Bayesian nonparametric latent feature modelling of river water quality

Abstract

Latent feature modelling is a class of multivariate techniques used to capture hidden structures underlying observed data sets. There are numerous instances in statistical literature where these techniques are applied to data sets, most notably from psychology. However, there is little to no published literature on the application of latent feature models to water quality data sets from scientific fields that attempt to identify and analyse scientific processes affecting our environment. In this study, two different latent feature modelling techniques known as Structural Equation Modelling (SEM) and Bayesian Nonparametric (BNP) latent feature modelling are fitted to a collection of data related to water quality. The former technique takes a finite-dimensional classical Frequentist approach, where the number of latent features extracted by the model is fixed and must be specified beforehand. In addition, hierarchical data is catered for by introducing a multi-level extension of SEM known as MSEM. However, these models encounter various problems when applied to larger, more complex data sets. To work with a more flexible and wider ranging family of models, we transition to infinite dimensions through the use of BNP models, which are the main protagonists of this study. In contrast to SEM, these models allow the number of latent features to be open-ended. The applicability of this technique to river water quality is determined by fitting a linear-Gaussian binary latent feature model using a Dirichlet Process (DP) prior. Studying the resulting posterior distribution allows us to identify and explain the possible sources affecting the water quality of rivers over time. Information on the provenance and timing of the observations comprising the data set shall be purposely left out during the estimation to enable us to test whether BNP models are able to detect, through the use of information contained within the data, inherent factors which are not measured directly but inferred statistically.