Statistical Analysis on Object Spaces with Applications

Most of the data encountered is bounded nonlinear data. The Universe is bounded, planets are sphere like shaped objects, and life growing on Earth comes in various shapes and colors that can hardly be represented as points on a linear space, and even if the object space they sit on is embedded in a Euclidean space, their mean vector can not be represented as a point on that object space, except for the case when such space is convex. To address this misgiving, since the mean vector is the minimizer of the expected square distance, following Fr\'echet (1948), on a compact metric space, one may consider both minimizers and maximizers of the expected square distance to a given point on the object space as mean, respectively {\bf anti-mean} of a given random point. Of all distances on a object space, one considers here the chord distance associated with an embedding of the object space, since for such distances one can give a necessary and sufficient condition for the existence of a unique Fr\'echet mean (respectively Fr\'echet anti-mean). For such distributions these location parameters are called extrinsic mean (respectively extrinsic anti-mean), and the corresponding sample statistics are consistent estimators of their population counterparts. Moreover one derives the limit distribution of such estimators around a mean located at a smooth extrinsic anti-mean. Extrinsic analysis is thus a general framework that allows one to run object data analysis on nonlinear object spaces that can be embedded in a numerical space. In particular one focuses on Veronese-Whitney (VW) means and anti-means of 3D projective shapes of configurations extracted from digital camera images. The 3D data extraction is greatly simplified by an RGB based algorithm followed by the Faugeras-Hartley-Gupta-Chen 3D reconstruction method. In particular one derives two sample tests for face analysis based on projective shapes, and more generally a MANOVA on manifolds method to be used in 3D projective shape analysis. The manifold based approach is also applicable to financial data analysis for exchange rates.