---
title: "Dating Methods"
author: "N. Frerebeau"
date: "`r Sys.Date()`"
output:
  rmarkdown::html_vignette:
    number_sections: yes
    fig_caption: yes
    toc: true
header-includes:
   - \usepackage{amsmath}
   - \usepackage{amssymb}
bibliography: bibliography.bib
vignette: >
  %\VignetteIndexEntry{Dating Methods}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  chunk_output_type: console
---

```{r setup, include = FALSE, echo=FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  out.width = NULL
)
```

```{r packages}
# Load packages
library(tabula)
library(magrittr)
```

# Not All Dates Are Created Equal
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" [@dean1978]. 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) [@obrien2002]. 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 [@obrien2002]: absolute dates "are expressed as points on standard scales of time measurement" [@dean1978].

We will keep here the distinction between a date an age as formulated by @colman1987: "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:

* An archaeological *event* is determined by its unknown calendar date $\theta$.
* $\theta$ can be addressed by one or more of calendar dates $t$ with associated error $\delta t$.
* $t \pm \delta t$ can be provided by different means and is assumed to be related to the event.

This implies that:

* There are no error-free dates in archaeology (although uncertainties cannot always be quantified).
* Errors are assumed here to be symmetrical. This is true for most physical dating methods, but may be false after some data processing (e. g. 14C calibration).

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. 

# Mean Ceramic Date
## Definition
The Mean Ceramic Date (MCD) is a point estimate of the occupation of an archaeological site [@south1977]. 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 error is defined as:
$$
\delta t^{MCD}_i = \sqrt{ \sum_{j = 1}^{p} \left( \delta t_j \times \frac{x_{ij}}{x_{i \cdot}} \right)^2 }
$$

## Limitation
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.

## Usage
```{r mcd-model, fig.width=7, fig.height=3.5, fig.align="center"}
## Coerce the zuni dataset to an abundance (count) matrix
zuni_count <- as_count(zuni)

## Set the start and end dates for each ceramic type
zuni_dates <- list(
  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
zuni_mid <- vapply(X = zuni_dates, FUN = mean, FUN.VALUE = numeric(1))
zuni_error <- vapply(X = zuni_dates, FUN = diff, FUN.VALUE = numeric(1)) / 2

## Calculate MCD
## (we use a bootstrapping procedure to estimate the confidence interval)
zuni_mcd <- date_mcd(zuni_count, dates = zuni_mid, errors = zuni_error)
head(zuni_mcd)

## Plot dates
set_dates(zuni_count) <- list(value = zuni_mcd$date, error = zuni_mcd$error)
plot_date(zuni_count, select = 1:15, sort = "asc") +
  ggplot2::labs(title = "Mean Ceramic Date") +
  ggplot2::theme_bw() +
  ggplot2::theme(legend.position = "none")
```

# Event & Accumulation Date
## Definition
Event and accumulation dates are density estimates of the occupation and duration of an archaeological site [@bellanger2006; @bellanger2008; @bellanger2012].

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 @bellanger2012, 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 @poblome2003. 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:
<!-- https://stats.stackexchange.com/questions/147242/how-to-calculate-the-prediction-interval-for-an-ols-multiple-regression -->
$$
\widehat{V_i} = \widehat{\sigma}^2 \left( f_i^T \left( F'F \right)^{-1} f_i + 1 \right)
$$

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 @dye2016.

## Limitation
Event and accumulation dates estimation relies on the same conditions and assumptions as the matrix seriation problem. @dunnell1970 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
```{r event-model}
## Bellanger et al. did not publish the data supporting their
## demonstration: no replication of their results is possible. 
## Here is a pseudo-replication using the zuni dataset and results of the 
## previous example.

## Coerce the zuni dataset to an abundance (count) matrix
zuni_count <- as_count(zuni)

## Randomly picks 100 assemblages and assume that they are reliably dated
## (this is NOT a real example)
set.seed(12345)
train_index <- sample(seq_len(nrow(zuni_count)), size = 100, replace = FALSE)
train_dates <- zuni_mcd[train_index, 2]
names(train_dates) <- zuni_mcd[train_index, 1]
set_dates(zuni_count) <- train_dates

## Model the event and accumulation date for each assemblage
model <- date_event(zuni_count, cutoff = 90)
summary(model[["model"]])

## Estimated event dates
head(model[["rows"]])
```

```{r event-plot, fig.show="hold"}
## Activity plot
plot_date(model, type = "activity", select = "LZ1105") +
  ggplot2::theme_bw()
## Tempo plot
plot_date(model, type = "tempo", select = "LZ1105") +
  ggplot2::theme_bw()
```

Resampling methods can be used to check the stability of the resulting model. If `jackknife` is `TRUE`, 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 `TRUE`, 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.

```{r event-refine}
## Check model variability
## Jackknife fabrics
refined_jack <- refine_dates(model, method = "jackknife", n = 1000)
head(refined_jack)

## Bootstrap of assemblages
refined_boot <- refine_dates(model, method = "bootstrap", n = 1000)
head(refined_boot)
```

# References
