Localization processes for functional data analysis
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We propose an alternative to k-nearest neighbors for functional data whereby the approximating neighbor curves are piecewise functions built from a functional sample. Instead of a distance on a function space we use a locally defined distance function that satisfies stabilization criteria. We exploit this feature to develop the asymptotic theory when the number of curves is large enough or when a finite number of curves is observed at time-points coinciding with the realization of a point process with intensity increasing to infinity. We use these results to investigate the problem of estimating unobserved segments of a partially observed functional data sample as well as to study the problem of functional classification and outlier detection. For these problems, we discuss methods that are competitive with and often superior to benchmark predictions in the field.
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