k-NN Regression on Functional Data with Incomplete Observations

In this paper we study a general version of regression where each covariate itself is a functional data such as distributions or functions. In real applications, however, typically we do not have direct access to such data; instead only some noisy estimates of the true covariate functions/distributions are available to us. For example, when each covariate is a distribution, then we might not be able to directly observe these distributions, but it can be assumed that i.i.d. sample sets from these distributions are available. In this paper we present a general framework and a k-NN based estimator for this regression problem. We prove consistency of the estimator and derive its convergence rates. We further show that the proposed estimator can adapt to the local intrinsic dimension in our case and provide a simple approach for choosing k. Finally, we illustrate the applicability of our framework with numerical experiments.