Mixture models

The program automatically performs a χ²-test for statistical homogeneity, based on the χ²-statistic:

for fission track data, or

for other data (where z_j=log[t_j] and σ_j=σ[t_j]/t_j). The corresponding p-value is shown on the radial plot. A p-value is the probability of observing a χ²_stat-value greater or equal than the observed value under the null assumption of age homogeneity. If the p-value is less than 0.05, say, then the underlying age distribution contains more than one mode and a mixture modelling algorithm may be employed (Galbraith and Green, 1990). To do this, click Options → Mixture models and select the desired number of peaks (between 0 and 5). The radial plot will display the peak centres as radial lines and summarise the numerical results (ages and proportions) under the radial scale.

The algorithm used for the mixture modelling is a hybrid one using both deterministic parts (taken from Galbraith, 2005) and MCMC ingredients. Because of the stochastic nature of the method, the user is advised to run the mixture modelling function several times, to make sure that local minima have been avoided. The auto option minimises the Bayes Information Criterion (BIC) to select the optimal (i.e., parsimonious) number of age components:

where L_max is the maximum likelihood, p is the number of age components and n is the number of grains (Galbraith, 2005).

For Other data, a different algorithm is used, based on the normal mixture modelling algorithm of Galbraith (2005, section 5.5.2). The output comprises the peak locations and proportions. All data are log-transformed prior to the mixture modelling regardless of the transformation used for the plotting.

Minimum age model

Sometimes one is interested in just the youngest age component of a finite mixture. This may be the case for detrital samples with a volcanic component, to estimate the depositional age. This can be modelled by a four parameter probability distribution, in which a proportion π of the population belongs to a discrete peak at time γ and the remaining (1-π) belong to a truncated normal distribution with mean μ and standard deviation σ. DensityPlotter and RadialPlotter implement a simplified version of this model, in which it is assumed that γ=μ:

The log likelihood function is maximized by a robust and deterministic but time consuming grid search algorithm, and the standard deviation of the minimum age is calculated by inverting the Fisher Information matrix, which requires numerically calculating the second derivatives of the log-likelihood. These calculations are computationally intensive and it may take a few seconds before the results are shown. As with the finite mixture models, the results will be depend on the chosen data transformation (linear or logarithmic). As discussed by Galbraith (2005, p.107), the three parameter model may produce slightly biased results, particularly for fission track data with low counts, in which the normal approximation is not valid. In these cases, it may be better to use other computational tools, or to employ a finite mixture model.