Luminescence (version 0.8.6)

# calc_FiniteMixture: Apply the finite mixture model (FMM) after Galbraith (2005) to a given De distribution

## Description

This function fits a k-component mixture to a De distribution with differing known standard errors. Parameters (doses and mixing proportions) are estimated by maximum likelihood assuming that the log dose estimates are from a mixture of normal distributions.

## Usage

```calc_FiniteMixture(data, sigmab, n.components, grain.probability = FALSE,
dose.scale, pdf.weight = TRUE, pdf.sigma = "sigmab",
pdf.colors = "gray", pdf.scale, plot.proportions = TRUE,
plot = TRUE, ...)```

## Arguments

data

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

sigmab

numeric (required): spread in De values given as a fraction (e.g. 0.2). This value represents the expected overdispersion in the data should the sample be well-bleached (Cunningham & Wallinga 2012, p. 100).

n.components

numeric (required): number of components to be fitted. If a vector is provided (e.g. `c(2:8)`) the finite mixtures for 2, 3 ... 8 components are calculated and a plot and a statistical evaluation of the model performance (BIC score and maximum log-likelihood) is provided.

grain.probability

logical (with default): prints the estimated probabilities of which component each grain is in

dose.scale

numeric: manually set the scaling of the y-axis of the first plot with a vector in the form of `c(min, max)`

pdf.weight

logical (with default): weight the probability density functions by the components proportion (applies only when a vector is provided for `n.components`)

pdf.sigma

character (with default): if `"sigmab"` the components normal distributions are plotted with a common standard deviation (i.e. `sigmab`) as assumed by the FFM. Alternatively, `"se"` takes the standard error of each component for the sigma parameter of the normal distribution

pdf.colors

character (with default): color coding of the components in the the plot. Possible options are `"gray"`, `"colors"` and `"none"`

pdf.scale

numeric: manually set the max density value for proper scaling of the x-axis of the first plot

plot.proportions

logical (with default): plot barplot showing the proportions of components

plot

logical (with default): plot output

...

further arguments to pass. See details for their usage.

## 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

covariance matrices of the log likelhoods

.\$BIC

BIC score

.\$llik

maximum log likelihood

.\$grain.probability

probabilities of a grain belonging to a component

.\$components

matrix estimates of the de, de error and proportion for each component

.\$single.comp

data.frame single componente FFM estimate

If a vector for n.components is provided (e.g. c(2:8)), mle and grain.probability are lists containing matrices of the results for each iteration of the model.

The output should be accessed using the function get_RLum

## Function version

0.4 (2018-04-19 13:18:48)

## How to cite

Burow, C. (2018). calc_FiniteMixture(): Apply the finite mixture model (FMM) after Galbraith (2005) to a given De distribution. Function version 0.4. 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

This model uses the maximum likelihood and Bayesian Information Criterion (BIC) approaches.

Indications of overfitting are:

• increasing BIC

• repeated dose estimates

• covariance matrix not positive definite

• covariance matrix produces NaNs

• convergence problems

Plot

If a vector (`c(k.min:k.max)`) is provided for `n.components` a plot is generated showing the the k components equivalent doses as normal distributions. By default `pdf.weight` is set to `FALSE`, so that the area under each normal distribution is always 1. If `TRUE`, the probability density functions are weighted by the components proportion for each iteration of k components, so the sum of areas of each component equals 1. While the density values are on the same scale when no weights are used, the y-axis are individually scaled if the probability density are weighted by the components proportion. The standard deviation (sigma) of the normal distributions is by default determined by a common `sigmab` (see `pdf.sigma`). For `pdf.sigma = "se"` the standard error of each component is taken instead. The stacked barplot shows the proportion of each component (in per cent) calculated by the FFM. The last plot shows the achieved BIC scores and maximum log-likelihood estimates for each iteration of k.

## References

Galbraith, R.F. & Green, P.F., 1990. Estimating the component ages in a finite mixture. Nuclear Tracks and Radiation Measurements 17, 197-206.

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

Roberts, R.G., Galbraith, R.F., Yoshida, H., Laslett, G.M. & Olley, J.M., 2000. Distinguishing dose populations in sediment mixtures: a test of single-grain optical dating procedures using mixtures of laboratory-dosed quartz. Radiation Measurements 32, 459-465.

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

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.

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 finite mixture model
## NOTE: the data set is not suitable for the finite mixture model,
## which is why a very small sigmab is necessary
calc_FiniteMixture(ExampleData.DeValues\$CA1,
sigmab = 0.2, n.components = 2,
grain.probability = TRUE)

## (2) repeat the finite mixture model for 2, 3 and 4 maximum number of fitted
## components and save results
## NOTE: The following example is computationally intensive. Please un-comment
## the following lines to make the example work.
FMM<- calc_FiniteMixture(ExampleData.DeValues\$CA1,
sigmab = 0.2, n.components = c(2:4),
pdf.weight = TRUE, dose.scale = c(0, 100))

## show structure of the results
FMM

## show the results on equivalent dose, standard error and proportion of
## fitted components
get_RLum(object = FMM, data.object = "components")

# }
```