## Load packages
library(kairos)
library(folio) # Datasets
This vignette presents different methods for dating archaeological assemblages using artifact count data. Here, dating refers to “the placement in time of events relative to one another or to any established scale of temporal measurement” (Dean 1978). This involves distinguishing between relative (that provide only a chronological sequence of events) and absolute dating methods (that yield a calendric indication and may provide the duration of an event) (O’Brien and Lyman 2002). Strictly speaking, there is no absolute dating given how dates are produced and given that any date refers to a scale. The distinction between absolute and relative time can be rephrased more clearly as quantifiable vs non-quantifiable (O’Brien and Lyman 2002): absolute dates “are expressed as points on standard scales of time measurement” (Dean 1978).
We will keep here the distinction between a date an age as formulated by Colman, Pierce, and Birkeland (1987): “a date is a specific point in time, whereas an age is an interval of time measured back from the present.” Dealing with dates in archaeology can be tricky if one does not take into account the sources of the chronological information. In most cases, a date represents a terminus for a given archaeological assemblage. That is, a date before (terminus ante-quem) or after (terminus post-quem) which the formation process of the assemblage took place. This in mind, one obvious question that should underlie any investigation is what does the date represent?
First, let’s be more formal:
This implies that:
For a set of \(m\) assemblages in which \(p\) different types of artifact were recorded, let \(X = \left[ x_{ij} \right] ~\forall i \in \left[ 1,m \right], j \in \left[ 1,p \right]\) be the \(m \times p\) count matrix with row and column sums:
\[ \begin{align} x_{i \cdot} = \sum_{j = 1}^{p} x_{ij} && x_{\cdot j} = \sum_{i = 1}^{m} x_{ij} && x_{\cdot \cdot} = \sum_{i = 1}^{m} x_{i \cdot} = \sum_{j = 1}^{p} x_{\cdot j} && \forall x_{ij} \in \mathbb{N} \end{align} \]
Note that all \(x_{ij}\) are assumed to be error-free.
The Mean Ceramic Date (MCD) is a point estimate of the occupation of an archaeological site (South 1977). The MCD is estimated as the weighted mean of the date midpoints of the ceramic types \(t_j\) (based on absolute dates or the known production interval) found in a given assemblage. The weights are the conditional frequencies of the respective types in the assemblage.
The MCD is defined as: \[ t^{MCD}_i = \sum_{j = 1}^{p} t_j \times \frac{x_{ij}}{x_{i \cdot}} \]
The MCD is a point estimate: knowing the mid-date of an assemblage and not knowing the time span of accumulation might be short sighted. MCD offers a rough indication of the chronological position of an assemblage, but does not tell if an assemblage represents ten or 100 years.
## Coerce the zuni dataset to an abundance (count) matrix
data("zuni", package = "folio")
## Set the start and end dates for each ceramic type
<- list(
zuni_dates LINO = c(600, 875), KIAT = c(850, 950), RED = c(900, 1050),
GALL = c(1025, 1125), ESC = c(1050, 1150), PUBW = c(1050, 1150),
RES = c(1000, 1200), TULA = c(1175, 1300), PINE = c(1275, 1350),
PUBR = c(1000, 1200), WING = c(1100, 1200), WIPO = c(1125, 1225),
SJ = c(1200, 1300), LSJ = c(1250, 1300), SPR = c(1250, 1300),
PINER = c(1275, 1325), HESH = c(1275, 1450), KWAK = c(1275, 1450)
)
## Calculate date midpoint
<- vapply(X = zuni_dates, FUN = mean, FUN.VALUE = numeric(1))
zuni_mid
## Calculate MCD
<- mcd(zuni, dates = zuni_mid)
zuni_mcd head(zuni_mcd)
#> LZ1105 LZ1103 LZ1100 LZ1099 LZ1097 LZ1096
#> 1162 1138 1154 1091 1092 841
plot(zuni_mcd, select = 100:125)
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:
The mathematical assumptions state that the distribution of any historical or temporal class:
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 summary(get_model(model))
#>
#> Call:
#> stats::lm(formula = date ~ ., data = contexts, na.action = stats::na.omit)
#>
#> Residuals:
#> 1 2 3 4 5 6 7 8
#> 0.517235 -4.017534 -0.279200 0.662137 -1.246499 0.576044 2.634482 -4.383683
#> 9 10 11 12 13 14 15
#> -1.093837 -0.005002 2.543773 -0.032706 3.480918 -0.759429 1.403301
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 1164.350 1.892 615.459 2.15e-13 ***
#> F1 -158.314 1.472 -107.582 1.32e-09 ***
#> F2 25.629 1.444 17.753 1.04e-05 ***
#> F3 -5.546 1.905 -2.912 0.0333 *
#> F4 11.416 3.407 3.351 0.0203 *
#> F5 -2.713 2.448 -1.108 0.3183
#> F6 2.697 1.181 2.285 0.0711 .
#> F7 3.966 3.001 1.322 0.2435
#> F8 11.132 2.941 3.785 0.0128 *
#> F9 -4.886 2.020 -2.418 0.0602 .
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 3.669 on 5 degrees of freedom
#> (405 observations effacées parce que manquantes)
#> Multiple R-squared: 0.9997, Adjusted R-squared: 0.9992
#> F-statistic: 1979 on 9 and 5 DF, p-value: 2.456e-08
## Estimate event dates
<- predict_event(model, margin = 1, level = 0.95)
event head(event)
#> date lower upper error
#> LZ1105 1168 1158 1178 4
#> LZ1103 1143 1139 1147 1
#> LZ1100 1156 1148 1164 3
#> LZ1099 1099 1092 1106 3
#> LZ1097 1088 1080 1097 3
#> LZ1096 839 829 849 4
## 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 1457 1447 1466 4 4913
#> LZ1103 948 945 952 1 -3315
#> LZ1100 1094 1086 1102 3 -1054
#> LZ1099 1253 1246 1260 3 2618
#> LZ1097 917 908 925 3 -2907
#> LZ1096 1060 1050 1070 4 3757
## Bootstrap of assemblages
<- bootstrap(model, n = 30)
boot head(boot)
#> min mean max Q5 Q95
#> LZ1105 1131 1167.133 1192 1144.35 1190.1
#> LZ1103 1081 1139.667 1190 1091.80 1173.1
#> LZ1100 1111 1159.267 1210 1115.95 1202.6
#> LZ1099 1089 1097.033 1107 1089.00 1105.1
#> LZ1097 972 1106.033 1255 1010.55 1179.3
#> LZ1096 726 847.200 964 726.00 942.7