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Register a new userGeneralized Matrix Decomposition Regression: Estimation and Inference for Two-way Structured Data
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Added by
Yue Wang
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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This paper studies high-dimensional regression with two-way structured data. To estimate the high-dimensional coefficient vector, we propose the generalized matrix decomposition regression (GMDR) to efficiently leverage any auxiliary information on row and column structures. The GMDR extends the principal component regression (PCR) to two-way structured data, but unlike PCR, the GMDR selects the components that are most predictive of the outcome, leading to more accurate prediction. For inference on regression coefficients of individual variables, we propose the generalized matrix decomposition inference (GMDI), a general high-dimensional inferential framework for a large family of estimators that include the proposed GMDR estimator. GMDI provides more flexibility for modeling relevant auxiliary row and column structures. As a result, GMDI does not require the true regression coefficients to be sparse; it also allows dependent and heteroscedastic observations. We study the theoretical properties of GMDI in terms of both the type-I error rate and power and demonstrate the effectiveness of GMDR and GMDI on simulation studies and an application to human microbiome data.
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In this article, we introduce a two-way factor model for a high-dimensional data matrix and study the properties of the maximum likelihood estimation (MLE). The proposed model assumes separable effects of row and column attributes and captures the correlation across rows and columns with low-dimensional hidden factors. The model inherits the dimension-reduction feature of classical factor models but introduces a new framework with separable row and column factors, representing the covariance or correlation structure in the data matrix. We propose a block alternating, maximizing strategy to compute the MLE of factor loadings as well as other model parameters. We discuss model identifiability, obtain consistency and the asymptotic distribution for the MLE as the numbers of rows and columns in the data matrix increase. One interesting phenomenon that we learned from our analysis is that the variance of the estimates in the two-way factor model depends on the distance of variances of row factors and column factors in a way that is not expected in classical factor analysis. We further demonstrate the performance of the proposed method through simulation and real data analysis.
Motivated by recent data analyses in biomedical imaging studies, we consider a class of image-on-scalar regression models for imaging responses and scalar predictors. We propose using flexible multivariate splines over triangulations to handle the irregular domain of the objects of interest on the images, as well as other characteristics of images. The proposed estimators of the coefficient functions are proved to be root-n consistent and asymptotically normal under some regularity conditions. We also provide a consistent and computationally efficient estimator of the covariance function. Asymptotic pointwise confidence intervals and data-driven simultaneous confidence corridors for the coefficient functions are constructed. Our method can simultaneously estimate and make inferences on the coefficient functions while incorporating spatial heterogeneity and spatial correlation. A highly efficient and scalable estimation algorithm is developed. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed method, which is then applied to the spatially normalized positron emission tomography data of the Alzheimers Disease Neuroimaging Initiative.
There are many scenarios such as the electronic health records where the outcome is much more difficult to collect than the covariates. In this paper, we consider the linear regression problem with such a data structure under the high dimensionality. Our goal is to investigate when and how the unlabeled data can be exploited to improve the estimation and inference of the regression parameters in linear models, especially in light of the fact that such linear models may be misspecified in data analysis. In particular, we address the following two important questions. (1) Can we use the labeled data as well as the unlabeled data to construct a semi-supervised estimator such that its convergence rate is faster than the supervised estimators? (2) Can we construct confidence intervals or hypothesis tests that are guaranteed to be more efficient or powerful than the supervised estimators? To address the first question, we establish the minimax lower bound for parameter estimation in the semi-supervised setting. We show that the upper bound from the supervised estimators that only use the labeled data cannot attain this lower bound. We close this gap by proposing a new semi-supervised estimator which attains the lower bound. To address the second question, based on our proposed semi-supervised estimator, we propose two additional estimators for semi-supervised inference, the efficient estimator and the safe estimator. The former is fully efficient if the unknown conditional mean function is estimated consistently, but may not be more efficient than the supervised approach otherwise. The latter usually does not aim to provide fully efficient inference, but is guaranteed to be no worse than the supervised approach, no matter whether the linear model is correctly specified or the conditional mean function is consistently estimated.
We consider problems in which a system receives external emph{perturbations} from time to time. For instance, the system can be a train network in which particular lines are repeatedly disrupted without warning, having an effect on passenger behavior. The goal is to predict changes in the behavior of the system at particular points of interest, such as passenger traffic around stations at the affected rails. We assume that the data available provides records of the system functioning at its natural regime (e.g., the train network without disruptions) and data on cases where perturbations took place. The inference problem is how information concerning perturbations, with particular covariates such as location and time, can be generalized to predict the effect of novel perturbations. We approach this problem from the point of view of a mapping from the counterfactual distribution of the system behavior without disruptions to the distribution of the disrupted system. A variant on emph{distribution regression} is developed for this setup.
The focus of modern biomedical studies has gradually shifted to explanation and estimation of joint effects of high dimensional predictors on disease risks. Quantifying uncertainty in these estimates may provide valuable insight into prevention strategies or treatment decisions for both patients and physicians. High dimensional inference, including confidence intervals and hypothesis testing, has sparked much interest. While much work has been done in the linear regression setting, there is lack of literature on inference for high dimensional generalized linear models. We propose a novel and computationally feasible method, which accommodates a variety of outcome types, including normal, binomial, and Poisson data. We use a splitting and smoothing approach, which splits samples into two parts, performs variable selection using one part and conducts partial regression with the other part. Averaging the estimates over multiple random splits, we obtain the smoothed estimates, which are numerically stable. We show that the estimates are consistent, asymptotically normal, and construct confidence intervals with proper coverage probabilities for all predictors. We examine the finite sample performance of our method by comparing it with the existing methods and applying it to analyze a lung cancer cohort study.
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