| dc.description.abstract | Functional data—data observed continuously over time, space, or other domains—arise in numerous modern applications such as biomedical signal analysis, handwriting and speech recognition, and remote sensing. With advances in sensing technologies, multivariate functional data, consisting of multiple functional predictors possibly defined on heterogeneous domains (e.g., time series and images), have become increasingly prevalent. However, existing classification methods have largely focused on univariate functional predictors and offer limited capability for high-dimensional, heterogeneous, and multimodal functional inputs. Moreover, reliable uncertainty quantification for functional classification remains underdeveloped, despite its importance in high-stakes settings such as medicine and finance.
This thesis addresses these gaps by developing new methodologies for multivariate functional data classification and distribution-free uncertainty quantification. First, we introduce the Multivariate Functional Deep Neural Network (MFDNN), a flexible deep learning framework capable of learning from multiple functional covariates defined on domains of potentially different dimensions. The MFDNN unifies functional representation learning and modern neural architectures, allowing scalable and accurate classification of complex multimodal functional data.
Second, we advance uncertainty quantification for functional classification through the conformal prediction (CP) framework. We establish general theoretical results characterizing the efficiency of conformal prediction sets for a broad class of nonconformity functions, offering guidance for designing effective uncertainty measures. Building on these insights, we develop score-based conformal methods for functional data that integrate dimension-reduction techniques with CP to achieve valid, efficient, and computationally tractable uncertainty sets.
Finally, we propose Functional Mahalanobis Conformal Prediction (FMCP), a novel distance-based conformal approach that operates directly in the functional domain without requiring projection into finite-dimensional feature spaces. By leveraging the functional Mahalanobis semi-distance, FMCP preserves intrinsic temporal–spatial structure and provides interpretable, finite-sample valid prediction sets tailored to functional geometry. Theoretical results confirm both the validity and efficiency of the method, and empirical studies on synthetic and real spectrometric data demonstrate strong predictive and uncertainty-quantification performance.
Collectively, this work contributes new theory, methodology, and practical tools for reliable and interpretable learning from complex functional data, bridging key gaps between functional data analysis, deep learning, and rigorous uncertainty quantification. | en_US |
