Abstract
We consider the problem of classifying curves when they are observed only partially on their parameter domains. We propose computational methods for (i) completion of partially observed curves; (ii) assessment of completion variability through a nonparametric multiple imputation procedure; (iii) development of nearest neighbor classifiers compatible with the completion techniques. Our contributions are founded on exploiting the geometric notion of shape of a curve, defined as those aspects of a curve that remain unchanged under translations, rotations and reparameterizations. Explicit incorporation of shape information into the computational methods plays the dual role of limiting the set of all possible completions of a curve to those with similar shape while simultaneously enabling more efficient use of training data in the classifier through shape-informed neighborhoods. Our methods are then used for taxonomic classification of partially observed curves arising from images of fossilized Bovidae teeth, obtained from a novel anthropological application concerning paleoenvironmental reconstruction.
Highlights
Modern functional and curve data come in all shapes and sizes
We presented two algorithms to complete and classify partially observed planar curves and simultaneously assess variability involved with the completion through a multiple imputation procedure
Through the application of the proposed framework on real data, we have found that hot-deck imputation can sometimes deteriorate classification performance; there is an intuitive explanation for these findings
Summary
Modern functional and curve data come in all shapes and sizes (pun intended!). A particular type of functional data that is starting to receive attention in recent years consists of univariate functions that are only observed in sub-intervals of their interval domains. Fundamental to the routine task of comparing and identifying objects by humans or a computer is an implicit understanding of a set of symmetries or transformations pertaining to its shape: those properties or features of the object that are unaffected by nuisance information (e.g., orientation of the camera under which the object is imaged). Such an understanding assumes added importance when the object is only partially observed (e.g., identifying a chair hidden behind a table based on the backrest only) since it eliminates the need to consider a substantially large class of possible completions of the object. 3) We propose two nearest neighbor classification procedures for partially observed curves based on shape distances by utlizing completions obtained from any of the above two algorithms
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