Fitting regression models with many multivariate responses and covariates can be challenging, but such responses and covariates sometimes have tensor-variate structure. We extend the classical multivariate regression model to exploit such structure in two ways: first, we impose four types of low-rank tensor formats on the regression coefficients. Second, we model the errors using the tensor-variate normal distribution that imposes a Kronecker separable format on the covariance matrix. We obtain maximum likelihood estimators via block-relaxation algorithms and derive their computational complexity and asymptotic distributions. Our regression framework enables us to formulate tensor-variate analysis of variance (TANOVA) methodology. This methodology, when applied in a one-way TANOVA layout, enables us to identify cerebral regions significantly associated with the interaction of suicide attempters or non-attemptor ideators and positive-, negative- or death-connoting words in a functional Magnetic Resonance Imaging study. Another application uses three-way TANOVA on the Labeled Faces in the Wild image dataset to distinguish facial characteristics related to ethnic origin, age group and gender. A R package totr implements the methodology.

Commonly-used clustering algorithms usually find ellipsoidal, spherical or other regular-structured clusters, but are more challenged when the underlying groups lack formal structure or definition. Syncytial clustering is the name that we introduce for methods that merge groups obtained from standard clustering algorithms in order to reveal complex group structure in the data. Here, we develop a distribution-free fully-automated syncytial clustering algorithm that can be used with $k$-means and other algorithms. Our approach estimates the cumulative distribution function of the normed residuals from an appropriately fit $k$-groups model and calculates the estimated nonparametric overlap between each pair of clusters. Groups with high pairwise overlap are merged as long as the estimated generalized overlap decreases. Our methodology is always a top performer in identifying groups with regular and irregular structures in several datasets and can be applied to datasets with scatter or incomplete records. The approach is also used to identify the distinct kinds of gamma ray bursts in the Burst and Transient Source Experiment 4Br catalog and the distinct kinds of activation in a functional Magnetic Resonance Imaging study.

A generalized cross-validation approach to estimate the reconstruction filter bandwidth in 2D filtered backprojection is presented. The method writes the reconstruction equation in equivalent backprojected filtering form, derives results on eigendecomposition of symmetric 2D circulant matrices, and applies them to make bandwidth estimation a computationally efficient operation within the context of standard backprojected filtering reconstruction. Performance evaluations on a range of simulated emission tomography experiments give promising results. The superior performance holds at both low and high total expected counts, pointing to the method's applicability even in weak signal-to-noise-ratio situations. The approach also applies to the more general class of elliptically symmetric filters, with the reconstructed estimate's performance often better than even that obtained with the true optimal radially symmetric filter.

Functional magnetic resonance imaging is a noninvasive tool for studying cerebral function. Many factors challenge activation detection, especially in low-signal scenarios that arise in the performance of high-level cognitive tasks. We provide a fully automated fast adaptive smoothing and thresholding (FAST) algorithm that uses smoothing and extreme value theory on correlated statistical parametric maps for thresholding. Performance on experiments spanning a range of low-signal settings is very encouraging. The methodology also performs well in a study to identify the cerebral regions that perceive only-auditory-reliable or only-visual-reliable speech stimuli.

Synthetic magnetic resonance (MR) imaging is an approach suggested in the literature to predict MR images at different design parameter settings from at least three observed MR scans. However, performance is poor when no regularization is used in the estimation and otherwise computationally impractical to implement for 3-D imaging methods. We propose a method which accounts for spatial context in MR images by the imposition of a Gaussian Markov random field (MRF) structure on a transformation of the spin-lattice relaxation time, the spin-spin relaxation time and the proton density at each voxel. The MRF structure is specified through a matrix normal distribution. We also model the observed magnitude images using the more accurate but computationally challenging Rice distribution. A one-step-late expectation-maximization approach is adopted to make our approach computationally practical. We evaluate predictive performance in generating synthetic MR images in a clinical setting: our results indicate that our suggested approach is not only computationally feasible to implement but also shows excellent performance.

Clustering datasets is a challenging problem needed in a wide array of applications. Partition-optimization approaches, such as k-means or expectation-maximization (EM) algorithms, are sub-optimal and find solutions in the vicinity of their initialization. This paper proposes a staged approach to specifying initial values by finding a large number of local modes and then obtaining representatives from the most separated ones. Results on test experiments are excellent. We also provide a detailed comparative assessment of the suggested algorithm with many commonly-used initialization approaches in the literature. Finally, the methodology is applied to two datasets on diurnal microarray gene expressions and industrial releases of mercury.

Estimating the noise parameter in magnitude magnetic resonance (MR) images is important in a wide range of applications. We propose an automatic noise estimation method that does not rely on a substantial proportion of voxels being from the background. Specifically, we model the magnitude of the observed signal as a mixture of Rice distributions with common noise parameter. The expectation-maximization (EM) algorithm is used to estimate all parameters, including the common noise parameter. The algorithm needs initializing values for which we provide some strategies that work well. The number of components in the mixture model also needs to be estimated en route to noise estimation and we provide a novel approach to doing so. Our methodology performs very well on a range of simulation experiments and physical phantom data. Finally, the methodology is demonstrated on four clinical datasets.