Journal of Multivariate Analysis

We introduce a new approach to nonlinear sufficient dimension reduction in cases where both the predictor and the response are distributional data, modeled as members of a metric space. Our key step is to build universal kernels (cc-universal) on the metric spaces, which results in reproducing kernel Hilbert spaces for the predictor and response that are rich enough to characterize the conditional independence that determines sufficient dimension reduction. For univariate distributions, we construct the universal kernel using the Wasserstein distance, while for multivariate distributions, we resort to the sliced Wasserstein distance...

Network estimation has been a critical component of high-dimensional data analysis and can provide an understanding of the underlying complex dependence structures. Among the existing studies, Gaussian graphical models have been highly popular. However, they still have limitations due to the homogeneous distribution assumption and the fact that they are only applicable to small-scale data. For example, cancers have various levels of unknown heterogeneity, and biological networks, which include thousands of molecular components, often differ across subgroups while also sharing some commonalities...

We study the limiting behavior of singular values of a lag-<mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"><mml:mi>τ</mml:mi></mml:math> sample auto-correlation matrix <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>τ</mml:mi></mml:mrow><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow></mml:msubsup></mml:math> of large dimensional vector white noise process, the error term <mml:math xmlns:mml="https://www...

When sample sizes are small, it becomes challenging for an asymptotic test requiring diverging sample sizes to maintain an accurate Type I error rate. In this paper, we consider one-sample, two-sample and ANOVA tests for mean vectors when data are high-dimensional but sample sizes are very small. We establish asymptotic <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:math>-distributions of the proposed <mml:math xmlns:mml="https://www...

We present the simplicial sampler , a class of parallel MCMC methods that generate and choose from multiple proposals at each iteration. The algorithm's multiproposal randomly rotates a simplex connected to the current Markov chain state in a way that inherently preserves symmetry between proposals. As a result, the simplicial sampler leads to a simplified acceptance step: it simply chooses from among the simplex nodes with probability proportional to their target density values. We also investigate a multivariate Gaussian-based symmetric multiproposal mechanism and prove that it also enjoys the same simplified acceptance step...

Given a functional central limit (fCLT) for an estimator and a parameter transformation, we construct random processes, called functional delta residuals, which asymptotically have the same covariance structure as the limit process of the functional delta method. An explicit construction of these residuals for transformations of moment-based estimators and a multiplier bootstrap fCLT for the resulting functional delta residuals are proven. The latter is used to consistently estimate the quantiles of the maximum of the limit process of the functional delta method in order to construct asymptotically valid simultaneous confidence bands for the transformed functional parameters...

In this paper, we study statistical inference in functional quantile regression for scalar response and a functional covariate. Specifically, we consider a functional linear quantile regression model where the effect of the covariate on the quantile of the response is modeled through the inner product between the functional covariate and an unknown smooth regression parameter function that varies with the level of quantile. The objective is to test that the regression parameter is constant across several quantile levels of interest...

In biomedical data analysis, clustering is commonly conducted. Biclustering analysis conducts clustering in both the sample and covariate dimensions and can more comprehensively describe data heterogeneity. In most of the existing biclustering analyses, scalar measurements are considered. In this study, motivated by time-course gene expression data and other examples, we take the "natural next step" and consider the biclustering analysis of functionals under which, for each covariate of each sample, a function (to be exact, its values at discrete measurement points) is present...

It is quite common for functional data arising from imaging data to assume values in infinite-dimensional manifolds. Uncovering associations between two or more such nonlinear functional data extracted from the same object across medical imaging modalities can assist development of personalized treatment strategies. We propose a method for canonical correlation analysis between paired probability densities or shapes of closed planar curves, routinely used in biomedical studies, which combines a convenient linearization and dimension reduction of the data using tangent space coordinates...

Change point analysis aims to detect structural changes in a data sequence. It has always been an active research area since it was introduced in the 1950s. In modern statistical applications, however, high-throughput data with increasing dimensions are ubiquitous in fields ranging from economics, finance to genetics and engineering. For those problems, the earlier works are typically no longer applicable. As a result, the problem of testing a change point for high dimensional data sequences has been an important yet challenging task...

We propose computationally efficient methods for estimating stationary multivariate spatial and spatial-temporal spectra from incomplete gridded data. The methods are iterative and rely on successive imputation of data and updating of model estimates. Imputations are done according to a periodic model on an expanded domain. The periodicity of the imputations is a key feature that reduces edge effects in the periodogram and is facilitated by efficient circulant embedding techniques. In addition, we describe efficient methods for decomposing the estimated cross spectral density function into a linear model of coregionalization plus a residual process...

Most existing methods of variable selection in partially linear models (PLM) with ultrahigh dimensional covariates are based on partial residuals, which involve a two-step estimation procedure. While the estimation error produced in the first step may have an impact on the second step, multicollinearity among predictors adds additional challenges in the model selection procedure. In this paper, we propose a new Bayesian variable selection approach for PLM. This new proposal addresses those two issues simultaneously as (1) it is a one-step method which selects variables in PLM, even when the dimension of covariates increases at an exponential rate with the sample size, and (2) the method retains model selection consistency, and outperforms existing ones in the setting of highly correlated predictors...

Canonical correlation analysis (CCA) is a common method used to estimate the associations between two different sets of variables by maximizing the Pearson correlation between linear combinations of the two sets of variables. We propose a version of CCA for transelliptical distributions with an elliptical copula using pairwise Kendall's tau to estimate a latent scatter matrix. Because Kendall's tau relies only on the ranks of the data this method does not make any assumptions about the marginal distributions of the variables, and is valid when moments do not exist...

Independent Component Analysis (ICA) offers an effective data-driven approach for blind source extraction encountered in many signal and image processing problems. Although many ICA methods have been developed, they have received relatively little attention in the statistics literature, especially in terms of rigorous theoretical investigation for statistical inference. The current paper aims at narrowing this gap and investigates the statistical sampling properties of the colorICA (cICA) method. The cICA incorporates the correlation structure within sources through parametric time series models in the frequency domain and outperforms several existing ICA alternatives numerically...

The aim of this paper is to develop a new framework of surface functional models (SFM) for surface functional data which contains repeated observations in two domains (typically, time-location). The primary problem of interest is to investigate the relationship between a response and the two domains, where the numbers of observations in both domains within a subject may be diverging. The SFMs are far beyond the multivariate functional models with two-dimensional predictor variables. Unprecedented complexity presented in the surface functional models, such as possibly distinctive sampling designs and the dependence between the two domains, makes our models more complex than the existing ones...

We propose a distributed method for simultaneous inference for datasets with sample size much larger than the number of covariates, i.e., N ≫ p , in the generalized linear models framework. When such datasets are too big to be analyzed entirely by a single centralized computer, or when datasets are already stored in distributed database systems, the strategy of divide-and-combine has been the method of choice for scalability. Due to partition, the sub-dataset sample sizes may be uneven and some possibly close to p , which calls for regularization techniques to improve numerical stability...

Dynamic networks are a general language for describing time-evolving complex systems, and discrete time network models provide an emerging statistical technique for various applications. It is a fundamental research question to detect a set of nodes sharing similar connectivity patterns in time-evolving networks. Our work is primarily motivated by detecting groups based on interesting features of the time-evolving networks (e.g., stability). In this work, we propose a model-based clustering framework for time-evolving networks based on discrete time exponential-family random graph models, which simultaneously allows both modeling and detecting group structure...

We propose a new class of generalized linear mixed models with Gaussian mixture random effects for clustered data. To overcome the weak identifiability issues, we fit the model using a penalized Expectation Maximization (EM) algorithm, and develop sequential locally restricted likelihood ratio tests to determine the number of components in the Gaussian mixture. Our work is motivated by an application to nationwide kidney transplant center evaluation in the United States, where the patient-level post-surgery outcomes are repeated measures of the care quality of the transplant centers...

The largest eigenvalue of a single or a double Wishart matrix, both known as Roy's largest root, plays an important role in a variety of applications. Recently, via a small noise perturbation approach with fixed dimension and degrees of freedom, Johnstone and Nadler derived simple yet accurate approximations to its distribution in the real valued case, under a rank-one alternative. In this paper, we extend their results to the complex valued case for five common single matrix and double matrix settings. In addition, we study the finite sample distribution of the leading eigenvector...

With the development of high-throughput technologies, principal component analysis (PCA) in the high-dimensional regime is of great interest. Most of the existing theoretical and methodological results for high-dimensional PCA are based on the spiked population model in which all the population eigenvalues are equal except for a few large ones. Due to the presence of local correlation among features, however, this assumption may not be satisfied in many real-world datasets. To address this issue, we investigate the asymptotic behavior of PCA under the generalized spiked population model...