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Abstract: We present a method to quantify abrupt changes (or changepoints) in data series, represented as a function of depth or time. These changes are often the result of climatic or environmental variations and can be manifested in multiple datasets as different responses, but all datasets can have the same changepoint locations/timings. The method we present uses transdimensional Markov chain Monte Carlo to infer probability distributions on the number and locations (in depth or time) of changepoints, the mean values between changepoints and, if required, the noise variance associated with each dataset being considered. This latter point is important as we generally will have limited information on the noise, such as estimates only of measurement uncertainty, and in most cases it is not practical to make repeat sampling/measurement to assess other contributions to the variation in the data. We describe the main features of the approach (and describe the mathematical formulation in supplementary material), and demonstrate its validity using synthetic datasets, with known changepoint structure (number and locations of changepoints) and distribution of noise variance for each dataset. We show that when using multiple data, we expect to achieve better resolution of the changepoint structure than when we use each dataset individually. This is conditional on the validity of the assumption of common changepoints between different datasets. We then apply the method to two sets of real geochemical data, both from peat cores, taken from NE Australia and eastern Tibet. Under the assumption that changes occur at the same time for all datasets, we recover solutions consistent with those previously inferred qualitatively from independent data and interpretations. However, our approach provides a quantitative estimate of the relative probability of the inferred changepoints, allowing an objective assessment of the significance of each change. [Copyright &y& Elsevier]

Abstract: We present an overview of Markov chain Monte Carlo, a sampling method for model inference and uncertainty quantification. We focus on the Bayesian approach to MCMC, which allows us to estimate the posterior distribution of model parameters, without needing to know the normalising constant in Bayes'' theorem. Given an estimate of the posterior, we can then determine representative models (such as the expected model, and the maximum posterior probability model), the probability distributions for individual parameters, and the uncertainty about the predictions from these models. We also consider variable dimensional problems in which the number of model parameters is unknown and needs to be inferred. Such problems can be addressed with reversible jump (RJ) MCMC. This leads us to model choice, where we may want to discriminate between models or theories of differing complexity. For problems where the models are hierarchical (e.g. similar structure but with a different number of parameters), the Bayesian approach naturally selects the simpler models. More complex problems require an estimate of the normalising constant in Bayes'' theorem (also known as the evidence) and this is difficult to do reliably for high dimensional problems. We illustrate the applications of RJMCMC with 3 examples from our earlier working involving modelling distributions of geochronological age data, inference of sea-level and sediment supply histories from 2D stratigraphic cross-sections, and identification of spatially discontinuous thermal histories from a suite of apatite fission track samples distributed in 3D. [Copyright &y& Elsevier]