John R. Tipton | USGS Science

USGS2021

A Bayesian nonparametric approach to unmixing detrital geochronologic data

Sedimentary deposits constitute the primary record of changing environmental conditions that have acted on Earth’s surface over geologic time. Clastic material is eroded from source locations (parents) in sediment routing systems and deposited at sink locations (children). Both parents and children have characteristics that vary across many different dimensions, including grain size, chemical composition, and the geochronologic age of constituent detrital minerals. During transport, sediment from different parents is mixed together to form a child, which in turn may serve as the parent for other sediment farther down-system or later in time when buried sediment is exhumed. The distribution of detrital mineral ages observed in parent and child sediments allows for investigation of the proportion of each parent in the child sediment, which reflects the properties of the sediment routing system. To model the proportion of dates in a child sample that comes from each of the parent distributions, we use a Bayesian mixture of Dirichlet processes. This model enables us to estimate the mixing proportions with associated uncertainty while making minimal assumptions. We also present an extension to the model whereby we reconstruct unobserved parent distributions from multiple observed child distributions using mixtures of Dirichlet processes. The model accounts for uncertainty in both the number of mineral formation events that constitute each parent distribution and the mixing proportions of each parent distribution that constitutes a child distribution. To demonstrate the model, we perform analyses using simulated data where the true age distribution is known as well as using a real-world case study from the coast of central California, USA.

Mathematical Geosciences↗

USGS2019

Predicting paleoclimate from compositional data using multivariate Gaussian process inverse prediction

Multivariate compositional count data arise in many applications including ecology, microbiology, genetics and paleoclimate. A frequent question in the analysis of multivariate compositional count data is what underlying values of a covariate(s) give rise to the observed composition. Learning the relationship between covariates and the compositional count allows for inverse prediction of unobserved covariates given compositional count observations. Gaussian processes provide a flexible framework for modeling functional responses with respect to a covariate without assuming a functional form. Many scientific disciplines use Gaussian process approximations to improve prediction and make inference on latent processes and parameters. When prediction is desired on unobserved covariates given realizations of the response variable, this is called inverse prediction. Because inverse prediction is often mathematically and computationally challenging, predicting unobserved covariates often requires fitting models that are different from the hypothesized generative model. We present a novel computational framework that allows for efficient inverse prediction using a Gaussian process approximation to generative models. Our framework enables scientific learning about how the latent processes co-vary with respect to covariates while simultaneously providing predictions of missing covariates. The proposed framework is capable of efficiently exploring the high dimensional, multi-modal latent spaces that arise in the inverse problem. To demonstrate flexibility, we apply our method in a generalized linear model framework to predict latent climate states given multivariate count data. Based on cross-validation, our model has predictive skill competitive with current methods while simultaneously providing formal, statistical inference on the underlying community dynamics of the biological system previously not available.

Annals of Applied Statistics↗