Author: Claire Christophe, Anne Philippe, Sebastian Kreutzer, Guillaume Guérin, Frederik Baumgarten

# BayLum: An Introduction

#### Claire Christophe, Anne Philippe, Sebastian Kreutzer, Guillaume Guérin, Frederik Baumgarten

#### Updated for BayLum package version 0.3.2 (2024-09-18)

# 1 Introduction

`'BayLum'` provides a collection of various R functions for Bayesian analysis of luminescence data. Amongst others, this includes data import, export, application of age models and palaeodose modelling.

Data can be processed simultaneously for various samples, including the input of multiple BIN/BINX-files per sample for single grain (SG) or multi-grain (MG) OSL measurements. Stratigraphic constraints and systematic errors can be added to constrain the analysis further.

For those who already know how to use R, `'BayLum'` won’t be difficult to use, for all others, this brief introduction may be of help to make the first steps with R and the package `'BayLum'` as convenient as possible.

## 1.1 Installing `BayLum’ package

If you read this document before having installed R itself, you should first visit the R project website and download and install R. You may also consider installing Rstudio, which provides an excellent desktop working environment for R; however it is not a prerequisite.

You will also need the external software JAGS (Just Another Gibs Sampler). Please visit the JAGS webpage and follow the installation instructions. Now you are nearly ready to work with ‘BayLum’.

If you have not yet installed ‘BayLum’, please run the following two R code lines to install ‘BayLum’ on your computer.

    install.packages("BayLum", dependencies = TRUE)

Alternatively, you can load an already installed R package (here ‘BayLum’) into your session by using the following R call.

    library(BayLum)

# 2 First steps: age analysis of one sample

Measurement data can be imported using two different options as detailed in the following:

- Option 1: Using the conventional ‘BayLum’ folder structure (old)
- Option 2: Using a single-setting config file (new)

## 2.1 Option 1: Import information from a BIN/BINX-file.

Let us consider the sample named samp1, which is the example dataset coming with the package. All information related to this sample is stored in a subfolder called also samp1. To test the package example, first, we add the path of the example dataset to the object `path`.

    path <- paste0(system.file("extdata/", package = "BayLum"), "/")

Please note that for your own dataset (i.e. not included in the package) you have to replace this call by something like:

    path <- "Users/Master_of_luminescence/Documents/MyFamousOSLData"

In our example the folder contains the following subfolders and files:

1 | example.yml
2 | FER1/bin.bin
3 | FER1/Disc.csv
4 | FER1/DoseEnv.csv
5 | FER1/DoseSource.csv
6 | FER1/rule.csv
7 | samp1/bin.bin
8 | samp1/DiscPos.csv
9 | samp1/DoseEnv.csv
10 | samp1/DoseSource.csv
11 | samp1/rule.csv
12 | samp2/bin.bin
13 | samp2/DiscPos.csv
14 | samp2/DoseEnv.csv
15 | samp2/DoseSource.csv
16 | samp2/rule.csv
17 | yaml_config_reference.yml

See “What are the required files in each subfolder?” in the manual of `Generate_DataFile()` function for the meaning of these files.

To import your data, simply call the function `Generate_DataFile()`:

    DATA1 <-
      Generate_DataFile(
        Path = path,
        FolderNames = "samp1",
        Nb_sample = 1,
        verbose = FALSE)

    Warning in Generate_DataFile(Path = path, FolderNames = "samp1", Nb_sample = 1, : 'Generate_DataFile' est obsolète.
    Utilisez plutôt ‘create_DataFile()’.
    Voir help("Deprecated")

### 2.1.1 Remarks

#### 2.1.1.1 Data import/export

The import may take a while, in particular for large BIN/BINX-files. This can become annoying if you want to play with the data. In such situations, it makes sense to save your imported data somewhere else before continuing.

To save the obove imported data on your hardrive use

    save(DATA1, file = "YourPath/DATA1.RData")

To load the data use

    load(DATA1, file = "YourPath/DATA1.RData")

#### 2.1.1.2 Data structure

To see the overall structure of the data generated from the BIN/BINX-file and the associated CSV-files, the following call can be used:

    str(DATA1)

    List of 9
     $ LT            :List of 1
      ..$ : num [1, 1:7] 2.042 0.842 1.678 3.826 4.258 ...
     $ sLT           :List of 1
      ..$ : num [1, 1:7] 0.344 0.162 0.328 0.803 0.941 ...
     $ ITimes        :List of 1
      ..$ : num [1, 1:6] 15 30 60 100 0 15
     $ dLab          : num [1:2, 1] 1.53e-01 5.89e-05
     $ ddot_env      : num [1:2, 1] 2.512 0.0563
     $ regDose       :List of 1
      ..$ : num [1, 1:6] 2.3 4.6 9.21 15.35 0 ...
     $ J             : num 1
     $ K             : num 6
     $ Nb_measurement: num 16

It reveals that `DATA1` is basically a list with 9 elements:

Element | Content
--- | ---
`DATA1$LT` | \(L_x\)/\(T_x\) values from each sample
`DATA1$sLT` | \(L_x\)/\(T_x\) error values from each sample
`DATA1$ITimes` | Irradiation times
`DATA1$dLab` | The lab dose rate
`DATA1$ddot_env` | The environmental dose rate and its variance
`DATA1$regDose` | The regenerated dose points
`DATA1$J` | The number of aliquots selected for each BIN-file
`DATA1$K` | The number of regenerated dose points
`DATA1$Nb_measurement` | The number of measurements per BIN-file

#### 2.1.1.3 Visualise Lx/Tx values and dose points

To get an impression on how your data look like, you can visualise them by using the function `LT_RegenDose()`:

    LT_RegenDose(
      DATA = DATA1,
      Path = path,
      FolderNames = "samp1",
      SampleNames = "samp1",
      Nb_sample = 1,
      nrow = NULL
    )

    Warning in LT_RegenDose(DATA = DATA1, Path = path, FolderNames = "samp1", : 'LT_RegenDose' est obsolète.
    Utilisez plutôt ‘plot_RegDosePoints()’.
    Voir help("Deprecated")

plot of chunk unnamed-chunk-10

Note that here we consider only one sample, and the name of the folder is the name of the sample. For that reason the argumetns were set to `FolderNames = samp1` and `SampleNames = samp1`.

### 2.1.2 Generate data file from BIN/BINX-files of multi-grain OSL measurements

For a multi-grain OSL measurements, instead of `Generate_DataFile()`, the function `Generate_DataFile_MG()` should be used with similar parameters. The functions differ by their expectations: Disc.csv instead of DiscPos.csv file for Single-grain OSL Measurements. Please check type `?Generate_DataFile_MG` for further information.

## 2.2 Option 2: Import data using `create_DataFile()`

With `'BayLum'` >= v0.3.2 we introduced a new function called `create_DataFile()`, which will at some point in time replace the function `Generate_DataFile()` and `Generate_DataFile_MG()`. `create_DataFile()` works conceptionally very different from the approach detailed above. Key differences are:

- The function uses a single configuration file for all samples and all measurement files
- The very error prone subfolder structure is no longer needed
- Measurement data can be imported with `create_DataFile()`, but also outside of the function and then passed on the functions. This enables the possibility of extensive pre-processing and selection of measurement data.

The configuration follows the so-called YAML format specification. For single sample the file looks as follows:

    - sample: "samp1"
    files: null
    settings:
    dose_source: { value: 0.1535, error: 0.00005891 }
    dose_env: { value: 2.512, error: 0.05626 }
    rules:
    beginSignal: 6
    endSignal: 8
    beginBackground: 50
    endBackground: 55
    beginTest: 6
    endTest: 8
    beginTestBackground: 50
    endTestBackground: 55
    inflatePercent: 0.027
    nbOfLastCycleToRemove: 1

In the case above, the configuration file assumes that data for `samp1` are already imported and treated and a R object called `samp1` is available in the global environment. The following example code reproduces this case:

    ## get example file path from package
    yaml_file <- system.file("extdata/example.yml", package = "BayLum")
    samp1_file <- system.file("extdata/samp1/bin.bin", package = "BayLum")

    ## read YAML manually and select only the first record
    config_file <- yaml::read_yaml(yaml_file)[[1]]

    ## import BIN/BINX files and select position 2 and grain 32 only
    samp1 <- Luminescence::read_BIN2R(samp1_file, verbose = FALSE) |>
      subset(POSITION == 2 & GRAIN == 32)

    ## create the data file
    DATA1 <- create_DataFile(config_file, verbose = FALSE)

## 2.3 Age computation

To compute the age of the sample samp1, you can run the following code:

    Age <- Age_Computation(
      DATA = DATA1,
      SampleName = "samp1",
      PriorAge = c(10, 100),
      distribution = "cauchy",
      LIN_fit = TRUE,
      Origin_fit = FALSE,
      Iter = 10000
    )

    Compiling model graph
       Resolving undeclared variables
       Allocating nodes
    Graph information:
       Observed stochastic nodes: 6
       Unobserved stochastic nodes: 9
       Total graph size: 139

    Initializing model

plot of chunk unnamed-chunk-12

    >> Sample name <<
    ----------------------------------------------
    samp1

    >> Results of the Gelman and Rubin criterion of convergence <<
    ----------------------------------------------
         Point estimate Uppers confidence interval
    A    1.043       1.102
    D    1.042       1.098
    sD   1.036       1.058

    ---------------------------------------------------------------------------------------------------
     *** WARNING: The following information are only valid if the MCMC chains have converged  ***
    ---------------------------------------------------------------------------------------------------

    parameter    Bayes estimate       Credible interval
    ----------------------------------------------
    A        25.061
                             lower bound     upper bound
                     at level 95%    10          50.294
                     at level 68%    10          21.841
    ----------------------------------------------
    D        62.212
                             lower bound     upper bound
                     at level 95%    19.675          125.768
                     at level 68%    22.385          55.487
    ----------------------------------------------
    sD       44.674
                             lower bound     upper bound
                     at level 95%    0.15        126.445
                     at level 68%    0.263       28.94

This also works if `DATA1` is the output of `Generate_DataFile_MG()`.

##### 2.3.0.0.1 Remark 1: MCMC trajectories

- If MCMC trajectories did not converge, you can add more iteration with the parameter `Iter` in the function `Age_Computation()`, for example `Iter = 20000` or `Iter = 50000`. If it is not desirable to re-run the model from scratch, read the
- To increase the precision of prior distribution, if not specified before you can use the argument `PriorAge`. For example: `PriorAge= c(0.01,10)` for a young sample and `PriorAge = c(10,100)` for an old sample.
- If the trajectories are still not convergering, you should whether the choice you made with the argument `distribution` and dose-response curves are meaningful.

##### 2.3.0.0.2 Remark 2: `LIN_fit` and `Origin_fit`, dose-response curves option

- By default, a saturating exponential plus linear dose response curve is expected. However, you choose other formula by changing arguments `LIN_fit` and `Origin_fit` in the function.

##### 2.3.0.0.3 Remark 3: `distribution`, equivalent dose dispersion option

By default, a cauchy distribution is assumed, but you can choose another distribution by replacing the word `cauchy` by `gaussian`, `lognormal_A` or `lognormal_M` for the argument `distribution`.

The difference between the models: lognormal_A and lognormal_M is that the equivalent dose dispersion are distributed according to:

- a log-normal distribution with mean or average equal to the palaeodose for the first model
- a log-normal distribution with median equal to the palaeodose for the second model.

##### 2.3.0.0.4 Remark 4: `SavePdf` and `SaveEstimates` option

These two arguments allow to save the results to files.

- `SavePdf = TRUE` create a PDF-file with MCMC trajectories of parameters `A` (age), `D` (palaeodose), `sD` (equivalent doses dispersion). You have to specify `OutputFileName` and `OutputFilePath` to define name and path of the PDF-file.
- `SaveEstimates = TRUE` saves a CSV-file containing the Bayes estimates, the credible interval at 68% and 95% and the Gelman and Rudin test of convergence of the parameters `A`, `D`, `sD`. For the export the arguments `OutputTableName` and `OutputTablePath` have to be specified.

##### 2.3.0.0.5 Remark 4: `PriorAge` option

By default, an age between 0.01 ka and 100 ka is expected. If the user has more informations on the sample, `PriorAge` should be modified accordingly.

For example, if you know that the sample is an older, you can set `PriorAge=c(10,120)`. In contrast, if you know that the sample is younger, you may want to set `PriorAge=c(0.001,10)`. Ages of \(<=0\) are not possible. The minimum bound is 0.001.

Please note that the setting of `PriorAge` is not trivial, wrongly set boundaries are likely biasing your results.

## 2.4 Multiple BIN/BINX-files for one sample

In the previous example we considered only the simplest case: one sample, and one BIN/BINX-file. However, ‘BayLum’ allows to process multiple BIN/BINX-files for one sample. To work with multiple BIN/BINX-files, the names of the subfolders need to beset in argument `Names` and both files need to be located unter the same `Path`.

For the case

    Names <- c("samp1", "samp2")

the call `Generate_DataFile()` (or `Generate_DataFile_MG()`) becomes as follows:

    ##argument setting
    nbsample <- 1
    nbbinfile <- length(Names)
    Binpersample <- c(length(Names))

    ##call data file generator
    DATA_BF <- Generate_DataFile(
      Path = path,
      FolderNames = Names,
      Nb_sample = nbsample,
      Nb_binfile = nbbinfile,
      BinPerSample = Binpersample,
      verbose = FALSE
    )

    Warning in Generate_DataFile(Path = path, FolderNames = Names, Nb_sample = nbsample, : 'Generate_DataFile' est obsolète.
    Utilisez plutôt ‘create_DataFile()’.
    Voir help("Deprecated")

    ##calculate the age
    Age <- Age_Computation(
      DATA = DATA_BF,
      SampleName = Names,
      BinPerSample = Binpersample
    )

    Compiling model graph
       Resolving undeclared variables
       Allocating nodes
    Graph information:
       Observed stochastic nodes: 12
       Unobserved stochastic nodes: 15
       Total graph size: 221

    Initializing model

plot of chunk unnamed-chunk-14

    >> Sample name <<
    ----------------------------------------------
    samp1 samp2

    >> Results of the Gelman and Rubin criterion of convergence <<
    ----------------------------------------------
         Point estimate Uppers confidence interval
    A    1.018       1.023
    D    1.022       1.027
    sD   1.044       1.057

    ---------------------------------------------------------------------------------------------------
     *** WARNING: The following information are only valid if the MCMC chains have converged  ***
    ---------------------------------------------------------------------------------------------------

    parameter    Bayes estimate       Credible interval
    ----------------------------------------------
    A        2.312
                             lower bound     upper bound
                     at level 95%    0.86        3.819
                     at level 68%    1.65        2.728
    ----------------------------------------------
    D        5.75
                             lower bound     upper bound
                     at level 95%    2.602       9.545
                     at level 68%    4.453       6.886
    ----------------------------------------------
    sD       0.881
                             lower bound     upper bound
                     at level 95%    0.003       3.318
                     at level 68%    0.003       0.846

# 3 Age analysis of various samples

## 3.1 Generate data file from BIN/BINX-files

The function `Generate_DataFile()` (or `Generate_DataFile_MF()`) can process multiple files simultaneously including multiple BIN/BINX-files per sample.

We assume that we are interested in two samples named: sample1 and sample2. In addition, we have two BIN/BINX-files for the first sample named: sample1-1 and sample1-2, and one BIN-file for the 2nd sample named sample2-1. In such case, we need three subfolders named sample1-1, sample1-2 and sample2-1; which each subfolder containing only one BIN-file named bin.bin, and its associated files DiscPos.csv, DoseEnv.csv, DoseSourve.csv and rule.csv. All of these 3 subfolders must be located in path.

To fill the argument corectly `BinPerSample`: \(binpersample=c(\underbrace{2}_{\text{sample 1: 2 bin files}},\underbrace{1}_{\text{sample 2: 1 bin file}})\)

    Names <-
      c("sample1-1", "sample1-2", "sample2-1") # give the name of the folder datat
    nbsample <- 2    # give the number of samples
    nbbinfile <- 3   # give the number of bin files
    DATA <- Generate_DataFile(
      Path = path,
      FolderNames = Names,
      Nb_sample = nbsample,
      Nb_binfile = nbbinfile,
      BinPerSample = binpersample
    )

### 3.1.1 Combine files using the function `combine_DataFiles()`

If the user has already saved informations imported with `Generate_DataFile()` function (or `Generate_DataFile_MG()` function) these data can be concatenate with the function `combine_DataFiles()`.

For example, if `DATA1` is the output of sample named “GDB3”, and `DATA2` is the output of sample “GDB5”, both data can be merged with the following call:

    data("DATA1", envir = environment())
    data("DATA2", envir = environment())
    DATA3 <- combine_DataFiles(L1 = DATA2, L2 = DATA1)
    str(DATA3)

    List of 11
     $ LT            :List of 2
      ..$ : num [1:188, 1:6] 4.54 2.73 2.54 2.27 1.48 ...
      ..$ : num [1:101, 1:6] 5.66 6.9 4.05 3.43 4.97 ...
     $ sLT           :List of 2
      ..$ : num [1:188, 1:6] 0.333 0.386 0.128 0.171 0.145 ...
      ..$ : num [1:101, 1:6] 0.373 0.315 0.245 0.181 0.246 ...
     $ ITimes        :List of 2
      ..$ : num [1:188, 1:5] 40 40 40 40 40 40 40 40 40 40 ...
      ..$ : num [1:101, 1:5] 160 160 160 160 160 160 160 160 160 160 ...
     $ dLab          : num [1:2, 1:2] 1.53e-01 5.89e-05 1.53e-01 5.89e-05
     $ ddot_env      : num [1:2, 1:2] 2.512 0.0563 2.26 0.0617
     $ regDose       :List of 2
      ..$ : num [1:188, 1:5] 6.14 6.14 6.14 6.14 6.14 6.14 6.14 6.14 6.14 6.14 ...
      ..$ : num [1:101, 1:5] 24.6 24.6 24.6 24.6 24.6 ...
     $ J             : num [1:2] 188 101
     $ K             : num [1:2] 5 5
     $ Nb_measurement: num [1:2] 14 14
     $ SampleNames   : chr [1:2] "samp 1" "samp 1"
     $ Nb_sample     : num 2
     - attr(*, "originator")= chr "create_DataFile"

The data structure should become as follows

- 2 `list`s (1 `list` per sample) for `DATA$LT`, `DATA$sLT`, `DATA1$ITimes` and `DATA1$regDose`
- A `matrix` with 2 columns (1 line per sample) for `DATA1$dLab`, `DATA1$ddot_env`
- 2 `integer`s (1 `integer` per BIN files here we have 1 BIN-file per sample) for `DATA1$J`, `DATA1$K`, `DATA1$Nb_measurement`.

Single-grain and multiple-grain OSL measurements can be merged in the same way. To plot the \(L/T\) as a function of the regenerative dose the function `LT_RegenDose()` can be used again:

    plot_RegDosePoints(DATA3)

Note: In the example `DATA3` contains information from the samples ‘GDB3’ and ‘GDB5’, which are single-grain OSL measurements. For a correct treatment the argument `SG` has to be manually set by the user. Please see the function manual for further details.

## 3.2 Age analysis without stratigraphic constraints

If no stratigraphic constraints were set, the following code can be used to analyse the age of the sample GDB5 and GDB3 simultaneously.

    priorage = c(1, 10, 10, 100)
    Age <- AgeS_Computation(
      DATA = DATA3,
      Nb_sample = 2,
      SampleNames = c("GDB5", "GDB3"),
      PriorAge = priorage,
      distribution = "cauchy",
      LIN_fit = TRUE,
      Origin_fit = FALSE,
      Iter = 1000,
      jags_method = "rjags"
    )

    Warning: No initial values were provided - JAGS will use the same initial values for all chains

    Compiling rjags model...
    Calling the simulation using the rjags method...
    Adapting the model for 1000 iterations...
    Running the model for 5000 iterations...
    Simulation complete
    Calculating summary statistics...
    Calculating the Gelman-Rubin statistic for 6 variables....
    Finished running the simulation

plot of chunk unnamed-chunk-18

plot of chunk unnamed-chunk-18

    >> Results of the Gelman and Rubin criterion of convergence <<
    ----------------------------------------------
     Sample name:  GDB5
    ---------------------
             Point estimate Uppers confidence interval
    A_GDB5   1.002       1.005
    D_GDB5   1.003       1.012
    sD_GDB5      1.006       1.021
    ----------------------------------------------
     Sample name:  GDB3
    ---------------------
             Point estimate Uppers confidence interval
    A_GDB3   1       1
    D_GDB3   1.001       1.002
    sD_GDB3      1.001       1.005

    ---------------------------------------------------------------------------------------------------
     *** WARNING: The following information are only valid if the MCMC chains have converged  ***
    ---------------------------------------------------------------------------------------------------

    >> Bayes estimates of Age, Palaeodose and its dispersion for each sample and credible interval <<
    ----------------------------------------------
     Sample name:  GDB5
    ---------------------
    Parameter    Bayes estimate       Credible interval
     A_GDB5      7.132
                             lower bound     upper bound
                     at level 95%    5.783       8.596
                     at level 68%    6.298       7.677

    Parameter    Bayes estimate       Credible interval
     D_GDB5      17.798
                             lower bound     upper bound
                     at level 95%    16.725          19.004
                     at level 68%    17.145          18.332

    Parameter    Bayes estimate       Credible interval
    sD_GDB5      4.53
                             lower bound     upper bound
                     at level 95%    3.544       5.782
                     at level 68%    4.028       5.142
    ----------------------------------------------
     Sample name:  GDB3
    ---------------------
    Parameter    Bayes estimate       Credible interval
     A_GDB3      46.979
                             lower bound     upper bound
                     at level 95%    36.343          57.758
                     at level 68%    40.774          51.082

    Parameter    Bayes estimate       Credible interval
     D_GDB3      104.689
                             lower bound     upper bound
                     at level 95%    96.694          112.104
                     at level 68%    101.184         108.653

    Parameter    Bayes estimate       Credible interval
    sD_GDB3      16.236
                             lower bound     upper bound
                     at level 95%    9.985       21.678
                     at level 68%    12.11       18.146

    ----------------------------------------------

plot of chunk unnamed-chunk-18

Note: For an automated parallel processing you can set the argument `jags_method = "rjags"` to `jags_method = "rjparallel"`.

### 3.2.1 Remarks

As for the function `Age_computation()`, the age for each sample is set by default between 0.01 ka and 100 ka. If you have more informations on your samples it is possible to change `PriorAge` parameters. `PriorAge` is a vector of `size = 2*$Nb_sample`, the two first values of `PriorAge` concern the 1st sample, the next two values the 2nd sample and so on.

For example, if you know that sample named GDB5 is a young sample whose its age is between 0.01 ka and 10 ka, and GDB3 is an old sample whose age is between 10 ka and 100 ka, \[PriorAge=c(\underbrace{0.01,10}_{GDB5\ prior\ age},\underbrace{10,100}_{GDB3\ prior\ age})\]

## 3.3 Age analysis with stratigraphic constraints

With the function `AgeS_Computation()` it is possible to take the stratigraphic relations between samples into account and define constraints.

For example, we know that GDB5 is in a higher stratigraphical position, hence it likely has a younger age than sample GDB3.

### 3.3.1 Ordering samples

To take into account stratigraphic constraints, the information on the samples need to be ordered. Either you enter a sample name (corresponding to subfolder names) in `Names` parameter of the function `Generate_DataFile()`, ordered by order of increasing ages or you enter saved .RData informations of each sample in `combine_DataFiles()`, ordered by increasing ages.

    # using Generate_DataFile function
    Names <- c("samp1", "samp2")
    nbsample <- 2
    DATA3 <- Generate_DataFile(
      Path = path,
      FolderNames = Names,
      Nb_sample = nbsample,
      verbose = FALSE
    )

    Warning in Generate_DataFile(Path = path, FolderNames = Names, Nb_sample = nbsample, : 'Generate_DataFile' est obsolète.
    Utilisez plutôt ‘create_DataFile()’.
    Voir help("Deprecated")

    # using the function combine_DataFiles()
    data(DATA1, envir = environment()) # .RData on sample GDB3
    data(DATA2, envir = environment()) # .RData on sample GDB5
    DATA3 <- combine_DataFiles(L1 = DATA1, L2 = DATA2)

### 3.3.2 Define matrix to set stratigraphic constraints

Let `SC` be the matrix containing all information on stratigraphic relations for this two samples. This matrix is defined as follows:

- matrix dimensions: the row number of `StratiConstraints` matrix is equal to `Nb_sample+1`, and column number is equal to \(Nb\_sample\).

- first matrix row: for all \(i\) in \(\{1,...,Nb\_Sample\}\), `StratiConstraints[1,i] <- 1`, means that the lower bound of the sample age given in `PriorAge[2i-1]` for the sample whose number ID is equal to \(i\) is taken into account

- sample relations: for all \(j\) in ${2,…,Nb_Sample+1}$ and all \(i\) in \(\{j,...,Nb\_Sample\}\), `StratiConstraints[j,i] <- 1` if the sample age whose ID is equal to \(j-1\) is lower than the sample age whose ID is equal to \(i\). Otherwise, `StratiConstraints[j,i] <- 0`.

To the define such matrix the function SCMatrix() can be used:

    SC <- SCMatrix(Nb_sample = 2,
                   SampleNames = c("samp1", "samp2"))

In our case: 2 samples, `SC` is a matrix with 3 rows and 2 columns. The first row contains `c(1,1)` (because we take into account the prior ages), the second line contains `c(0,1)` (sample 2, named samp2 is supposed to be older than sample 1, named samp1) and the third line contains `c(0,0)` (sample 2, named samp2 is not younger than the sample 1, here named samp1). We can also fill the matrix with the stratigraphic relations as follow:

    SC <- matrix(
      data = c(1, 1, 0, 1, 0, 0),
      ncol = 2,
      nrow = (2 + 1),
      byrow = T
    )

### 3.3.3 Age computation

    Age <-
      AgeS_Computation(
        DATA = DATA3,
        Nb_sample = 2,
        SampleNames = c("samp1", "samp2"),
        PriorAge = priorage,
        distribution = "cauchy",
        LIN_fit = TRUE,
        Origin_fit = FALSE,
        StratiConstraints = SC,
        Iter = 1000,
        jags_method = 'rjags')

    Warning: No initial values were provided - JAGS will use the same initial values for all chains

    Compiling rjags model...
    Calling the simulation using the rjags method...
    Adapting the model for 1000 iterations...
    Running the model for 5000 iterations...
    Simulation complete
    Calculating summary statistics...
    Calculating the Gelman-Rubin statistic for 6 variables....
    Finished running the simulation

plot of chunk unnamed-chunk-23

plot of chunk unnamed-chunk-23

    >> Results of the Gelman and Rubin criterion of convergence <<
    ----------------------------------------------
     Sample name:  samp1
    ---------------------
             Point estimate Uppers confidence interval
    A_samp1      1.003       1.005
    D_samp1      1       1.002
    sD_samp1     1.003       1.009
    ----------------------------------------------
     Sample name:  samp2
    ---------------------
             Point estimate Uppers confidence interval
    A_samp2      1.004       1.008
    D_samp2      1.005       1.017
    sD_samp2     1.002       1.01

    ---------------------------------------------------------------------------------------------------
     *** WARNING: The following information are only valid if the MCMC chains have converged  ***
    ---------------------------------------------------------------------------------------------------

    >> Bayes estimates of Age, Palaeodose and its dispersion for each sample and credible interval <<
    ----------------------------------------------
     Sample name:  samp1
    ---------------------
    Parameter    Bayes estimate       Credible interval
     A_samp1     9.711
                             lower bound     upper bound
                     at level 95%    9.126       10
                     at level 68%    9.677       10

    Parameter    Bayes estimate       Credible interval
     D_samp1     29.26
                             lower bound     upper bound
                     at level 95%    23.914          34.493
                     at level 68%    26.756          32.052

    Parameter    Bayes estimate       Credible interval
    sD_samp1     67.869
                             lower bound     upper bound
                     at level 95%    51.164          84.839
                     at level 68%    57.714          74.182
    ----------------------------------------------
     Sample name:  samp2
    ---------------------
    Parameter    Bayes estimate       Credible interval
     A_samp2     10.413
                             lower bound     upper bound
                     at level 95%    10          11.236
                     at level 68%    10          10.469

    Parameter    Bayes estimate       Credible interval
     D_samp2     18.343
                             lower bound     upper bound
                     at level 95%    17.089          19.48
                     at level 68%    17.62       18.846

    Parameter    Bayes estimate       Credible interval
    sD_samp2     4.619
                             lower bound     upper bound
                     at level 95%    3.588       5.669
                     at level 68%    4.003       5.09

    ----------------------------------------------

plot of chunk unnamed-chunk-23

Thee results can be also be used for an alternative graphical representation:

    plot_Ages(Age, plot_mode = "density")

plot of chunk unnamed-chunk-24

      SAMPLE    AGE HPD68.MIN HPD68.MAX HPD95.MIN HPD95.MAX ALT_SAMPLE_NAME AT
    1  samp1  9.711     9.677    10.000     9.126    10.000              NA  2
    2  samp2 10.413    10.000    10.469    10.000    11.236              NA  1

## 3.4 When MCMC trajectories did not converge

If MCMC trajectories did not converge, it means we should run additional MCMC iterations.
For `AgeS_computation()` and `Age_OSLC14()` models we can run additional iterations by supplying the function output back into the parent function. In the following, notice we are using the output of the previous `AgeS_computation()` example, namely `Age`. The key argument to set/change is `DATA`.

    Age <- AgeS_Computation(
      DATA = Age,
      Nb_sample = 2,
      SampleNames = c("GDB5", "GDB3"),
      PriorAge = priorage,
      distribution = "cauchy",
      LIN_fit = TRUE,
      Origin_fit = FALSE,
      Iter = 1000,
      jags_method = "rjags"
    )

    Calling the simulation using the rjags method...
    Note: the model did not require adaptation
    Burning in the model for 4000 iterations...
    Running the model for 5000 iterations...
    Simulation complete
    Calculating summary statistics...
    Calculating the Gelman-Rubin statistic for 6 variables....
    Finished running the simulation

plot of chunk unnamed-chunk-25

plot of chunk unnamed-chunk-25

    >> Results of the Gelman and Rubin criterion of convergence <<
    ----------------------------------------------
     Sample name:  GDB5
    ---------------------
             Point estimate Uppers confidence interval
    A_GDB5   1       1
    D_GDB5   1.001       1.004
    sD_GDB5      1.001       1.005
    ----------------------------------------------
     Sample name:  GDB3
    ---------------------
             Point estimate Uppers confidence interval
    A_GDB3   1.008       1.012
    D_GDB3   1.009       1.031
    sD_GDB3      1.007       1.026

    ---------------------------------------------------------------------------------------------------
     *** WARNING: The following information are only valid if the MCMC chains have converged  ***
    ---------------------------------------------------------------------------------------------------

    >> Bayes estimates of Age, Palaeodose and its dispersion for each sample and credible interval <<
    ----------------------------------------------
     Sample name:  GDB5
    ---------------------
    Parameter    Bayes estimate       Credible interval
     A_GDB5      9.724
                             lower bound     upper bound
                     at level 95%    9.154       10
                     at level 68%    9.687       10

    Parameter    Bayes estimate       Credible interval
     D_GDB5      29.372
                             lower bound     upper bound
                     at level 95%    23.658          34.505
                     at level 68%    26.635          32.055

    Parameter    Bayes estimate       Credible interval
    sD_GDB5      67.561
                             lower bound     upper bound
                     at level 95%    50.632          84.409
                     at level 68%    59.654          76.278
    ----------------------------------------------
     Sample name:  GDB3
    ---------------------
    Parameter    Bayes estimate       Credible interval
     A_GDB3      10.406
                             lower bound     upper bound
                     at level 95%    10          11.176
                     at level 68%    10          10.468

    Parameter    Bayes estimate       Credible interval
     D_GDB3      18.29
                             lower bound     upper bound
                     at level 95%    17.184          19.532
                     at level 68%    17.675          18.837

    Parameter    Bayes estimate       Credible interval
    sD_GDB3      4.557
                             lower bound     upper bound
                     at level 95%    3.526       5.591
                     at level 68%    3.975       5.059

    ----------------------------------------------

plot of chunk unnamed-chunk-25

# References

Combès, B., Philippe, A., Lanos, P., Mercier, N., Tribolo, C., Guerin, G., Guibert, P., Lahaye, C., 2015. A Bayesian central equivalent dose model for optically stimulated luminescence dating. Quaternary Geochronology 28, 62-70. doi: 10.1016/j.quageo.2015.04.001

Combès, B., Philippe, A., 2017. Bayesian analysis of individual and systematic multiplicative errors for estimating ages with stratigraphic constraints in optically stimulated luminescence dating. Quaternary Geochronology 39, 24–34. doi: 10.1016/j.quageo.2017.02.003

Philippe, A., Guérin, G., Kreutzer, S., 2019. BayLum - An R package for Bayesian analysis of OSL ages: An introduction. Quaternary Geochronology 49, 16-24. doi: 10.1016/j.quageo.2018.05.009

## Further reading

### For more details on the diagnostic of Markov chains

Robert and Casella, 2009. Introducing Monte Carlo Methods with R. Springer Science & Business Media.

### For details on the here used dataset

Tribolo, C., Asrat, A., Bahain, J. J., Chapon, C., Douville, E., Fragnol, C., Hernandez, M., Hovers, E., Leplongeon, A., Martin, L., Pleurdeau, D., Pearson, O., Puaud, S., Assefa, Z., 2017. Across the Gap: Geochronological and Sedimentological Analyses from the Late Pleistocene-Holocene Sequence of Goda Buticha, Southeastern Ethiopia. PloS one, 12(1), e0169418. doi: 10.1371/journal.pone.0169418