Joint Registration and Conformal Prediction for Partially Observed Functional Data

TL;DR

This paper introduces a unified method combining registration and conformal prediction to accurately predict missing segments in partially observed functional data, effectively handling amplitude and phase variations with finite-sample guarantees.

Contribution

It proposes a novel integrated approach for functional data prediction that separately addresses amplitude and phase, ensuring coverage guarantees without strong parametric assumptions.

Findings

Produces pointwise prediction bands with finite-sample coverage guarantees.

Efficient and easy-to-implement, suitable for parallel computing.

Demonstrates effectiveness on real-world and simulated data.

Abstract

Predicting missing segments in partially observed functions is challenging due to infinite-dimensionality, complex dependence within and across observations, and irregular noise. These challenges are further exacerbated by the existence of two distinct sources of variation in functional data, termed amplitude (variation along the $y$-axis) and phase (variation along the $x$-axis). While registration can disentangle them from complete functional data, the process is more difficult for partial observations. Thus, existing methods for functional data prediction often ignore phase variation. Furthermore, they rely on strong parametric assumptions, and require either precise model specifications or computationally intensive techniques, such as bootstrapping, to construct prediction intervals. To tackle this problem, we propose a unified registration and prediction approach for partially observed functions under the conformal prediction framework, which separately focuses on the amplitude and phase components. By leveraging split conformal methods, our approach integrates registration and prediction while ensuring exchangeability through carefully constructed predictor-response pairs. Using a neighborhood smoothing algorithm, the framework produces pointwise prediction bands with finite-sample marginal coverage guarantees under weak assumptions. The method is easy to implement, computationally efficient, and suitable for parallelization. Numerical studies and real-world data examples clearly demonstrate the effectiveness and practical utility of the proposed approach.