coffee

Chronological Ordering For Fossils and Environmental Events can be acronymised to coffee. While individual calibrated radiocarbon dates can span several centuries, combining multiple dates together with any chronological constraints can make a chronology much more robust and precise.

This package uses Bayesian methods to enforce the chronological ordering of radiocarbon and other dates, for example for trees with multiple radiocarbon dates spaced at exactly known intervals (e.g., 10 annual rings). Another example is sites where the relative chronological position of the dates is taken into account - the ages of dates further down a site or core must be older than those of dates further up.

On first usage of the package, it has to be installed:

install.packages('coffee')

To load the code:

library(coffee)
## Loading required package: rintcal

1. tree rings

Let’s simulate the radiocarbon dating of a tree with 400 rings, which started growing in AD 950 (1000 cal BP), with rings radiocarbon dated at regularly spaced 20-yr intervals, and then wiggle-match it using a Bayesian framework (Christen and Litton 19951):

sim.rings(age.start=1000, length=400, gaps=20)
mytree <- rings()
## The run's files will be put in this folder: trees/mytree
## best age: 995 cal BP, 95% range: 1004 to 992 cal BP, 12 yr

The above plot (Fig. 1) contains two panels: the top one shows the calibrated (blue) and wiggle-matched (grey) distributions of all radiocarbon dates, and the bottom one shows how the uncalibrated radiocarbon dates (steelblue dots with error bars) match the calibration curve (green), as well as the modelled age distribution of the ring of interest (by default the oldest, innermost one).

The variable mytree contains both the radiocarbon dating information and the estimated age distribution of the oldest ring.

The file containing the dates should have the following formatting (only the first five entries are shown):

lab ID age error ring cc
18881 601 18 400 1
18882 583 17 380 1
18883 606 19 360 1
18884 709 21 340 1
18885 733 23 320 1

The fourth column should contain the ring number, starting with the youngest, outermost ring and working down backwards in time toward the date of the oldest, innermost ring. The above dates are calibrated using IntCal20 (Reimer et al. 20202) through supplying cc=1; Marine20 (Heaton et al. 20203, cc=2), SHCal20 (Hogg et al. 20204, cc=3) or a tailor-made calibration curve (cc=4) can also be used. Dates that are already on the cal BP scale get cc=0. As with rbacon’s .csv files5, additional columns can be added to deal with reservoir offsets (mean and error; columns 6 and 7) and to provide parameters for the student-t distribution (t.a and t.b; columns 8 and 9).

For more information, ask for help:

?rings

2. stratigraphy

Now that the coffee is brewing, let’s imagine a tiramisu or spekkoek6. A site’s stratigraphy can be age-modelled by assuming chronological ordering of the layers and dates according to their relative position, with levels further down a site assumed to be older than those above them. This can too be done within a Bayesian framework (e.g., Buck et al. 19917):

set.seed(123)
sim.strat()
strat(its=7e5) # note that this run will take long!
# The run's files will be put in this folder: strats/mystrat
#   |>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>| 100%
# Done running...
# Removed a burn-in of 100
# Thinning the MCMC by storing every 40 iterations
# Mean overlap between calibrated and modelled dates 52.16%, ranging from 2.4% (date 2) to 59.2% (date 3)
# Run took 15.15 minutes
Fig. 2 - output of the strat function, from a simulated dataset of 5 chronologically ordered dated levels. The ‘islands’ or ‘swimming elephants’ reflect the age-modelled, chronologically constrained distributions (upward dark grey) and the individual calibrated distributions (downward light-grey). Black lines show the 95% modelled age ranges. Note that whereas the modelled distributions are in chronological order, some of the individually calibrated distributions show reversals.
Fig. 2 - output of the strat function, from a simulated dataset of 5 chronologically ordered dated levels. The ‘islands’ or ‘swimming elephants’ reflect the age-modelled, chronologically constrained distributions (upward dark grey) and the individual calibrated distributions (downward light-grey). Black lines show the 95% modelled age ranges. Note that whereas the modelled distributions are in chronological order, some of the individually calibrated distributions show reversals.

Beside the graph itself, the model also reports the degree of overlap between the calibrated and modelled ages (an overlap of 100% indicates that the calibrated and modelled ages agree perfectly, whereas a small overlap such as 5% indicates outlying dates). The model’s running time is also reported (we hope to be able to speed up the strats runs in the future).

Strat files are .csv files formatted like so:

lab ID age error position cc
8881 4147 83 1 1
8882 4326 87 2 1
8883 4159 83 3 1
8884 4530 91 4 1
8885 4521 90 5 1

The positions of the dates are indicated in the fourth column. See below for more options such as gaps, undated levels or blocks of dates in a layer.

Note that the above model uses 700,000 iterations and took around 13 minutes to run on a reasonably fast computer (it is hoped future versions of coffee will be able to run faster). It is recommendable to assess the influence of the MCMC settings on the model, and use sufficiently large sample sizes (at least 3,000 remaining iterations are recommended). Note that the thinning size is calculated from the MCMC run by default. In the following command, on purpose we are providing some bad initial point age estimates (but still in chronological order; 2 rows of initial ages are needed), storing every single iteration (both thinning and internal.thinning are set to 1), not removing the burn-in and performing a much-too-short run of only 2000 iterations:

set.seed(234)
sim.strat()
strat(burnin=0, thinning=1, internal.thinning=1, its=2000, 
  init.ages=rbind(seq(3000, 4000, length=5), rbind(seq(3010, 4010, length=5))))
Fig. 3 - output of a strat simulation with an overly short MCMC run.
Fig. 3 - output of a strat simulation with an overly short MCMC run.

As can be seen from the top panel, the burnin of the above example took around 500 to 1000 iterations, and also clearly visible are long horizontal stretches that indicate parts of the MCMC where none of the proposed iterations could be accepted. Hence the need for thinning and much longer runs. Note that a warning is given that the run has to be done for longer.

2.1 stratigraphical features

2.1.1 blocks

Sometimes, layers might contain material which cannot be assumed to be in chronological order within that layer (a bit like apple pie), even though the layer itself can safely be assumed to be older than the layers above it and younger than the layers below. In this case, we can assign this ‘block’ of dates to the same position (in this example, layer 4 contains four unordered dates, and we’re using SHCal20):

lab ID age error position cc
8881 3837 77 1 3
8882 4327 87 2 3
8883 4138 83 3 3
8884 4357 87 4 3
8885 3968 79 4 3
8886 4200 84 4 3
8887 4186 84 4 3
8888 4294 86 5 3
8889 4325 86 6 3
8890 4240 85 7 3

Such sites can be modelled too (the blue shading indicates the dates within the block):

strat("block_example", its=2e6, thinning=20, internal.thinning=20)
Fig. 4 - output of a strat simulation containing a ‘block’ of mixed dates, with the block in chronological order with the layers below and above it.
Fig. 4 - output of a strat simulation containing a ‘block’ of mixed dates, with the block in chronological order with the layers below and above it.

2.1.2 undated levels

If you’re interested in modelling the ages of undated levels, in-between dated levels, this can be done in two ways. The first option is to add an undated level, by adding an entry to your .csv file and providing the number 10 in the fifth column, for example:

lab ID age error position cc
12263 10091 202 1 1
12549 10505 210 2 1
12760 10986 220 3 1
12918 11410 228 4 1
undated 5 10
13326 11363 227 6 1
13594 12042 241 7 1
13759 11824 236 8 1

Any name can be given for the lab ID, and the age and error should be left empty. Make sure that the position of the undated level is such that the positions are increasing going down the file. Then run as:

strat("undated_example") # will need a longer run for reliable results (the example below used 4 million iterations)
# The run's files will be put in this folder: strats/undated_example
#   |>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>| 100%
# Done running...
# Removed a burn-in of 100
# Thinning the MCMC by storing every 10 iterations
# Mean overlap between calibrated and modelled dates 65.54%, ranging from 27.6% (date 8) to 99% (date 2)
# Run took 1.01 hours
Fig. 5 - output of a strat simulation containing an undated level, at position 5. This modelled age has no accompanying light-grey ‘shadow’ (which is for calibrated distributions only).
Fig. 5 - output of a strat simulation containing an undated level, at position 5. This modelled age has no accompanying light-grey ‘shadow’ (which is for calibrated distributions only).

Alternatively, just run your site as usual, and after the run model the ages between two of the dated levels (Fig. 6 below; for each iteration this will model random years that fit between the ages of the dates below and above it):

sim.strat(n=5)
strat(its=100, burnin=0, thinning=1, internal.thinning=1) # much too short
## The run's files will be put in this folder: strats/mystrat
## 
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## 
## Done running...
## Removed a burn-in of 0
## Thinning the MCMC by storing every 1 iterations
## Fewer than 3k MCMC iterations remaining, please consider a longer run by increasing 'its'
## Mean overlap between calibrated and modelled dates 26.36%, ranging from 6.6% (date 3) to 57.2% (date 1)
## Run took 0 seconds

undated_1.5 <- ages.undated(1.5) # positioned between positions 1 and 2
summary(undated_1.5)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    4619    4731    4747    4747    4765    4800

2.1.3 gaps

In some instances, there could be information about the time gaps between dated levels. These gaps could either be known exactly (e.g., tree rings), or with a degree of uncertainty (e.g., varves with counting uncertainty, or the development of soils which could be assumed to take an approximately-known amount of time). Coffee can model such gaps as follows:

lab ID age error position cc
12263 10091 202 1 1
exactgap 200 1.5 11
12549 10505 210 2 1
12760 10986 220 3 1
normalgap 100 20 3.5 12
12918 11410 228 4 1
13326 11363 227 5 1
gammagap 100 2 5.5 13
13594 12042 241 6 1
13759 11824 236 7 1

Here, an exact gap is indicated with cc=11, and the size of the gap is placed in the age field (no error to be added). For a normally distributed gap size, add cc=12, and the mean and standard error go in the age and error fields respectively. For a gamma distributed gap size, add cc=13, and enter the mean and shape in the age and error fields respectively.

strat("gaps_example") # will need a longer run for reliable results (the example below used 1 million iterations)
# The run's files will be put in this folder: strats/gaps_example
#   |>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>| 100%
# Done running...
# Removed a burn-in of 100
# Thinning the MCMC by storing every 15 iterations
# Mean overlap between calibrated and modelled dates 52.23%, ranging from 23% (date 6) to 70.4% (date 3)
# Run took 15.15 minutes
Fig. 6 - output of a strat simulation containing gaps between dated levels; an exact gap between positions 1 and 2 (Ex(200)), a normally distributed one between positions 3 and 4 (N(100,20)), and a gamma distributed one between positions 5 and 6 (Ga(100,2)). Blue diagonal lines indicate the size and nature of the gaps. Note that the modelled (dark grey) distributions of dates 1 and 2 are the exact same shape, but shifted by exactly 200 years. The shapes of the modelled distributions surrounding the normal and gamma gaps show only some similarity.
Fig. 6 - output of a strat simulation containing gaps between dated levels; an exact gap between positions 1 and 2 (Ex(200)), a normally distributed one between positions 3 and 4 (N(100,20)), and a gamma distributed one between positions 5 and 6 (Ga(100,2)). Blue diagonal lines indicate the size and nature of the gaps. Note that the modelled (dark grey) distributions of dates 1 and 2 are the exact same shape, but shifted by exactly 200 years. The shapes of the modelled distributions surrounding the normal and gamma gaps show only some similarity.

For more help, type ?strat.

  1. Christen JA, Litton CD, 1995. A Bayesian approach to wiggle-matching. Journal of Archaeological Science 22, 719-725↩︎

  2. Reimer PJ et al. 2020. The IntCal20 Northern Hemisphere radiocarbon age calibration curve (0-55 cal kBP). Radiocarbon 62, 725-757↩︎

  3. Heaton et al. 2020. Marine20-the marine radiocarbon age calibration curve (0-55,000 cal BP). Radiocarbon 62, 779-820↩︎

  4. Hogg et al. 2020. SHCal20 Southern Hemisphere calibration, 0-55,000 years cal BP. Radiocarbon 62, 759-778↩︎

  5. please check the vignettes at https://cran.r-project.org/package=rbacon↩︎

  6. https://en.wikipedia.org/wiki/Spekkoek↩︎

  7. Buck CE, Kenworthy JB, Litton CD, Smith AFM, 1991. Combining archaeological and radiocarbon information: a Bayesian approach to calibration. Antiquity 65, 808-821.↩︎