As data acquisition technologies advance, longitudinal analysis is facing challenges of exploring complex feature patterns from high-dimensional data and modeling potential temporally lagged effects of features on a response. We propose a tensor-based model to analyze multidimensional data. It simultaneously discovers patterns in features and reveals whether features observed at past time points have impact on current outcomes. The model coefficient, a k-mode tensor, is decomposed into a summation of k tensors of the same dimension. We introduce a so-called latent F-1 norm that can be applied to the coefficient tensor to performed structured selection of features. Specifically, features will be selected along each mode of the tensor. The proposed model takes into account within-subject correlations by employing a tensor-based quadratic inference function. An asymptotic analysis shows that our model can identify true support when the sample size approaches to infinity. To solve the corresponding optimization problem, we develop a linearized block coordinate descent algorithm and prove its convergence for a fixed sample size. Computational results on synthetic datasets and real-life fMRI and EEG datasets demonstrate the superior performance of the proposed approach over existing techniques.
Multi-classification is commonly encountered in data science practice, and it has broad applications in many areas such as biology, medicine, and engineering. Variable selection in multiclass problems is much more challenging than in binary classification or regression problems. In addition to estimating multiple discriminant functions for separating different classes, we need to decide which variables are important for each individual discriminant function as well as for the whole set of functions. In this paper, we address the multi-classification variable selection problem by proposing a new form of penalty, supSCAD, which first groups all the coefficients of the same variable associated with all the discriminant functions altogether and then imposes the SCAD penalty on the supnorm of each group. We apply the new penalty to both soft and hard classification and develop two new procedures: the supSCAD multinomial logistic regression and the supSCAD multi-category support vector machine. Our theoretical results show that, with a proper choice of the tuning parameter, the supSCAD multinomial logistic regression can identify the underlying sparse model consistently and enjoys oracle properties even when the dimension of predictors goes to infinity. Based on the local linear and quadratic approximation to the non-concave SCAD and nonlinear multinomial log-likelihood function, we show that the new procedures can be implemented efficiently by solving a series of linear or quadratic programming problems. Performance of the new methods is illustrated by simulation studies and real data analysis of the Small Round Blue Cell Tumors and the Semeion Handwritten Digit data sets.