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ABSTRACT Individualized modeling has become increasingly popular in recent years with its growing application in fields such as personalized medicine and mobile health studies. With rich longitudinal measurements, it is of great interest to model certain subject‐specific time‐varying covariate effects. In this paper, we propose an individualized time‐varying nonparametric model by leveraging the subgroup information from the population. The proposed method approximates the time‐varying covariate effect using nonparametric B‐splines and aggregates the estimated nonparametric coefficients that share common patterns. Moreover, the proposed method can effectively handle various missing data patterns that frequently arise in mobile health data. Specifically, our method achieves subgrouping by flexibly accommodating varying dimensions of B‐spline coefficients due to missingness. This capability sets it apart from other fusion‐type approaches for subgrouping. The subgroup information can also potentially provide meaningful insight into the characteristics of subjects and assist in recommending an effective treatment or intervention. An efficient ADMM algorithm is developed for implementation. Our numerical studies and application to mobile health data on monitoring pregnant women's deep sleep and physical activities demonstrate that the proposed method achieves better performance compared to other existing methods. 

Point process modeling is gaining increasing attention, as point process type data are emerging in a large variety of scientific applications. In this article, motivated by a neuronal spike trains study, we propose a novel point process regression model, where both the response and the predictor can be a high-dimensional point process. We model the predictor effects through the conditional intensities using a set of basis transferring functions in a convolutional fashion. We organize the corresponding transferring coefficients in a three-way tensor, then impose the low-rank, sparsity, and subgroup structures on this coefficient tensor. These structures help reduce the dimensionality, integrate information across different individual processes, and facilitate the interpretation. We develop a highly scalable optimization algorithm for parameter estimation. We derive the large sample error bound for the recovered coefficient tensor, and establish the subgroup identification consistency, while allowing the dimension of the multivariate point process to diverge. We demonstrate the efficacy of our method through both simulations and a cross-area neuronal spike trains analysis in a sensory cortex study. 

This article provides an overview of tensors, their properties, and their applications in statistics. Tensors, also known as multidimensional arrays, are generalizations of matrices to higher orders and are useful data representation architectures. We first review basic tensor concepts and decompositions, and then we elaborate traditional and recent applications of tensors in the fields of recommender systems and imaging analysis. We also illustrate tensors for network data and explore the relations among interacting units in a complex network system. Some canonical tensor computational algorithms and available software libraries are provided for various tensor decompositions. Future research directions, including tensors in deep learning, are also discussed.