Joint Registration and Conformal Prediction for Partially Observed Functional Data

2025-02-20

AI Summary

Existing methods for predicting missing segments in partially observed functional data often neglect phase variation, rely on strong parametric assumptions, or incur high computational costs due to resampling—compromising both modeling accuracy and reliable uncertainty quantification. Method: We propose the first unified framework integrating functional registration with conformal prediction, termed *segment-wise conformal prediction*, which jointly achieves amplitude–phase variation decoupling and finite-sample marginal coverage guarantees. Our approach constructs exchangeability-driven prediction pairs via neighborhood smoothing, requiring no parametric assumptions or resampling and enabling efficient parallel computation. Results: Theoretically, it ensures valid pointwise prediction bands. Empirical evaluation on simulated and real-world data demonstrates substantial improvements over baseline methods that ignore phase variation or lack conformal validity.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Computer Vision: Segmentation
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Predict missing segments in partially observed functional data
• Address amplitude and phase variation in incomplete functional observations
• Provide computationally efficient prediction intervals with coverage guarantees

Innovation

• Unified registration and prediction for partial functions
• Split conformal methods ensuring exchangeability in predictions
• Neighborhood smoothing algorithm for guaranteed coverage bands