Bayesian analysis of historical functional linear models with application to air pollution forecasting - Junglee, Yovna (Department of Statistical Sciences, 2023)
Historical functional linear models are used to analyse the relationship between a functional response and a functional predictor whereby only the past of the predictor process can affect the current outcome. In this work, we develop a Bayesian framework for the analysis of the historical functional linear model with multiple predictors. Different from existing Bayesian approaches to historical functional linear models, our proposed methodology is able to handle multiple functional covariates with measurement error and sparseness. The proposed model utilises the well-established connection between non-parametric smoothing and Bayesian methods to reduce sensitivity to the number of basis functions which are used to model the functional regression coefficients. We investigate two methods of estimation within the Bayesian framework. We first propose to smooth the functional predictors independently from the regression model in a two-stage analysis, and secondly, jointly with the regression model. The efficiency of the MCMC algorithms is increased by implementing a Cholesky decomposition to sample from high-dimensional Gaussian distributions and by taking advantage of the orthogonal properties of the functional principal components used to model the functional covariates. Our extensive simulation study shows substantial improvements in both the recovery of the functional regression surface and the true underlying functional response with higher coverage probabilities, when compared to a classical model under which the measurement error is unaccounted for. We further found that the Bayesian two-stage analysis outperforms the joint model under certain conditions. A major challenge with the collection of environmental data is that they are prone to measurement error, both random and systematic. Hence, our methodology provides a reliable functional data analytic framework for modelling environmental data. Our focus is on the application of our method to forecast the level of daily atmospheric pollutants using meteorological information such as hourly records of temperature, humidity and wind speed from data collected by the City of Cape Town, South Africa. The forecasts provided by the proposed Bayesian two-stage model are highly competitive against the functional autoregressive models which are traditionally used for functional time series.