Dynamic Linear Discriminant Analysis in High Dimensional Space

High-dimensional data that evolve dynamically feature predominantly in the modern data era. As a partial response to this, recent years have seen increasing emphasis to address the dimensionality challenge. However, the non-static nature of these datasets is largely ignored. This paper addresses both challenges by proposing a novel yet simple dynamic linear programming discriminant (DLPD) rule for binary classification. Different from the usual static linear discriminant analysis, the new method is able to capture the changing distributions of the underlying populations by modeling their means and covariances as smooth functions of covariates of interest. Under an approximate sparse condition, we show that the conditional misclassification rate of the DLPD rule converges to the Bayes risk in probability uniformly over the range of the variables used for modeling the dynamics, when the dimensionality is allowed to grow exponentially with the sample size. The minimax lower bound of the estimation of the Bayes risk is also established, implying that the misclassification rate of our proposed rule is minimax-rate optimal. The promising performance of the DLPD rule is illustrated via extensive simulation studies and the analysis of a breast cancer dataset.Comment: 34 pages; 3 figure

Senior Recital, Binyan Xu, piano

VCU DEPARTMENT OF MUSIC SENIOR RECITAL Binyan Xu, piano Tuesday, November 24 at 4:00 p.m. Sonia Vlahcevic Concert Hall W. E. Singleton Center for the Performing Art

Postmenopausal hormone and the risk of nephrolithiasis

Menopause is reported to be associated with increased urinary calcium excretion, which may enhance the risk for the development of calcium kidney stones. However, it remains controversial about whether high level of postmenopausal hormone (PMH) is a risk factor for formation of nephrolithiasis. Several observational studies have shown that PMH is protective based on 24-hour urinary parameters. Recent clinical trials provided evidence to conclude that estrogen therapy increases the risk of nephrolithiasis in healthy postmenopausal women. Our study aimed to comprehensively assess clinical evidence on the relationship between postmenopausal hormone level and risk of nephrolithiasis. To conduct systematic review, we pooled total 98 potentially related articles in Cochrane library, Medline, and Embase. Three studies with a total of 71101 study participants that included two clinical trials, 4 stratified and potentially usable results by the status of menopause and type of PMH use derived from one prospective cohort study, and one case-control studies were selected to pool relative risk using random-effect model. How the difference in menopause status, whether naturally menopausal or surgically menopausal, influenced the pooled relative risk was included in the subgroup analysis. The study population aged from 45 to 70 years old. The follow-up year and adjusted confounders differed across different studies. The pooled relative risk for the 7 stratified studies was 0.91 (95 % confidence interval (CI): [0.72, 1.14]). In the menopausal status-specific analysis, the pooled relative risk for naturally menopausal women was 0.92 (95 % CI, [0.64, 1.27]; I2 = 82.74 %) whereas the pooled relative risk for surgically postmenopausal women is 0.90 (95 % CI, [0.63, 1.29]; I2 = 78.47 %). The above results suggested that there was no significant association between PMH and the risk of nephrolithiasis. The difference in menopausal status did not influence the relationship between PMH and the risk of kidney stone formation

HiQR: An efficient algorithm for high dimensional quadratic regression with penalties

This paper investigates the efficient solution of penalized quadratic regressions in high-dimensional settings. We propose a novel and efficient algorithm for ridge-penalized quadratic regression that leverages the matrix structures of the regression with interactions. Building on this formulation, we develop an alternating direction method of multipliers (ADMM) framework for penalized quadratic regression with general penalties, including both single and hybrid penalty functions. Our approach greatly simplifies the calculations to basic matrix-based operations, making it appealing in terms of both memory storage and computational complexity.Comment: 18 page

Autoregressive Networks

We propose a first-order autoregressive model for dynamic network processes in which edges change over time while nodes remain unchanged. The model depicts the dynamic changes explicitly. It also facilitates simple and efficient statistical inference such as the maximum likelihood estimators which are proved to be (uniformly) consistent and asymptotically normal. The model diagnostic checking can be carried out easily using a permutation test. The proposed model can apply to any network processes with various underlying structures but with independent edges. As an illustration, an autoregressive stochastic block model has been investigated in depth, which characterizes the latent communities by the transition probabilities over time. This leads to a more effective spectral clustering algorithm for identifying the latent communities. Inference for a change point is incorporated into the autoregressive stochastic block model to cater for possible structure changes. The developed asymptotic theory as well as the simulation study affirms the performance of the proposed methods. Application with three real data sets illustrates both relevance and usefulness of the proposed models

MARS: A second-order reduction algorithm for high-dimensional sparse precision matrices estimation

Estimation of the precision matrix (or inverse covariance matrix) is of great importance in statistical data analysis. However, as the number of parameters scales quadratically with the dimension p, computation becomes very challenging when p is large. In this paper, we propose an adaptive sieving reduction algorithm to generate a solution path for the estimation of precision matrices under the $\ell_1$ penalized D-trace loss, with each subproblem being solved by a second-order algorithm. In each iteration of our algorithm, we are able to greatly reduce the number of variables in the problem based on the Karush-Kuhn-Tucker (KKT) conditions and the sparse structure of the estimated precision matrix in the previous iteration. As a result, our algorithm is capable of handling datasets with very high dimensions that may go beyond the capacity of the existing methods. Moreover, for the sub-problem in each iteration, other than solving the primal problem directly, we develop a semismooth Newton augmented Lagrangian algorithm with global linear convergence on the dual problem to improve the efficiency. Theoretical properties of our proposed algorithm have been established. In particular, we show that the convergence rate of our algorithm is asymptotically superlinear. The high efficiency and promising performance of our algorithm are illustrated via extensive simulation studies and real data applications, with comparison to several state-of-the-art solvers