Joint Registration and Conformal Prediction for Partially Observed Functional Data

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 treat phase variation as negligible. Furthermore, they typically require precise model specifications and/or rely on computationally intensive tools, such as bootstrapping, to construct prediction intervals. We propose a unified registration and prediction approach for partially observed functions using conformal prediction. Our method integrates registration and prediction in one algorithm 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 permits simple parallelization. Numerical studies and real-world data examples demonstrate the effectiveness and practical utility of our method.