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Abstract:
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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 focusses on developing tools for handling such data. Networks, high dimensional data, images, functions, surfaces, or shapes, all present data structures which are not well handled under a traditional univariate or multivariate statistical paradigm. 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 is a manifold. As a first step of MDA, we present the RACE algorithm for constructing functional units from manifolds based on modern techniques from manifold learning. Combined with spatial basis expansions, the smoothness of the manifolds can be exploited for increased inferential performance. We will further discuss conducting functional principal components on manifold data and illustrate these techniques through high frequency 3D facial imaging.
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