Modelling modern data objects: statistical methods for high-dimensionality and intricate correlation structures

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Advances in technology have been generating data with increased complexity. Modern data objects are often high-dimensional and can lay in 2D, 3D and even 4D Euclidean and non-Euclidean spaces. Examples of such functions can arise in a wide range of scenarios, such as wearable devices, imaging recordings, medical imaging studies, eye-tracking devices, custom made instruments, and others. It is usually of interest to associate these complex functions to other covariates of interest, often scalars, which is the main focus of this proposal. In the literature, these models are known as function-on-scalar regression models (functional response regression models) or scalar-on-function regression models (predictive models). Most of the methodological contributions in functional regression models were initially developed for the first generation functional data which consists of simple smooth functions, typically one dimensional. We build up on previous ideas of data decorrelation and dimensionality reduction to bring a new set of tools that are able to handle the intricate correlation structures inherent in more complex functional data, while simultaneously addressing data high-dimensionality. We handle the complex correlation structures by devising an adaptive basis strategy that provides the foundation for borrowing information within functions. We also propose to develop a criteria to perform basis selection that automatically reduces data dimensionality. The methodology will be embedded in a Bayesian framework with shrinkage priors, that allows us to obtain MCMC samples in the basis space that are easily converted into MCMC samples in the data space through the use of basis inverse transforms. We will also develop an algorithm that will scale up to large datasets and a software package that will be easily accessible and open source. The proposed program is suitable for all HQP expertise level and facilitate their learning of skills in a way that is appropriate for both academia and industry. The contributions that will result from this program will definitely have an impact in advancing the statistical methods to include the latest data advancement.