Luminescence (version 0.8.6)

# calc_MinDose: Apply the (un-)logged minimum age model (MAM) after Galbraith et al. (1999) to a given De distribution

## Description

Function to fit the (un-)logged three or four parameter minimum dose model (MAM-3/4) to De data.

## Usage

calc_MinDose(data, sigmab, log = TRUE, par = 3, bootstrap = FALSE,
init.values, level = 0.95, plot = TRUE, multicore = FALSE, ...)

## Arguments

data

'>RLum.Results or data.frame (required): for data.frame: two columns with De (data[ ,1]) and De error (data[ ,2]).

sigmab

numeric (required): additional spread in De values. This value represents the expected overdispersion in the data should the sample be well-bleached (Cunningham & Walling 2012, p. 100). NOTE: For the logged model (log = TRUE) this value must be a fraction, e.g. 0.2 (= 20 %). If the un-logged model is used (log = FALSE), sigmab must be provided in the same absolute units of the De values (seconds or Gray). See details.

log

logical (with default): fit the (un-)logged minimum dose model to De data.

par

numeric (with default): apply the 3- or 4-parametric minimum age model (par=3 or par=4). The MAM-3 is used by default.

bootstrap

logical (with default): apply the recycled bootstrap approach of Cunningham & Wallinga (2012).

init.values

numeric (optional): a named list with starting values for gamma, sigma, p0 and mu (e.g. list(gamma=100, sigma=1.5, p0=0.1, mu=100)). If no values are provided reasonable values are tried to be estimated from the data.

level

logical (with default): the confidence level required (defaults to 0.95).

plot

logical (with default): plot output (TRUE/FALSE)

multicore

logical (with default): enable parallel computation of the bootstrap by creating a multicore SNOW cluster. Depending on the number of available logical CPU cores this may drastically reduce the computation time. Note that this option is highly experimental and may not work on all machines. (TRUE/FALSE)

...

(optional) further arguments for bootstrapping (bs.M, bs.N, bs.h, sigmab.sd). See details for their usage. Further arguments are

• verbose to de-/activate console output (logical),

• debug for extended console output (logical) and

• cores (integer) to manually specify the number of cores to be used when multicore=TRUE.

## Value

Returns a plot (optional) and terminal output. In addition an '>RLum.Results object is returned containing the following elements:

.$summary data.frame summary of all relevant model results. .$data

data.frame original input data

args

list used arguments

call

call the function call

.$mle mle2 object containing the maximum log likelhood functions for all parameters BIC numeric BIC score .$confint

data.frame confidence intervals for all parameters

.$profile profile.mle2 the log likelihood profiles .$bootstrap

list bootstrap results

The output should be accessed using the function get_RLum

## Function version

0.4.4 (2018-02-13 12:57:49)

## How to cite

Burow, C. (2018). calc_MinDose(): Apply the (un-)logged minimum age model (MAM) after Galbraith et al. (1999) to a given De distribution. Function version 0.4.4. In: Kreutzer, S., Burow, C., Dietze, M., Fuchs, M.C., Schmidt, C., Fischer, M., Friedrich, J. (2018). Luminescence: Comprehensive Luminescence Dating Data Analysis. R package version 0.8.6. https://CRAN.R-project.org/package=Luminescence

## Details

Parameters

This model has four parameters:

 gamma: minimum dose on the log scale mu: mean of the non-truncated normal distribution sigma: spread in ages above the minimum p0: proportion of grains at gamma

If par=3 (default) the 3-parametric minimum age model is applied, where gamma=mu. For par=4 the 4-parametric model is applied instead.

(Un-)logged model

In the original version of the minimum dose model, the basic data are the natural logarithms of the De estimates and relative standard errors of the De estimates. The value for sigmab must be provided as a ratio (e.g, 0.2 for 20 %). This model will be applied if log = TRUE.

If log=FALSE, the modified un-logged model will be applied instead. This has essentially the same form as the original version. gamma and sigma are in Gy and gamma becomes the minimum true dose in the population. Note that the un-logged model requires sigmab to be in the same absolute unit as the provided De values (seconds or Gray).

While the original (logged) version of the mimimum dose model may be appropriate for most samples (i.e. De distributions), the modified (un-logged) version is specially designed for modern-age and young samples containing negative, zero or near-zero De estimates (Arnold et al. 2009, p. 323).

Initial values & boundaries

The log likelihood calculations use the nlminb function for box-constrained optimisation using PORT routines. Accordingly, initial values for the four parameters can be specified via init.values. If no values are provided for init.values reasonable starting values are estimated from the input data. If the final estimates of gamma, mu, sigma and p0 are totally off target, consider providing custom starting values via init.values. In contrast to previous versions of this function the boundaries for the individual model parameters are no longer required to be explicitly specified. If you want to override the default boundary values use the arguments gamma.lower, gamma.upper, sigma.lower, sigma.upper, p0.lower, p0.upper, mu.lower and mu.upper.

Bootstrap

When bootstrap=TRUE the function applies the bootstrapping method as described in Wallinga & Cunningham (2012). By default, the minimum age model produces 1000 first level and 3000 second level bootstrap replicates (actually, the number of second level bootstrap replicates is three times the number of first level replicates unless specified otherwise). The uncertainty on sigmab is 0.04 by default. These values can be changed by using the arguments bs.M (first level replicates), bs.N (second level replicates) and sigmab.sd (error on sigmab). With bs.h the bandwidth of the kernel density estimate can be specified. By default, h is calculated as

$$h = (2*\sigma_{DE})/\sqrt{n}$$

Multicore support

This function supports parallel computing and can be activated by multicore=TRUE. By default, the number of available logical CPU cores is determined automatically, but can be changed with cores. The multicore support is only available when bootstrap=TRUE and spawns n R instances for each core to get MAM estimates for each of the N and M boostrap replicates. Note that this option is highly experimental and may or may not work for your machine. Also the performance gain increases for larger number of bootstrap replicates. Also note that with each additional core and hence R instance and depending on the number of bootstrap replicates the memory usage can significantly increase. Make sure that memory is always availabe, otherwise there will be a massive perfomance hit.

Likelihood profiles

The likelihood profiles are generated and plotted by the bbmle package. The profile likelihood plots look different to ordinary profile likelihood as

"[...] the plot method for likelihood profiles displays the square root of the the deviance difference (twice the difference in negative log-likelihood from the best fit), so it will be V-shaped for cases where the quadratic approximation works well [...]." (Bolker 2016).

For more details on the profile likelihood calculations and plots please see the vignettes of the bbmle package (also available here: https://CRAN.R-project.org/package=bbmle).

## References

Arnold, L.J., Roberts, R.G., Galbraith, R.F. & DeLong, S.B., 2009. A revised burial dose estimation procedure for optical dating of young and modern-age sediments. Quaternary Geochronology 4, 306-325.

Galbraith, R.F. & Laslett, G.M., 1993. Statistical models for mixed fission track ages. Nuclear Tracks Radiation Measurements 4, 459-470.

Galbraith, R.F., Roberts, R.G., Laslett, G.M., Yoshida, H. & Olley, J.M., 1999. Optical dating of single grains of quartz from Jinmium rock shelter, northern Australia. Part I: experimental design and statistical models. Archaeometry 41, 339-364.

Galbraith, R.F., 2005. Statistics for Fission Track Analysis, Chapman & Hall/CRC, Boca Raton.

Galbraith, R.F. & Roberts, R.G., 2012. Statistical aspects of equivalent dose and error calculation and display in OSL dating: An overview and some recommendations. Quaternary Geochronology 11, 1-27.

Olley, J.M., Roberts, R.G., Yoshida, H., Bowler, J.M., 2006. Single-grain optical dating of grave-infill associated with human burials at Lake Mungo, Australia. Quaternary Science Reviews 25, 2469-2474.

Arnold, L.J. & Roberts, R.G., 2009. Stochastic modelling of multi-grain equivalent dose (De) distributions: Implications for OSL dating of sediment mixtures. Quaternary Geochronology 4, 204-230.

Bolker, B., 2016. Maximum likelihood estimation analysis with the bbmle package. In: Bolker, B., R Development Core Team, 2016. bbmle: Tools for General Maximum Likelihood Estimation. R package version 1.0.18. https://CRAN.R-project.org/package=bbmle

Bailey, R.M. & Arnold, L.J., 2006. Statistical modelling of single grain quartz De distributions and an assessment of procedures for estimating burial dose. Quaternary Science Reviews 25, 2475-2502.

Cunningham, A.C. & Wallinga, J., 2012. Realizing the potential of fluvial archives using robust OSL chronologies. Quaternary Geochronology 12, 98-106.

Rodnight, H., Duller, G.A.T., Wintle, A.G. & Tooth, S., 2006. Assessing the reproducibility and accuracy of optical dating of fluvial deposits. Quaternary Geochronology 1, 109-120.

Rodnight, H., 2008. How many equivalent dose values are needed to obtain a reproducible distribution?. Ancient TL 26, 3-10.

## Examples

# NOT RUN {
data(ExampleData.DeValues, envir = environment())

# (1) Apply the minimum age model with minimum required parameters.
# By default, this will apply the un-logged 3-parametric MAM.
calc_MinDose(data = ExampleData.DeValues$CA1, sigmab = 0.1) # } # NOT RUN { # (2) Re-run the model, but save results to a variable and turn # plotting of the log-likelihood profiles off. mam <- calc_MinDose(data = ExampleData.DeValues$CA1,
sigmab = 0.1,
plot = FALSE)

# Show structure of the RLum.Results object
mam

# Show summary table that contains the most relevant results
res <- get_RLum(mam, "summary")
res

# Plot the log likelihood profiles retroactively, because before
# we set plot = FALSE
plot_RLum(mam)

# Plot the dose distribution in an abanico plot and draw a line
# at the minimum dose estimate
plot_AbanicoPlot(data = ExampleData.DeValues$CA1, main = "3-parameter Minimum Age Model", line = mam,polygon.col = "none", hist = TRUE, rug = TRUE, summary = c("n", "mean", "mean.weighted", "median", "in.ci"), centrality = res$de,
line.col = "red",
grid.col = "none",
line.label = paste0(round(res$de, 1), "\U00B1", round(res$de_err, 1), " Gy"),
bw = 0.1,
ylim = c(-25, 18),
summary.pos = "topleft",
mtext = bquote("Parameters: " ~
sigma[b] == .(get_RLum(mam, "args")$sigmab) ~ ", " ~ gamma == .(round(log(res$de), 1)) ~ ", " ~
sigma == .(round(res$sig, 1)) ~ ", " ~ rho == .(round(res$p0, 2))))

# (3) Run the minimum age model with bootstrap
# NOTE: Bootstrapping is computationally intensive
# (3.1) run the minimum age model with default values for bootstrapping
calc_MinDose(data = ExampleData.DeValues$CA1, sigmab = 0.15, bootstrap = TRUE) # (3.2) Bootstrap control parameters mam <- calc_MinDose(data = ExampleData.DeValues$CA1,
sigmab = 0.15,
bootstrap = TRUE,
bs.M = 300,
bs.N = 500,
bs.h = 4,
sigmab.sd = 0.06,
plot = FALSE)

# Plot the results
plot_RLum(mam)

# save bootstrap results in a separate variable
bs <- get_RLum(mam, "bootstrap")

# show structure of the bootstrap results
str(bs, max.level = 2, give.attr = FALSE)

# print summary of minimum dose and likelihood pairs
summary(bs$pairs$gamma)

# Show polynomial fits of the bootstrap pairs
bs$poly.fits$poly.three

# Plot various statistics of the fit using the generic plot() function
par(mfcol=c(2,2))
plot(bs$poly.fits$poly.three, ask = FALSE)

# Show the fitted values of the polynomials
summary(bs$poly.fits$poly.three\$fitted.values)
# }
# NOT RUN {
# }