# calc_FiniteMixture: Apply the finite mixture model (FMM) after Galbraith (2005)... In R-Lum/Luminescence: Comprehensive Luminescence Dating Data Analysis

## 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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```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.

## 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.

## 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

0.4.1

## How to cite

Burow, C., 2019. calc_FiniteMixture(): Apply the finite mixture model (FMM) after Galbraith (2005) to a given De distribution. Function version 0.4.1. In: Kreutzer, S., Burow, C., Dietze, M., Fuchs, M.C., Schmidt, C., Fischer, M., Friedrich, J., 2019. Luminescence: Comprehensive Luminescence Dating Data Analysis. R package version 0.9.6. https://CRAN.R-project.org/package=Luminescence

## Author(s)

Christoph Burow, University of Cologne (Germany)
Based on a rewritten S script of Rex Galbraith, 2006. , RLum Developer Team

## 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.

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24``` ```## load example data 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") ```