README.Rmd

---
bibliography: "inst/REFERENCES.bib"
output: github_document
---

<!-- README.md is generated from README.Rmd. Please edit that file -->

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.path = "man/figures/README-",
  out.width = "100%"
)
```

# <img src="man/figures/logo.png" align="right" width="150px"/> eratosthenes: Archaeological Synchronism

<!-- badges: start -->
<a href="https://joss.theoj.org/papers/75ab124a2cbb3b7125a9458650544020"><img src="https://joss.theoj.org/papers/75ab124a2cbb3b7125a9458650544020/status.svg"></a>
<!-- badges: end -->

The `R` package `eratosthenes` aims to provide a general, flexible toolkit for archaeological chronology-building by incorporating, computationally, all relevant sources of information on uncertain archaeological or historical dates. Archaeological dates are subject to relational conditions (via seriation or stratigraphic relationships) and absolute constraints (such as radiocarbon dates, datable artifacts, or other known historical events, as _termini post_ or _ante quos_), which prompt the use of a joint conditional probability density to convey those relationships. The date of any one event can then be marginalized from that full, joint conditional distribution. `Rcpp` is used for faster sampling [@eddelbuettel_extending_2018].

While software exists for calibrating and conditioning radiocarbon dates upon relative constraints, such as `OxCal` [@bronk_ramsey_bayesian_2009] and `BCal` [@buck_bcal_1999], as well as R packages `oxcAAR` [@hinz_oxcaar_2021], `Bchron` [@haslett_simple_2008], and `rcarbon` [@crema_spatio-temporal_2017], along with software for general chronological modeling like `Chronomodel` [@lanos_hierarchical_2017] and `ChronoLog` [@levy_chronological_2021], formal methods for dating artifacts and artifact types are lacking. One of the major goals of  `eratosthenes` is to advance the synchronism of chronologies and the crafting of large-scale chronological models that rely heavily upon artifact typologies. The package therefore facilitates the marginalization of dates of a type's production, use, and deposition. The method of sampling employed in `eratosthenes` involves a two-step process of Gibbs sampling, using consistent batch means (CBM) and Monte Carlo standard errors (MCSE) to determine convergence [@jones_fixed-width_2006; @flegal_markov_2008]. Finaly, `eratosthenes` provides tools for analyzing the impact of events on each other with the conditional structure stipulated by the investigator, by implementing a jackknife-style estimator of squared displacement (how much the date of one event shifts when another is omitted). Ancillary functions include checking for discrepancies in sequences of events and constraining optimal seriations to known sequences. 

The package is motivated by a philosophy of generalism and minimalism, eschewing the following:

* intervals or durative events. If desired, such instances can be asserted as two separate point events in sequences, e.g., `"X - Start"` and `"X - End"`.
* periods and phases. Periods and phases are not actual material or behavioral events, but ideal (and often contested) constructs used to make sense of the past. If desired, an investigator can always enter period-related events, e.g., `"Archaic Period - Start"`, into their list of sequences.
* discretization of time into intervals. Samples are drawn along the continuum.
* overly cumbersome chronological relationships. As `eratosthenes` samples points along the continuum, there is only before and after. If desired, overlaping events can be expressed in sequence construction: e.g., for sequences `c("A", "B", "C")` and `c("A", "D", "E", "C")`, events `"B"` and `"D", "E"` will overlap with each other.

The focus of the package is on the structure of the joint conditional, rather than specific probability models. Hence, `eratosthenes` relies on the continuous uniform for estimating relative events. Any model can however be used for absolute constraints, from single points to customized densities.

The package is named after Eratosthenes of Cyrene, author of the _Chronographiai_.

## Installation

To obtain the current development version of `eratosthenes` from GitHub, install the package in the `R` command line with `devtools`:

``` r
library(devtools)
install_github("scollinselliott/eratosthenes", dependencies = TRUE, build_vignettes = TRUE) 
```

## Tutorial: Archaeological Example

The Dressel 1B type of amphora (a ceramic shipping container typical of the ancient Mediterranean) is a ceramic type defined on the basis of morphology, comprising a tall, long-necked, two-handled vessel with a concave collared rim and sharply defined, angular shoulder. To give just three chronological summaries of this type, the Dressel 1B type was produced/used/deposited:

* in the "last quarter of the second until the last decade of the first century BC" [@southampton_roman_2014];
* "shortly before the middle of the 1st century BC" and "until c. 10 BC" [@tyers_roman_1996, 2.2];
* at the earliest "during _c_.100-80 BC", disappearing "by 30 BC" [@loughton_arverni_2014, 43].

Such determinations are the result of many comparisons of stratigraphic and single contexts where the Dressel 1B type has appeared in conjunction with absolute constraints (e.g., coinage, datable stamps, historical events in the occupation/destruction of sites).  It is evident in the above statements, too, that there will be greater or lesser chronological discrepancies among investigators.

In order to obtain a probability density for any aspect of dating the Dressel 1B (that is, its production, use, and/or deposition), `eratosthenes` avoids the awkward synthesis of this chronological information by estimating dates directly from archaeological record itself, in the information provided by contexts (in **[sequences](#sequences)**, howsoever established), **[finds](#finds)**, which are contained in those contexts, and **[absolute constraints](#absolute-constraints)** for those contexts, as inputs. In other words, it formalizes the logic of dating finds which is already in practice, yielding a probability density instead of a qualitative appraisal of the object's dates of production, use, and deposition.

The following aims to be a concrete tutorial that gives step-by-step instructions to obtain probability density functions on the production, use, and deposition of the Dressel 1B type using `eratosthenes`. Further information on the data for this tutorial is found **[here](https://volweb.utk.edu/~scolli46/eratosthenes/eda20250628.html)**.

### Creating a Sequences Object

First, sequences are given as vectors, with contexts entered as elements earlier to later, from left to right. The sequence `seq1` below gives a sequence of just two depositional contexts, from the archaeological site of Rirha in Morocco, `Rirha US 5182` and `Rirha US 5154`, with `Rirha US 5182` being the earlier of the two:

``` r
library(eratosthenes)
seq1 <- c("Rirha US 5182", "Rirha US 5154") 
```

Multiple sequences can, and typically will, be given. To define seven more sequences for this tutorial, we have additional deposits from the site of Carthage (Byrsa Hill) and also several shipwrecks ([further information here](https://volweb.utk.edu/~scolli46/eratosthenes/eda20250628.html)):

``` r
seq2 <- c("Byrsa II B 19.4", "Byrsa II B 19.2")
seq3 <- c("Rirha US 5154", "Planier A")
seq4 <- c("El Sec", "Filicudi F", "Tour Fondue", "Cabrera 2", "Tour d'Agnello", "Sanguinaires A", "Lazaret")
seq5 <- c("Madrague de Giens", "Planier C", "Cap Béar C", "Planier A")
seq6 <- c("Cabrera 2", "Grand Congloué A", "Lazaret", "Byrsa II B 19.2", "Punta Scaletta", "Isla Pedrosa", "Cavalière", "Madrague de Giens")
seq7 <- c("Mazotos", "El Sec")
seq8 <- c("Grand Congloué A", "Héliopolis B", "Punta Scaletta")
```

A single `list` object is then created which contains all of the sequences:

``` r
contexts <- list(seq1, seq2, seq3, seq4, seq5, seq6, seq7, seq8)
```

In order to check that all sequences accord with one another, we run the `seq_check()` function, which will return a value of `TRUE` if there are no conflicts:

``` r
seq_check(contexts)
```

The `contexts` object therefore contains all of the information about the relationships of the contexts, in terms of which come before or after one another, which should be based on a stated rationale.

### Creating a Finds Object

Next, a separate object for the finds data must be created. If a particular find has absolute chronological information associated with it (just the object itself, not the type), it should not be entered here, but rather as an absolute constraint (on which see below). To start, each find is a `list` object with the following information:

``` r 
id1 <- list(id = "id 1",
            assoc = "Isla Pedrosa",
            type = "AMPH Dressel 1B",
            residual = TRUE)
```

Each find, indexed with an `id`, must be linked to a context given in the sequences above. Its appertaining context is entered into the `assoc` field. This particular entry comprises the find of Dressel 1B amphorae associated with the Isla Pedrosa shipwreck. These amphorae may in fact relate to another shipwreck (i.e., their association with this context may be spurious). By indicating `residual = TRUE`, the find remains associated with its context, but it will not be taken into account when estimating its date. This option applies whether finds are residual to a deposit (i.e., much earlier finds redeposited into a later deposit) or whether they are intrusive (i.e., much later finds have been associated with an earlier deposit). The `type` field provides information on any aspects of artifact classification or typology: here, the type `AMPH Dressel 1B` is entered. It should be noted that `type` is optional, and more than one entry can be given for `type`, as the next four types show:

``` r
id865 <- list(id = "id 865", assoc = "Rirha US 5154", type = "AMPH Dressel 1B")
id1202 <- list(id = "id 1202", assoc = "Madrague de Giens", type = "AMPH Dressel 1B")
id1285 <- list(id = "id 1285", assoc = "Planier C", type = "AMPH Dressel 1B")
id1364 <- list(id = "id 1364", assoc = "Cap Béar C", type = c("AMPH Dressel 1B", "AMPH Dressel 1B Tarraconensis"))
```

The last find, `id1364`, belongs to the production group of Dressel 1B amphorae produced in Spain, and so indicating its production subtype as another element in the `type` fields ensures that the find and its associated context will be used in estimating dates if an investigator is obtaining dates for either `AMPH Dressel 1B` or `AMPH Dressel 1B Tarraconensis`.

Finally, a single `list` object is then created which contains all of the finds:

``` r
finds <- list(id1, id865, id1202, id1285, id1364)
```

### Creating Absolute Constraints Objects

The last step is to create `lists` of the absolute constraints which pertain to contexts as either _termini post quos_ or _termini ante quos_. Absolute constraints have the same fields as a find, but include another field, `samples`, which contains the absolute dates (see [Absolute Constraints](#absolute-constraints) below). Two separate `lists`, one for _t.p.q._ and the other for _t.a.q._, are created as follows.

The _termini post quos_ will include radiocarbon dates. A vector of samples of calibrated dates, e.g., `c(375, 371, 370, ...)` may be entered, but it is more expedient to call one of the R packages available to calibrate dates and then assign their output into the `list`. Here, uncalibrated dates were uploaded as a `csv` file which was processed using `Bchron` [@haslett_simple_2008]. The following code shows how to loop over a table of uncalibrated dates in order to draw samples from the calibrated date and insert them into a list of _t.p.q._ with an `id` and associated context (`assoc`):

``` r
# create an empty list to contain all tpq
tpq_info <- list()

# The csv file at https://volweb.utk.edu/~scolli46/eratosthenes/data/rc20250628.csv
# contains a table of uncalibrated radiocarbon dates
eratosthenes_rcdates <- read.csv("rc20250628.csv")

library(Bchron)

# calibrate and insert rc dates into tpq by looping over rows
for (i in 1:nrow(eratosthenes_rcdates)) {
    calib <- BchronCalibrate(ages = eratosthenes_rcdates$mu[i],
                             ageSds = eratosthenes_rcdates$sigma[i],
                             calCurves = "intcal20")
    x <- 1950 - sampleAges(calib)

    # this 
    tpq_info[[i]] <- list(id = eratosthenes_rcdates$id[i],
                     assoc = eratosthenes_rcdates$assoc[i],
                     samples = x)
}
```

The _termini ante quos_ include the destruction of Carthage in 146 BCE, which the deposit B 19.2 on Byrsa Hill predates, as well as a range of absolute dates, ca. 1-15 CE, for the Planier A shipwreck (to give a finite endpoint for this tutorial).

``` r
Carthage_Destr_1 <- list(id = "Carthage_Destr_1",
                         assoc = "Byrsa II B 19.2",
                        samples = -146)
Planier_A_abs <- list(id = "Planier_A_abs",
                      assoc = "Planier A",
                      samples = seq(1, 15, length.out = 100))

taq_info <- list(Carthage_Destr_1, Planier_A_abs)
```

With the inputs of these sequences, finds, and absolute constraints, we can estimate the dates of the production, use, and deposition of the Dressel 1B type by calling the `gibbs_ad_type()` function from `eratosthenes`. The `contexts` object containing the sequences is entered first and the `finds` object second. We then specify what `type` we want to obtain dates for (here, `AMPH Dressel 1B`). Finally, the absolute constraints of `tpq_info` and `taq_info` are given as inputs:

``` r
dr_1b <- gibbs_ad_type(contexts,
                       finds,
                       type = "AMPH Dressel 1B",
                       tpq = tpq_info,
                       taq = taq_info)
```

The resulting `dr_1b` object contains samples of dates (production, use, and depositional dates are estimated separately), and the console output will give information on the structure of that object as well as a table listing the mean dates. The densities can be visualized using the `histogram()` function:

``` r
histogram(dr_1b, xlim = c(-300,20), ylim = c(0, 0.010), legend = "topleft") 
```

Which shows the probable dates of production, use, and deposition separately (see [Fig. 3 of the _JOSS_ paper](https://raw.githubusercontent.com/openjournals/joss-papers/joss.08559/joss.08559/10.21105.joss.08559.pdf) awaiting review), indicating which dates are more probable than others.

To determine a highest density region (HDR) of the dates, i.e., the range of the most probable dates according to a given percentage, we can construct a histogram of the dates with 100 bins (i.e., as a percentage) and sort them by their density, choosing, for example, the 10% and 95% regions of the most probable dates of the production of the type. 

``` r
dat <- dr_1b$type$production
perc <- seq(min(dat), max(dat), length.out = 101)

x <- hist(dat, perc)$mids
y <- hist(dat, perc)$density

hpd <- rev(x[order(y)])
hpd_10 <- c(min(hpd[1:10]), max(hpd[1:10]))
hpd_95 <- c(min(hpd[1:95]), max(hpd[1:95]))
```

It should be noted that if the resulting distribution is multimodal, this code will need to be modified to identify multiple regions. But here, with a simple unimodal case, the lower and upper bounds or either a narrower (10% HDR) or broader (95% HDR) interval of dates for the production of a Dressel 1B type amphorae are, approximately:

```
> hpd_10
[1] -103.7998  -70.4462
> hpd_95
[1] -340.980962    7.378864
```

The dates of production will be earlier than those of deposition, which can be seen in the histogram and by selecting `dr_1b$type$deposition` rather than `dr_1b$type$production`. 

The estimates naturally depend on the inputs, and this tutorial has used only a small portion of the available information on the contexts and constraints pertaining to the Dressel 1B type. The estimates given by `eratosthenes` roughly accord with that of the conventional typology, as would be expected, but it gives investigators the benefit of numerical precision afforded by probability theory. Chronological typologies depend on any number of conditional statements about depositional contexts, their similarity to one another, and their sequencing. New information is always being added, and older views are always being reassessed. Hence, `eratosthenes` allows for a more rapid reassessment of dates, as can be shown next.

### Chronological Interventions and Revisions

Revising chronologies, or evaluating the impact of choices upon or interventions within a chronology, is performed directly on the inputs. If we reconsider  the view that the Dressel 1B is part of the Isla Pedrosa wreck, we can set `residual = FALSE` and reassign the `id1` object, reassign the finds object, and re-run the `gibbs_ad_type()` function:

``` r
id1 <- list(id = "id 1", assoc = "Isla Pedrosa", type = "AMPH Dressel 1B", residual = FALSE)
finds <- list(id1, id865, id1202, id1285, id1364)
dr_1b <- gibbs_ad_type(contexts, finds, type = "AMPH Dressel 1B", tpq = tpq_info, taq = taq_info)
```

Performing the same steps as above to estimate the 10% and 95% HDR of the dates of production, we can see that associating the Dressel 1B amphorae with the cargo of the Isla Pedrosa wreck has substantially changed the allocation of probability for what the most probable dates are, shifting them earlier:

```
> hpd_10
[1] -137.09992  -99.68446
> hpd_95
[1] -342.884905    8.820347
```

Having a probablistically determined date for artifacts that works directly from the basis of their archaeological relationships evades the need to account for chronological discrepancies, and moreover affords the ability to work entirely within a formal, mathematical environment when it comes to topics such as the quantification of finds with uncertain dating (since each find will have its own particular density function).  Futher tools to evaluate the influence of events on one another for determining dates within in the chronology are discussed below (see [Evaluating Displacement](#evaluating-displacement)).

## Usage

The basic items of interest in `eratosthenes` are:

* **sequences** of relative events, typically stratigraphic deposits, but also isolated contexts such as may be part of a frequency or contextual seriation
* **finds**, elements which belong to those events, typically artifacts
* **absolute constraints**, as either _termini post_ or _ante quos_, expressed as samples from a probability density

Information related to these three items must be formatted in objects of a `list` class, as follows.

### Sequences

Relative sequences should run in order from left (earliest) to right (latest). All sequences should consist of vectors, contained in a `list`. In the following example, the object `contexts` contains three sequences of events.

``` r 
x <- c("A", "B", "C", "D", "E", "F", "G", "H", "I", "J")
y <- c("B", "D", "G", "H", "K")
z <- c("F", "K", "L", "M")
contexts <- list(x, y, z)
```

See also the section [Evaluating Sequences](#evaluating-sequences) below.

### Finds

Finds should be formatted as a `list` of `lists`, each of which contains the entries of the following:

* `id` : a unique identification number or code 
* `assoc` : an element in the sequences `list` to which that find or element pertains 
* `type` : optional -- one or more types, attributes, features, or aspects that pertain to that find (`NULL` if none)
* `residual` : optional -- if `TRUE`, it means that the find is considered residual to the context (the sequential event) in which it was found, and will not be considered when estimating aspects of the date of a type (production, use, and deposition).

In the following example, the `artifacts` object contains six artifacts which pertain to elements of the sequences contained in `contexts` above"

``` r
f1 <- list(id = "find01", assoc = "D", type = c("type1", "form1"))
f2 <- list(id = "find02", assoc = "E", type = c("type1", "form2"))
f3 <- list(id = "find03", assoc = "G", type = c("type1", "form1"), residual = TRUE)
f4 <- list(id = "find04", assoc = "H", type = c("type2", "form1"))
f5 <- list(id = "find05", assoc = "I", type = "type2")
f6 <- list(id = "find06", assoc = "H", type = NULL)
artifacts <- list(f1, f2, f3, f4, f5, f6)
```

### Absolute Constraints

Constraints should be given as two separate `lists`, one for _termini post quos_ and the other for _termini ante quos_. These take the same form as the finds object, as a `list` of `lists`, with the same headings, but include one additional heading of `samples` which contains the absolute dates pertinent to that _t.p.q._ or _t.a.q_.

``` r
coin1 <- list(id = "coin1", assoc = "B", type = NULL, samples = runif(100, -320, -300))
coin2 <- list(id = "coin2", assoc = "G", type = NULL, samples = runif(100, 37, 41))
destr <- list(id = "destr", assoc = "J", type = NULL, samples = 79)

tpq_info <- list(coin1, coin2)
taq_info <- list(destr)
```

It can be noted that absolute constraints can belong to a type. Any artifact which carries absolute dating information (i.e., extrinsic to the joint conditional density) should be assigned as an absolute constraint. It assumed that if a _t.p.q_ has a type, it refers to the artifact's date of production, and is treated as such (see the section [Dates of the Production, Use, and Deposition of a Type](#dates-of-the-production-use-and-deposition of-a-type) below).

Absolute dates can take any form:

* Single dates, e.g., `79` for 79 CE.
* Samples between two potential dates for a date range, e.g., `-91:-88`, `seq(-91, -88, length = 10^5)`, or `runif(10^5, -91, -88)` for 91-88 BCE.
* Samples from a bespoke density, e.g., from a calibrated radiocarbon date. `eratosthenes` does not provide functionality for calibrating dates, which can be accomplished using preexisting software or directly from a calibration curve. As a brief example, given an uncalibrated date and its standard deviation, a crude sample of calibrated dates can be drawn from the IntCal20 curve data, available from IntCal [here](https://www.intcal.org/curves/intcal20.14c) [@reimer_intcal20_2020], using the following script:

``` r
intcal20 <- read.csv("../path/to/intcal20.14c")

# 14c date mean and st.dev.
mu <- 2040  
sigma <- 30

# samples of 14c dates
uncalib <- round(rnorm(10^5, mu, sigma))

calib <- c()

for (i in 1:length(uncalib)) {
  x <- intcal20$CAL.BP[ intcal20$X14C.age == uncalib[i] ] 
  #g <- intcal20$Sigma[ intcal20$X14C.age == uncalib[i] ]

  if (length(x) > 0) {
    for (j in 1:length(x)) {
      calib <- c(calib, x[j])  
    }
  }
}

# samples of cal BC date
calBC <- 1950 - calib
hist(calBC, breaks = 100)
```

## Estimating Dates

The core approach of `eratosthenes` is a Gibbs sampler, a common Markov Chain Monte Carlo (MCMC) technique used for dating archaeological events, above all radiocarbon dates [@geman_stochastic_1984; @buck_bayesian_1996; @bronk_ramsey_bayesian_2009]. Gibbs sampling however can take a number of different forms, and so it is worthwhile to describe explicitly how it is conducted in `eratosthenes`. The precise method is as follows:

* To initialize, the earliest possible _t.p.q._ and latest possible _t.a.q_ dates are selected.
* Relative events are indexed along a single sequence for the purpose of sampling (this does not change their conditional relationships).
* To select the initial date for each relative event, a sample is drawn uniformly at random between its upper and lower constraints (absolute and relative).
  * For each initial date, a subroutine of Gibbs sampling is performed in order to avoid catastrophic collapse of dates due to floating point errors (e.g., if one has a high number of events compressed into a brief span of time).
* After all dates are initialized, the main Gibbs sampler is performed for a specified maximum number of samples, which will stop automatically if convergence in distribution has been achieved.
  * Given that dates have already been initialized via Gibbs subroutines, the need to discard initial samples due to burn-in is obviated (or reduced).
  * Convergence is determined using consistent batch means (CBM), which divides the samples into batches. For all events (i.e., variates), the Monte Carlo standard errors (MCSE) of their batch means are computed. If the mean MCSE falls below a specified criterion (by default 0.5, to determine the date of an event +/- 1 year), the main Gibbs sampler will stop. See @jones_fixed-width_2006 and  @flegal_markov_2008 for details.
  * Given that this is the mean MCSE of all events, certain events will have higher or lower MCSE, and so each event's MCSE should be reported.

There are two functions in `eratosthenes` for estimating dates:

* `gibbs_ad()` estimates the marginal density of the date of events in sequences and absolute constraints (_t.p./a.q_).
* `gibbs_ad_type()` estimates densities of the date of the production, use, and deposition of a specified artifact type, given seuqences and cosntraints.

See the section [Evaluating Displacement](#evaluating-displacement) below for tools on assessing the effective influence of events upon each other within the joint conditional density.

### Dates of Events in Sequences and Absolute Constraints

The function `gibbs_ad()` takes as inputs the following objects:

* `sequences` : A `list` of relative sequences of contexts or events.
* `max_samples` : The maximum number of samples to run, which will stop the main sampling routine even if convergence has not been achieved (default is `10^5`).
* `size` : How many samples to take between each check for convergence (default is `10^3`). 
* `mcse_crit` : The criterion of the mean MCSE at which to stop the sampler (default is `0.5`)
* `tpq` and `taq`: Separate `lists` that indicate any elements that provide extrinsic (i.e., absolute) chronological information, as _termini post_ and _ante quos_. Format must follow that illustrated in the Section above on [Absolute Constraints](#absolute-constraints).
* `alpha_` and `omega_`: lowest and highest bounds within which to sample.
* `trim`: whether to remove contexts from the output that are before or after user-provided _t.p.q._ and _t.a.q._ (i.e., those which depend on `alpha_` and `omega_`).

For example, to sample from the sequences, finds, and constraints given above, the following inputs are entered into the `gibbs_ad()` function: 

``` r 
result <- gibbs_ad(contexts, finds = artifacts, tpq = tpq_info, taq = taq_info)
```

The output is a `list` object of class `marginals` containing the following objects:

* `deposition` : a `list` of the marginal densities of the date of the final deposition of contexts.
* `externals` : a `list` of the marginal densities of date of any _terminus post quem_ or _terminus ante quem_, as affected by depositional variates in the joint conditional distribution.
* `mcse` :  a vector of the MCSE of all events.

Information on the `marginals` object can be accessed with `print()` and `summary()`. Density plots and density histograms of one more events can be produced using `plot()` and `histogram()` respectively (see packag documentation for details).

### Dates of the Production, Use, and Deposition of a Type 

Determining the date of the production, use, and deposition of an artifact type uses the same method of Gibbs sampling discussed above, i.e., consistent batch means to determine convergence. Given that types are ideal constructs used to categorize artifacts, the notion of a "type" has flexibility. While only one "type" at a time can be estimated with `gibbs_ad_type()`, here, a "type" can be defined on the basis of:

* One or more `id` in the finds list.
* One or more `type` in the finds list.

That is, one can pool together multiple finds as a type on the basis of their `id`, even if they were not so explicitly given a `type` in the finds object. Similarly, one can pool together more than one type of artifact, e.g., if one is dealing with multiple subtypes and one wants to evaluate them as a single type (e.g., pooling the labels of "Late Greco-Italic amphora", "MGS V amphora", "MGS VI amphora" into a single type).

The function works on the principle of the presence/absence of the specified type in a given context. First, it identifies all contexts in the sequences to which it has been assigned (i.e., been deposited). Then, it uses a stipulated rule to identify the earliest moment of production, contingent upon its earliest absence within the joint conditional density (see the argument `rule` below). Finally, dates of use are sampled between production and deposition. 

The `gibbs_ad_type()` function takes the following inputs, similar to `gibbs_ad()`, but with some additional fields:

* `sequences` : A `list` of relative sequences of contexts or events.
* `finds` : Either the `list` object of finds originally used as input to produce `gibbs`, or a `data.frame` of two columns, the first column listing the context and the second the incidence of the id or type in that context. 
  * If a find entry contains the expression `"residual = TRUE"`, it indicates that its association with the context should not be taken into account. Primarily, this indicates that a finds depositional date occured prior to the context it pertains to (i.e., it has been redeposited from an earlier time), but it can also be used to suppress the association of finds which may be spurious.
* `id` : A vector of the `id` of one or more specific finds whose use date is to be estimated. The values of `id` must match those in the `list` of `finds`. If `type` is used, `id` is ignored.
* `type` : A vector of one or more types to estimate a use density for. Must contain a value if `id` is left as `NULL`.
* `type_name` : A customized label for the type (e.g., if one is pooling together multiple `id`/`type` entries). If only one `type` has been entered, that label is used. Otherwise it defaults to just `"Type"`.
* `max_samples`, `size`, `mcse_crit`, `trim` : The same information used for determining the maximum length of the Gibbs sampler and when convergence has been achieved, as well as whether to trim events, [as above](#dates-of-events-in-sequences-and-absolute-constraints). Note that the `mcse_crit`, as a stopping rule, applies still to the sequential events/absolute constraints, but MCSE will still be reported for the estimates of the production, use, and depositional dates.
* `tpq` and `taq` :  Format must follow that illustrated in the Section above on [Absolute Constraints](#absolute-constraints). 
* `rule`: the rule for determining the earliest date of production of an artifact type. Initial threshold boundaries are first established between the earliest depositional context containing an artifact of that type and the next earliest context which lacks it. Then, the following rules will sample a date accordingly:
  * `naive`: samples are drawn between the initial threshold sample and the depositional date of that artifact.
  * `earliest`: samples are drawn within the initial threshold boundaries.

As use dates are drawn between production and depositional dates, if one chooses `"earliest"` as the rule, then the use density is equivalent to that of the `"naive"` production density. It should also be noted that, for this function, Gibbs sampling is only used for the depositional sequences and absolute constraints, not for production, use, and deposition (i.e, the use date does not affect the production date, nor is the depositional date affected by the production date).

Using the `result` object above, the densities of the use dates of the following types is computed using the `gibbs_ad_type()` function as follows:

``` r
# use dates by specifying ids
gibbs_ad_type(contexts, artifacts, id = c("find04", "find05"), tpq = tpq_info, taq = taq_info)
# use dates by specifying types
gibbs_ad_type(contexts, artifacts, type = "type1", tpq = tpq_info, taq = taq_info)
```

Adjusting the values of `max_samples` and `mcse_crit` is recommended to reduce computational time, as needed.

The result is a `list` object of the class `type_marginals`, which contains information on the densities of the dates of production, use, and deposition, as well as the MCSE, of the type specified.

## Graphics

Base R graphics are provided by `eratosthenes` to generate traceplots of the results of `gibbs_ad()` and produce density histograms of the results of `gibbs_ad()` and `gibbs_ad_type()`. For  `gibbs_ad()`, histograms may contain up to 12 distinct events. For  `gibbs_ad_type()`, the production, use, and deposition of the stipulated artifact type are shown.

## Evaluating Sequences

Managing and evaluating the validity of relative sequences consists of checking multiple partial sequences against one another. Not all relative sequences are of the same informational validity, and not all sequences will contain the same elements. An investigator may seek to constrain one sequence against another, i.e., keeping elements of sequence as close as possible to one another while reordering only some of the elements.

Some functions related to relative sequences:

* `seq_check()` sees whether partial sequences agree in their relative ordering of elements.
* `seq_adj()` provides the means to coerce an "input" sequence to a discrepant "target" sequence which contains fewer elements. E.g., if one has obtained an optimal seriation of contexts (of both single, unrelated deposits and stratigraphic deposits) as determined by the presence/absence of find-types, which conflicts with a sequence obtained from a stratigraphic sequence whose physical relationships are certain, this function will reorder the optimal seriation, fitting any single deposits missing from the stratigraphic sequence accordingly.

The package `eratosthenes` does not have functionality to produce seriations or ordinations, since R packages such as `seriation` [@hahsler_getting_2008], `vegan` [@oksanen_vegan_2024], `boral` [@hui_boral_2016], `ecoCopula` [@popovic_fast_2022], `VGAM` [@yee_new_2004], and `lakhesis` [@collins-elliott_lakhesis_underreview] can perform this task already.

## Evaluating Displacement 

As real-world joint conditional densities will comprise hundreds of events or more, it is easy for an investigator to loose track of which relative/absolute events are determinative or influential upon others, in terms of the estimation of their date. `eratosthenes` assesses such influence within the conditional structure via the estimation of "displacement." That is, given the omission of an event $j$ (either a depositional event or an absolute constraint) from the set of all events, how much does the estimation of the date of another event change?

The squared displacement $\delta^2(i,j)$ of a target event $i$ caused by the omission of $j$ is computed as follows. Let $\tilde{x}_i$ be the estimated marginalized Monte Carlo mean date using all events within the full joint conditional, and then let $\tilde{x}_i^{(-j)}$ be the "jackknife" estimated date, when event $j$ has been omitted from all sequences and absolute constraints. Squared displacement of $j$ upon $i$ is then:

$$
\delta^2(i,j) = (\tilde{x}_i^{(-j)} - \tilde{x}_i)^2
$$

If squared displacement is high, then the omission of $j$ has greatly shifted the date of $i$. If squared displacement is low, then the omission of $j$ has not altered the date of $i$ much. Squared displacement is measured in continuous time, whichever scale the investigator is using (typically years). 

Conversely, one can estimate the effective influence of an event $j$ upon all others by taking the mean squared 
displacement (MSD). This involves taking the mean of the squared displacements of all other events when $j$ is omitted. Where $\Theta$ represents the set of all relative and absolute events, the MSD is defined as

$$
\text{MSD}(j) = \frac{1}{n-1} \sum_{i \in \Theta, i \neq j} \delta^2 (i,j)
$$

The squared displacement and MSD are computed in `eratosthenes` for relative events and absolute constraints after running the `gibbs_ad()` function, and for an artifact type after running the `gibbs_ad_type()` function. Note that squared displacement may be computed for any event $i$ that represents a relative or absolute constraint, as well as a type (the use date is used to compute displacement for finds, as it is affected by both production and deposition) production date, while $j$ can only be a relative event or absolute constraint (it would make no sense to omit e.g. an artifact production date, since these are conditional upon relative/absolute dates to begin with). Similarly, MSD can only be computed for relative/absolute events.

Objects in the example below are provided from the section [Usage](#usage) above. As these routines are fairly intensive, computational time can be reduced by lowering the values of `max_samples` and/or raising `mcse_crit`.

``` r
# run gibbs_ad() first
result <- gibbs_ad(contexts, tpq = tpq_info, taq = taq_info)

# squared displacement for depositional context "E" as the target event ("j" above)
sq_disp(result, target = "E", sequences = contexts, 
        max_samples = 20000, mcse_crit = 2, tpq = tpq_info, taq = taq_info)

# mean squared displacement (MSD) is estimated for all relative and absolute dates
msd(result, contexts, finds = artifacts,
    mcse_crit = 1, tpq = tpq_info, taq = taq_info)

# squared displacement for production of artifact type "type1"
# run gibbs_ad_type() first
result_type1 <- gibbs_ad_type(contexts, finds = artifacts, type = "type1",
                              tpq = tpq_info, taq = taq_info)
sq_disp(result_type1, sequences = contexts, finds = artifacts,
        max_samples = 3000, mcse_crit = 2, tpq = tpq_info, taq = taq_info)
```

## References