  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. This paper
presents 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 framework
that converts manifolds into functional objects, an efficient 2-step functional
principal component method, and a manifold-on-scalar regression model. This
work is motivated by an anthropological application involving 3D facial imaging
data, which is discussed extensively throughout the paper. The proposed
framework is used to understand how individual characteristics, such as age and
genetic ancestry, influence the shape of the human face.
