Event Date Model
## Install extra packages (if needed):
# install.packages("folio") # datasets
# Load packages
library(kairos)
#> Loading required package: dimensioDefinition
Event and accumulation dates are density estimates of the occupation and duration of an archaeological site (Bellanger, Husi and Tomassone 2006; Bellanger, Tomassone and Husi 2008; 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 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 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 fik ∀i ∈ [1, m], k ∈ [1, q].
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 tiE is the known date point estimate of the ith assemblage, βk are the regression coefficients and ϵi are normally, identically and independently distributed random variables, ϵi ∼ 𝒩(0, σ2).
These n equations are stacked together and written in matrix notation as
tE = Fβ + ϵ
where ϵ ∼ 𝒩n(0, σ2In), β = [β0⋯βq]′ ∈ ℝq + 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 β is:
β̂ = (F′F)−1F′tE
Finally, for a given vector of CA coordinates fi, the predicted event date of an assemblage tiE is:
$$ \widehat{t^E_i} = f_i \hat{\beta} $$
The endpoints of the 100(1 − α)% 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 tiE 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 tjE can be predicted following the same procedure as above.
Then, the accumulation date tiA 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 tiA 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).
Limitation
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.
Usage
This package provides an implementation of the chronological modeling method developed by 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 an example using the Zuni dataset from Peeples and Schachner 2012
data("zuni", package = "folio")
## 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
zuni_dates <- c(
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
model <- event(zuni, dates = zuni_dates, rank = 10)
## Extract model coefficients
## (convert results to Gregorian years)
coef(model, calendar = CE())
#> (Intercept) F1 F2 F3 F4 F5
#> 1163.6525183 -158.2876592 25.7026078 -5.6049778 10.7570247 -3.1382078
#> F6 F7 F8 F9 F10
#> 2.7245577 4.5641627 11.2966954 -5.1721456 -0.3965122
## Extract residual standard deviation
## (convert results to Gregorian years)
sigma(model, calendar = CE())
#> [1] 4.090834
## Extract model residuals
## (convert results to Gregorian years)
resid(model, calendar = CE())
#> LZ0852 LZ0610 LZ0578 LZ0569 LZ0563 LZ0329 LZ0322
#> -1.3123347 0.9623218 3.5141430 0.4478299 -4.4537670 -0.7964654 2.6011154
#> LZ0279 LZ0227 LZ0067 LZ0066 LZ0005Q CS16 CS144
#> -3.9079020 -0.7682236 -0.1375450 0.7283483 0.1254681 -0.2887566 2.7823817
#> LZ1209
#> 0.5033862
## Extract model fitted values
## (convert results to Gregorian years)
fitted(model, calendar = CE())
#> [1] 1217.3105 1073.0370 1176.4849 1096.5531 1210.4540 1076.7948 1106.3999
#> [8] 1122.9078 1104.7666 863.1376 1110.2712 858.8744 1328.2882 1259.2185
#> [15] 1250.4963## Estimate event dates
eve <- predict_event(model, margin = 1, level = 0.95)
head(eve)
#> date lower upper error
#> LZ1105 1168.5901 1152.1407 1185.0396 6.923526
#> LZ1103 1143.1063 1136.3822 1149.8294 3.420707
#> LZ1100 1156.3091 1141.0492 1171.5681 6.494020
#> LZ1099 1098.8294 1086.9674 1110.6942 5.269872
#> LZ1097 1088.4302 1078.1029 1098.7571 4.717296
#> LZ1096 839.8136 824.5635 855.0621 6.490568
## Estimate accumulation dates (median)
acc <- predict_accumulation(model, level = 0.95)
head(acc)
#> date lower upper
#> LZ1105 1172.1053 1089.4725 1237.140
#> LZ1103 1120.9272 776.5453 1285.652
#> LZ1100 1205.6884 951.9349 1251.002
#> LZ1099 1091.0728 1058.5542 1118.796
#> LZ1097 1093.2039 776.0133 1251.002
#> LZ1096 782.9428 770.6816 1118.796
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
## Jackknife fabrics
jack <- jackknife(model)
head(jack)
#> date lower upper
#> LZ1105 1035.1315 1018.6809 1051.5821
#> LZ1103 1280.8505 1274.1270 1287.5759
#> LZ1100 1120.9870 1105.7288 1136.2471
#> LZ1099 1028.1890 1016.3265 1040.0514
#> LZ1097 320.6905 310.3640 331.0181
#> LZ1096 769.9842 754.7357 785.2327
## Bootstrap of assemblages
boot <- bootstrap(model, n = 30)
head(boot)
#> min mean max Q5 Q95
#> LZ1105 1129.5585 1166.7813 1205.589 1140.2983 1195.293
#> LZ1103 1099.4150 1140.3967 1199.296 1103.7162 1185.345
#> LZ1100 1084.6255 1152.8636 1205.709 1104.7819 1195.482
#> LZ1099 1089.0251 1098.6631 1110.938 1090.1436 1108.529
#> LZ1097 972.6925 1087.5469 1180.767 1022.4584 1167.340
#> LZ1096 726.0654 854.0992 1124.189 743.9818 1009.727References
Bellanger, L., Husi, P. & Tomassone, R. (2006). Une approche statistique pour la datation de contextes archéologiques. Revue de statistique appliquée, 54(2), 65-81. URL: http://www.numdam.org/item/RSA_2006__54_2_65_0/.
Bellanger, L. & Husi, P. (2012). Statistical Tool for Dating and Interpreting Archaeological Contexts Using Pottery. Journal of Archaeological Science, 39(4), 777-790. DOI: 10.1016/j.jas.2011.06.031.
Bellanger, L. & Tomassone, R. & Husi, P. (2008). A Statistical Approach for Dating Archaeological Contexts. Journal of Data Science, 6, 135-154.
Dunnell, R. C. (1970). Seriation Method and Its Evaluation. American Antiquity, 35(3), 305-319. DOI: 10.2307/278341.
Dye, T. S. (2016). Long-Term Rhythms in the Development of Hawaiian Social Stratification. Journal of Archaeological Science, 71, 1-9. DOI: 10.1016/j.jas.2016.05.006.
Poblome, J. & Groenen, P. J. F. (2003). Constrained Correspondence Analysis for Seriation of Sagalassos Tablewares. In M. Doerr & A. Sarris (Eds.), The Digital Heritage of Archaeology Athens: Hellenic Ministry of Culture.