Model selection during sample-standard-bracketing using reversible jump Markov chain Monte Carlo

Abstract

For instrument calibration where interpolation between reference materials is required, there exists a need for a generally applicable technique to determine: (1) the value of the calibration (e.g., the mass bias at an arbitrary time); (2) the uncertainty of this value; and, (3) the degree to which the uncertainties in the reference material analyses account for their scatter about the calibration. Here, we show that an implementation of the reversible-jump Markov chain Monte Carlo (rj-MCMC) technique can provide all three values for a reasonable range of complexity. Using this algorithm we treat a drifting calibration value as a function of time by a series of straight line segments. The benefit of the rj-MCMC technique is that the number of straight line segments does not need to be specified a priori, but is a parameter that is estimated. This technique is also able to simultaneously determine the presence and magnitude of overdispersion (the amount of scatter in the data not accounted for by estimated uncertainties) even in the presence of complex, non-linear drift. The result of this data treatment is a probability distribution in calibration-time space that, despite having an origin in line segments, smoothly follows the data and therefore yields the calibration value and its uncertainty. We validate this technique using synthetic data from prescribed distributions, and demonstrate its utility and flexibility by applying it to data collected by multi-collector inductively coupled plasma mass spectrometry that display complex non-linear drift.