Keywords: Functional data analysis; Manifold data analysis; Shape analysis; Manifold learning; Functional principal component analysis; Functional regression
Many scientific areas are faced with the challenge of extracting information from large, complex, and highly structured data sets. A great deal of modern statistical work focuses on developing tools for handling such data. We present a new subfield of functional data analysis, FDA, which we call Manifold Data Analysis, or MDA. MDA is concerned with the statistical analysis of samples where one or more variables measured on each unit is a manifold, thus resulting in as many manifolds as we have units. We propose a novel framework that converts manifolds into functional objects, a computationally efficient 2-step functional principal component method, and a manifold-on-scalar regression model with an application to 3D facial imaging data.