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Functional Principal Component Analysis (FPCA) has become a widely used dimension reduction tool for functional data analysis. When additional covariates are available, existing FPCA models integrate them either in the mean function or in both the mean function and the covariance function. However, methods of the first kind are not suitable for data that display second-order variation, while those of the second kind are time-consuming and make it difficult to perform subsequent statistical analyses on the dimension-reduced representations. To tackle these issues, we introduce an eigen-adjusted FPCA model that integrates covariates in the covariance function only through its eigenvalues. In particular, different structures on the covariate-specific eigenvalues—corresponding to different practical problems—are discussed to illustrate the model’s flexibility as well as utility. To handle functional observations under … 

Unlike standard prediction tasks, survival analysis requires modeling right censored data, which must be treated with care. While deep neural networks excel in traditional supervised learning, it remains unclear how to best utilize these models in survival analysis. A key question asks which data-generating assumptions of traditional survival models should be retained and which should be made more flexible via the function-approximating capabilities of neural networks. Rather than estimating the survival function targeted by most existing methods, we introduce a Deep Extended Hazard (DeepEH) model to provide a flexible and general framework for deep survival analysis. The extended hazard model includes the conventional Cox proportional hazards and accelerated failure time models as special cases, so DeepEH subsumes the popular Deep Cox proportional hazard (DeepSurv) and Deep Accelerated Failure Time (DeepAFT) models. We additionally provide theoretical support for the proposed DeepEH model by establishing consistency and convergence rate of the survival function estimator, which underscore the attractive feature that deep learning is able to detect low-dimensional structure of data in high-dimensional space. Numerical experiments also provide evidence that the proposed methods outperform existing statistical and deep learning approaches to survival analysis.