Multiple matrix Gaussian graphs estimation Zhu, Yunzhang; Li, Lexin
Journal of the Royal Statistical Society. Series B, Statistical methodology,
November 2018, Letnik:
80, Številka:
5
Journal Article
Recenzirano
Odprti dostop
Matrix-valued data, where the sampling unit is a matrix consisting of rows and columns of measurements, are emerging in numerous scientific and business applications. Matrix Gaussian graphical models ...are a useful tool to characterize the conditional dependence structure of rows and columns. We employ non-convex penalization to tackle the estimation of multiple graphs from matrix-valued data under a matrix normal distribution. We propose a highly efficient non-convex optimization algorithm that can scale up for graphs with hundreds of nodes. We establish the asymptotic properties of the estimator, which requires less stringent conditions and has a sharper probability error bound than existing results. We demonstrate the efficacy of our proposed method through both simulations and real functional magnetic resonance imaging analyses.
With the recent explosion of scientific data of unprecedented size and complexity, feature ranking and screening are playing an increasingly important role in many scientific studies. In this ...article, we propose a novel feature screening procedure under a unified model framework, which covers a wide variety of commonly used parametric and semiparametric models. The new method does not require imposing a specific model structure on regression functions, and thus is particularly appealing to ultrahigh-dimensional regressions, where there are a huge number of candidate predictors but little information about the actual model forms. We demonstrate that, with the number of predictors growing at an exponential rate of the sample size, the proposed procedure possesses consistency in ranking, which is both useful in its own right and can lead to consistency in selection. The new procedure is computationally efficient and simple, and exhibits a competent empirical performance in our intensive simulations and real data analysis.
Existing sufficient dimension reduction methods suffer from the fact that each dimension reduction component is a linear combination of all the original predictors, so that it is difficult to ...interpret the resulting estimates. We propose a unified estimation strategy, which combines a regression-type formulation of sufficient dimension reduction methods and shrinkage estimation, to produce sparse and accurate solutions. The method can be applied to most existing sufficient dimension reduction methods such as sliced inverse regression, sliced average variance estimation and principal Hessian directions. We demonstrate the effectiveness of the proposed method by both simulations and real data analysis.
We introduce a principal support vector machine (PSVM) approach that can be used for both linear and nonlinear sufficient dimension reduction. The basic idea is to divide the response variables into ...slices and use a modified form of support vector machine to find the optimal hyperplanes that separate them. These optimal hyperplanes are then aligned by the principal components of their normal vectors. It is proved that the aligned normal vectors provide an unbiased, $\sqrt n $ -consistent, and asymptotically normal estimator of the sufficient dimension reduction space. The method is then generalized to nonlinear sufficient dimension reduction using the reproducing kernel Hubert space. In that context, the aligned normal vectors become functions and it is proved that they are unbiased in the sense that they are functions of the true nonlinear sufficient predictors. We compare PSVM with other sufficient dimension reduction methods by simulation and in real data analysis, and through both comparisons firmly establish its practical advantages.
Parsimonious Tensor Response Regression Li, Lexin; Zhang, Xin
Journal of the American Statistical Association,
07/2017, Letnik:
112, Številka:
519
Journal Article
Recenzirano
Odprti dostop
Aiming at abundant scientific and engineering data with not only high dimensionality but also complex structure, we study the regression problem with a multidimensional array (tensor) response and a ...vector predictor. Applications include, among others, comparing tensor images across groups after adjusting for additional covariates, which is of central interest in neuroimaging analysis. We propose parsimonious tensor response regression adopting a generalized sparsity principle. It models all voxels of the tensor response jointly, while accounting for the inherent structural information among the voxels. It effectively reduces the number of free parameters, leading to feasible computation and improved interpretation. We achieve model estimation through a nascent technique called the envelope method, which identifies the immaterial information and focuses the estimation based upon the material information in the tensor response. We demonstrate that the resulting estimator is asymptotically efficient, and it enjoys a competitive finite sample performance. We also illustrate the new method on two real neuroimaging studies. Supplementary materials for this article are available online.
Alzheimer's disease (AD) is a progressive and irreversible neurodegenerative disorder that has recently seen serious increase in the number of affected subjects. In the last decade, neuroimaging has ...been shown to be a useful tool to understand AD and its prodromal stage, amnestic mild cognitive impairment (MCI). The majority of AD/MCI studies have focused on disease diagnosis, by formulating the problem as classification with a binary outcome of AD/MCI or healthy controls. There have recently emerged studies that associate image scans with continuous clinical scores that are expected to contain richer information than a binary outcome. However, very few studies aim at modeling multiple clinical scores simultaneously, even though it is commonly conceived that multivariate outcomes provide correlated and complementary information about the disease pathology. In this article, we propose a sparse multi-response tensor regression method to model multiple outcomes jointly as well as to model multiple voxels of an image jointly. The proposed method is particularly useful to both infer clinical scores and thus disease diagnosis, and to identify brain subregions that are highly relevant to the disease outcomes. We conducted experiments on the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset, and showed that the proposed method enhances the performance and clearly outperforms the competing solutions.
Regularized matrix regression Zhou, Hua; Li, Lexin
Journal of the Royal Statistical Society. Series B, Statistical methodology,
March 2014, Letnik:
76, Številka:
2
Journal Article
Recenzirano
Odprti dostop
Modern technologies are producing a wealth of data with complex structures. For instance, in two‐dimensional digital imaging, flow cytometry and electroencephalography, matrix‐type covariates ...frequently arise when measurements are obtained for each combination of two underlying variables. To address scientific questions arising from those data, new regression methods that take matrices as covariates are needed, and sparsity or other forms of regularization are crucial owing to the ultrahigh dimensionality and complex structure of the matrix data. The popular lasso and related regularization methods hinge on the sparsity of the true signal in terms of the number of its non‐zero coefficients. However, for the matrix data, the true signal is often of, or can be well approximated by, a low rank structure. As such, the sparsity is frequently in the form of low rank of the matrix parameters, which may seriously violate the assumption of the classical lasso. We propose a class of regularized matrix regression methods based on spectral regularization. A highly efficient and scalable estimation algorithm is developed, and a degrees‐of‐freedom formula is derived to facilitate model selection along the regularization path. Superior performance of the method proposed is demonstrated on both synthetic and real examples.
Multimodal data, where different types of data are collected from the same subjects, are fast emerging in a large variety of scientific applications. Factor analysis is commonly used in integrative ...analysis of multimodal data, and is particularly useful to overcome the curse of high dimensionality and high correlations. However, there is little work on statistical inference for factor analysis-based supervised modeling of multimodal data. In this article, we consider an integrative linear regression model that is built upon the latent factors extracted from multimodal data. We address three important questions: how to infer the significance of one data modality given the other modalities in the model; how to infer the significance of a combination of variables from one modality or across different modalities; and how to quantify the contribution, measured by the goodness of fit, of one data modality given the others. When answering each question, we explicitly characterize both the benefit and the extra cost of factor analysis. Those questions, to our knowledge, have not yet been addressed despite wide use of factor analysis in integrative multimodal analysis, and our proposal bridges an important gap. We study the empirical performance of our methods through simulations, and further illustrate with a multimodal neuroimaging analysis.
A central question in high-dimensional mediation analysis is to infer the significance of individual mediators. The main challenge is that the total number of potential paths that go through any ...mediator is super-exponential in the number of mediators. Most existing mediation inference solutions either explicitly impose that the mediators are conditionally independent given the exposure, or ignore any potential directed paths among the mediators. In this article, we propose a novel hypothesis testing procedure to evaluate individual mediation effects, while taking into account potential interactions among the mediators. Our proposal thus fills a crucial gap, and greatly extends the scope of existing mediation tests. Our key idea is to construct the test statistic using the logic of Boolean matrices, which enables us to establish the proper limiting distribution under the null hypothesis. We further employ screening, data splitting, and decorrelated estimation to reduce the bias and increase the power of the test. We show that our test can control both the size and false discovery rate asymptotically, and the power of the test approaches one, while allowing the number of mediators to diverge to infinity with the sample size. We demonstrate the efficacy of the method through simulations and a neuroimaging study of Alzheimer's disease. A Python implementation of the proposed procedure is available at
https://github.com/callmespring/LOGAN
.
We consider the problem of decomposing a higher-order tensor with binary entries. Such data problems arise frequently in applications such as neuroimaging, recommendation system, topic modeling, and ...sensor network localization. We propose a multilinear Bernoulli model, develop a rank-constrained likelihood-based estimation method, and obtain the theoretical accuracy guarantees. In contrast to continuous-valued problems, the binary tensor problem exhibits an interesting phase transition phenomenon according to the signal-to-noise ratio. The error bound for the parameter tensor estimation is established, and we show that the obtained rate is minimax optimal under the considered model. Furthermore, we develop an alternating optimization algorithm with convergence guarantees. The efficacy of our approach is demonstrated through both simulations and analyses of multiple data sets on the tasks of tensor completion and clustering.
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