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Brendan Williams

Publications and source records attributed to Brendan Williams.

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Berry-Esseen Bounds for Statistics of Non-Stationary, $\phi$-Mixing Random Variables

Using a modification of Stein's method, we generalize the results of Bentkus, G{\"o}tze, and Tikhomirov \cite{bentkus1997berry} to obtain Berry-Esseen bounds for a broad class of statistics of sequences of $\phi$-mixing, non-stationary random variables with polynomial mixing rates. %and linear variance. We then consider applications of this theorem to ensure Berry-Esseen rates for various classes of non-stationary $\phi$-mixing random variables, including rates for a general class of processes of $\phi$-mixing random variables satisfying an aggregate third moment bound.

math.PR

Thalamic nuclei segmentation from T$_1$-weighted MRI: unifying and benchmarking state-of-the-art methods with young and old cohorts

The thalamus and its constituent nuclei are critical for a broad range of cognitive and sensorimotor processes, and implicated in many neurological and neurodegenerative conditions. However, the functional involvement and specificity of thalamic nuclei in human neuroimaging is underappreciated and not well studied due, in part, to technical challenges of accurately identifying and segmenting nuclei. This challenge is further exacerbated by a lack of common nomenclature for comparing segmentation methods. Here, we use data from healthy young (Human Connectome Project, 100 subjects) and older healthy adults, plus those with minor cognitive impairment and Alzheimer$'$s disease (Alzheimer$'$s Disease Neuroimaging Initiative, 540 subjects), to benchmark four state of the art thalamic segmentation methods for T1 MRI (FreeSurfer, HIPS-THOMAS, SCS-CNN, and T1-THOMAS) under a single segmentation framework. Segmentations were compared using overlap and dissimilarity metrics to the Morel stereotaxic atlas. We also quantified each method$'$s estimation of thalamic nuclear degeneration across Alzheimer$'$s disease progression, and how accurately early and late mild cognitive impairment, and Alzheimers disease could be distinguished from healthy controls. We show that HIPS-THOMAS produced the most effective segmentations of individual thalamic nuclei and was also most accurate in discriminating healthy controls from those with mild cognitive impairment and Alzheimer$'$s disease using individual nucleus volumes. This work is the first to systematically compare the efficacy of anatomical thalamic segmentation approaches under a unified nomenclature. We also provide recommendations of which segmentation method to use for studying the functional relevance of specific thalamic nuclei, based on their overlap and dissimilarity with the Morel atlas.

eess.IV

Consequences of the packing problem

We study several consequences of the packing problem, a conjecture from combinatorial optimization, using algebraic invariants of square-free monomial ideals. While the packing problem is currently unresolved, we successfully settle the validity of its consequences. Our work prompts additional questions and conjectures, which are presented together with their motivation.

math.AC