Functional Classification in Infinite Dimensional Spaces with Incomplete Data via Filtering

The problem of statistical classification and pattern recognition with functional covariates has received considerable attention in recent years. In standard functional classification, the covariates are typically assumed to be fully observable. This research deals with the problem of functional classification with covariates taking values in a separable Hilbert space when some of the covariates may have missing or unobservable fragments. Here, we allow both the training sample as well as the new unclassified observation to have missing fragments in their functional covariates. Moreover, unlike most results in the literature where covariate fragments are typically assumed to be missing completely at random, we do not impose any such assumptions on the underlying missing data mechanisms. Given the observed segments of the curves, we reduce the infinite dimension of the covariates by the "filtering" method and then perform a kernel-type classification on R^d. The consistency of the proposed classifier is also established. Lastly, some numerical simulations are provided to assess the performance of our proposed classifier in practice.