034 1.6841 15.263 0.000107 ***
#> F3 -5.6058 2.1603 -2.595 0.060368 .
#> F4 10.7571 5.8231 1.847 0.138419
#> F5 -3.1388 3.9485 -0.795 0.471160
#> F6 2.7244 1.3288 2.050 0.109653
#> F7 4.5647 5.2198 0.874 0.431212
#> F8 11.2980 3.4629 3.263 0.031006 *
#> F9 -5.1715 2.9554 -1.750 0.155041
#> F10 -0.3957 2.6497 -0.149 0.888522
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 4.09 on 4 degrees of freedom
#> Multiple R-squared: 0.9997, Adjusted R-squared: 0.999
#> F-statistic: 1433 on 10 and 4 DF, p-value: 1.167e-06
## Estimate event dates
<- predict_event(model, margin = 1, level = 0.95)
event head(event)
#> date lower upper error
#> LZ1105 1169 1152 1185 6
#> LZ1103 1143 1136 1150 2
#> LZ1100 1156 1141 1172 5
#> LZ1099 1099 1087 1111 4
#> LZ1097 1088 1078 1099 4
#> LZ1096 840 825 855 5
## Estimate accumulation dates
<- predict_accumulation(model)
acc head(acc)
#> LZ1105 LZ1103 LZ1100 LZ1099 LZ1097 LZ1096
#> 1170 1140 1158 1087 1092 875
```

```
## Activity plot
plot(model, type = "activity", event = TRUE, select = "LZ1105")
```

```
## Tempo plot
plot(model, type = "tempo", select = "LZ1105")
```

Resampling methods can be used to check the stability of the
resulting model. If `jackknife()`

is used, one type/fabric is
removed at a time and all statistics are recalculated. In this way, one
can assess whether certain type/fabric has a substantial influence on
the date estimate. If `bootstrap()`

is used, a large number
of new bootstrap assemblages is created, with the same sample size, by
resampling the original assemblage with replacement. Then, examination
of the bootstrap statistics makes it possible to pinpoint assemblages
that require further investigation.

```
## Check model variability
## Warning: this may take a few seconds
## Jackknife fabrics
<- jackknife(model)
jack head(jack)
#> date lower upper error bias
#> LZ1105 1169 1153 1186 6 0
#> LZ1103 1143 1137 1150 2 0
#> LZ1100 1158 1142 1173 5 34
#> LZ1099 1097 1085 1109 4 -34
#> LZ1097 1094 1083 1104 4 102
#> LZ1096 842 827 857 5 34
## Bootstrap of assemblages
<- bootstrap(model, n = 30)
boot head(boot)
#> min mean max Q5 Q95
#> LZ1105 1131 1171.333 1215 1134.35 1200.40
#> LZ1103 1081 1144.667 1198 1101.35 1186.60
#> LZ1100 1129 1167.833 1226 1141.50 1213.15
#> LZ1099 1088 1099.000 1112 1089.45 1106.00
#> LZ1097 971 1095.633 1254 1005.00 1198.15
#> LZ1096 726 844.600 1124 726.00 993.05
```

Bellanger, L., Ph. Husi, and R. Tomassone. 2006. “Une
approche statistique pour la datation de contextes
archéologiques.” *Revue de statistique appliquée*
54 (2): 65–81. https://doi.org/10.1111/j.1475-4754.2006.00249.x.

Bellanger, Lise, and Philippe Husi. 2012. “Statistical
Tool for Dating and Interpreting
Archaeological Contexts Using Pottery.” *Journal of
Archaeological Science* 39 (4): 777–90. https://doi.org/10.1016/j.jas.2011.06.031.

Bellanger, L., R. Tomassone, and P. Husi. 2008. “A
Statistical Approach for Dating Archaeological
Contexts.” *Journal of Data Science* 6: 135–54.

Dunnell, Robert C. 1970. “Seriation Method and
Its Evaluation.” *American Antiquity* 35
(03): 305–19. https://doi.org/10.2307/278341.

Dye, Thomas S. 2016. “Long-Term Rhythms in the
Development of Hawaiian Social
Stratification.” *Journal of Archaeological
Science* 71 (July): 1–9. https://doi.org/10.1016/j.jas.2016.05.006.

Poblome, J., and P. J. F. Groenen. 2003. “Constrained
Correspondence Analysis for Seriation of
Sagalassos Tablewares.” In *The Digital
Heritage of Archaeology*, edited by M. Doerr and
A. Sarris. Hellenic Ministry of Culture.