Cross-fitted stacking for intrinsic partial functional linear regression

The cumulative explained variance criterion and cross-validation are widely used to determine the truncation number in Riemannian functional regression based on a Karhunen–Loève expansion. However, the former may omit low-variance components that are relevant to the response. Cross-validation incorporates response information, but selecting a truncation number large enough to retain higher-order response-relevant components necessarily includes all intervening components, even when some carry no predictive information. Moreover, the truncation number selected by cross-validation may be sensitive to sampling variability and random fold assignments. Motivated by model averaging, this paper proposes a regularized cross-fitted stacking procedure for intrinsic partial functional linear regression (iPLR) with a Riemannian functional predictor and Euclidean covariates. Specifically, iPLR models are fitted using randomly sampled subsets of Riemannian functional principal component (RFPC) indices from a common candidate pool, and their out-of-fold predictions are used as meta-features for a meta-learner, thereby combining predictive information across different RFPC subsets. Simulation studies and a real-data application show that the proposed method generally attains lower prediction errors and more stable performance than methods based on a single truncation number, especially when both lower- and higher-order RFPCs are relevant to the response while some intervening components are not.