Apply the finite mixture model (FMM) after Galbraith (2005) to a given De distribution
Source:R/calc_FiniteMixture.R
calc_FiniteMixture.RdThis 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): colour 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 graphics::barplot showing the proportions of components if
n.componentsa vector with a length > 1 (e.g.,n.components = c(2:3))- 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
- .$mle
covariance matrices of the log likelihoods
- .$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 component 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
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
NaNconvergence 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 graphics::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.
How to cite
Burow, C., 2024. calc_FiniteMixture(): Apply the finite mixture model (FMM) after Galbraith (2005) to a given De distribution. Function version 0.4.2. In: Kreutzer, S., Burow, C., Dietze, M., Fuchs, M.C., Schmidt, C., Fischer, M., Friedrich, J., Mercier, N., Philippe, A., Riedesel, S., Autzen, M., Mittelstrass, D., Gray, H.J., Galharret, J., Colombo, M., 2024. Luminescence: Comprehensive Luminescence Dating Data Analysis. R package version 0.9.26. https://r-lum.github.io/Luminescence/
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.
Further reading
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.
Author
Christoph Burow, University of Cologne (Germany)
Based on a rewritten S script of Rex Galbraith, 2006.
, RLum Developer Team
Examples
## 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)
#>
#> [calc_FiniteMixture]
#>
#> --- covariance matrix of mle's ---
#>
#> [,1] [,2] [,3]
#> [1,] 0.002144 0.001821 0.000283
#> [2,] 0.001821 0.013319 0.000877
#> [3,] 0.000283 0.000877 0.001118
#>
#> ----------- meta data ------------
#> n: 62
#> sigmab: 0.2
#> number of components: 2
#> llik: -20.3938
#> BIC: 53.169
#>
#> ----------- components -----------
#>
#> comp1 comp2
#>
#> dose (Gy) 31.5299 72.0333
#> rse(dose) 0.1154 0.0334
#> se(dose)(Gy) 3.6387 2.4082
#>
#> proportion 0.1096 0.8904
#>
#> -------- grain probability -------
#>
#> [,1] [,2]
#> [1,] 1.00 0.00
#> [2,] 1.00 0.00
#> [3,] 1.00 0.00
#> [4,] 0.99 0.01
#> [5,] 0.94 0.06
#> [6,] 0.91 0.09
#> [7,] 0.29 0.71
#> [8,] 0.13 0.87
#> [9,] 0.11 0.89
#> [10,] 0.10 0.90
#> [11,] 0.09 0.91
#> [12,] 0.06 0.94
#> [13,] 0.04 0.96
#> [14,] 0.03 0.97
#> [15,] 0.04 0.96
#> [16,] 0.02 0.98
#> [17,] 0.01 0.99
#> [18,] 0.01 0.99
#> [19,] 0.01 0.99
#> [20,] 0.00 1.00
#> [21,] 0.00 1.00
#> [22,] 0.00 1.00
#> [23,] 0.00 1.00
#> [24,] 0.00 1.00
#> [25,] 0.00 1.00
#> [26,] 0.00 1.00
#> [27,] 0.00 1.00
#> [28,] 0.00 1.00
#> [29,] 0.00 1.00
#> [30,] 0.00 1.00
#> [31,] 0.00 1.00
#> [32,] 0.00 1.00
#> [33,] 0.00 1.00
#> [34,] 0.00 1.00
#> [35,] 0.00 1.00
#> [36,] 0.00 1.00
#> [37,] 0.00 1.00
#> [38,] 0.00 1.00
#> [39,] 0.00 1.00
#> [40,] 0.00 1.00
#> [41,] 0.00 1.00
#> [42,] 0.00 1.00
#> [43,] 0.00 1.00
#> [44,] 0.00 1.00
#> [45,] 0.00 1.00
#> [46,] 0.00 1.00
#> [47,] 0.00 1.00
#> [48,] 0.00 1.00
#> [49,] 0.00 1.00
#> [50,] 0.00 1.00
#> [51,] 0.00 1.00
#> [52,] 0.00 1.00
#> [53,] 0.00 1.00
#> [54,] 0.00 1.00
#> [55,] 0.00 1.00
#> [56,] 0.00 1.00
#> [57,] 0.00 1.00
#> [58,] 0.00 1.00
#> [59,] 0.00 1.00
#> [60,] 0.00 1.00
#> [61,] 0.00 1.00
#> [62,] 0.00 1.00
#>
#> -------- single component --------
#> mu: 65.2273
#> sigmab: 0.2
#> llik: -44.96
#> BIC: 94.047
#> ----------------------------------
#>
## (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))
#>
#> [calc_FiniteMixture]
#>
#> ----------- meta data ------------
#> n: 62
#> sigmab: 0.2
#> number of components: 2-4
#>
#> -------- single component --------
#> mu: 65.2273
#> sigmab: 0.2
#> llik: -44.96
#> BIC: 94.047
#>
#> ----------- k components -----------
#> 2 3 4
#> c1_dose 31.53 29.91 29.91
#> c1_se 3.64 3.97 4.32
#> c1_prop 0.11 0.09 0.09
#> c2_dose 72.03 56.65 56.65
#> c2_se 2.41 13.05 33.16
#> c2_prop 0.89 0.25 0.07
#> c3_dose <NA> 77.49 56.65
#> c3_se <NA> 6.37 21.31
#> c3_prop <NA> 0.66 0.18
#> c4_dose <NA> <NA> 77.49
#> c4_se <NA> <NA> 7.68
#> c4_prop <NA> <NA> 0.66
#>
#> ----------- statistical criteria -----------
#> 2 3 4
#> BIC 53.169 59.719 67.973
#> llik -20.394 -19.542 -19.542
#>
#> Lowest BIC score for k = 2
#> No significant increase in maximum log likelihood estimates.
#>
## show structure of the results
FMM
#>
#> [RLum.Results-class]
#> originator: calc_FiniteMixture()
#> data: 10
#> .. $summary : data.frame
#> .. $data : data.frame
#> .. $args : list
#> .. $call : call
#> .. $mle : list
#> .. $BIC : data.frame
#> .. $llik : data.frame
#> .. $grain.probability : list
#> .. $components : matrix
#> .. $single.comp : data.frame
#> additional info elements: 0
## show the results on equivalent dose, standard error and proportion of
## fitted components
get_RLum(object = FMM, data.object = "components")
#> 2 3 4
#> c1_dose 31.53 29.91 29.91
#> c1_se 3.64 3.97 4.32
#> c1_prop 0.11 0.09 0.09
#> c2_dose 72.03 56.65 56.65
#> c2_se 2.41 13.05 33.16
#> c2_prop 0.89 0.25 0.07
#> c3_dose NA 77.49 56.65
#> c3_se NA 6.37 21.31
#> c3_prop NA 0.66 0.18
#> c4_dose NA NA 77.49
#> c4_se NA NA 7.68
#> c4_prop NA NA 0.66
