Online Estimation for Functional Data

TL;DR

This paper introduces an online nonparametric method for real-time updating of mean and covariance function estimates in functional data analysis, addressing challenges posed by streaming data.

Contribution

It develops a computationally efficient online approach using dynamic bandwidth selection and sufficient statistics, with theoretical guarantees and practical validation.

Findings

Method achieves asymptotic normality of estimates.

Provides bounds on relative efficiency of online estimates.

Validated with simulations and real data examples.

Abstract

Functional data analysis has attracted considerable interest and is facing new challenges, one of which is the increasingly available data in a streaming manner. In this article we develop an online nonparametric method to dynamically update the estimates of mean and covariance functions for functional data. The kernel-type estimates can be decomposed into two sufficient statistics depending on the data-driven bandwidths. We propose to approximate the future optimal bandwidths by a sequence of dynamically changing candidates and combine the corresponding statistics across blocks to form the updated estimation. The proposed online method is easy to compute based on the stored sufficient statistics and the current data block. We derive the asymptotic normality and, more importantly, the relative efficiency lower bounds of the online estimates of mean and covariance functions. This provides insight into the relationship between estimation accuracy and computational cost driven by the length of candidate bandwidth sequence. Simulations and real data examples are provided to support such findings.