The Supremum Norm of the Discrepancy Function: Recent Results and Connections

A great challenge in the analysis of the discrepancy function D_N is to obtain universal lower bounds on the L-infty norm of D_N in dimensions d \geq 3. It follows from the average case bound of Klaus Roth that the L-infty norm of D_N is at least (log N) ^{(d-1)/2}. It is conjectured that the L-infty bound is significantly larger, but the only definitive result is that of Wolfgang Schmidt in dimension d=2. Partial improvements of the Roth exponent (d-1)/2 in higher dimensions have been established by the authors and Armen Vagharshakyan. We survey these results, the underlying methods, and some of their connections to other subjects in probability, approximation theory, and analysis.Comment: 15 pages, 3 Figures. Reports on talks presented by the authors at the 10th international conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Sydney Australia, February 2011. v2: Comments of the referee are incorporate

Boundary non-crossings of Brownian pillow

Let B_0(s,t) be a Brownian pillow with continuous sample paths, and let h,u:[0,1]^2\to R be two measurable functions. In this paper we derive upper and lower bounds for the boundary non-crossing probability \psi(u;h):=P{B_0(s,t)+h(s,t) \le u(s,t), \forall s,t\in [0,1]}. Further we investigate the asymptotic behaviour of $\psi(u;\gamma h)$ with $\gamma$ tending to infinity, and solve a related minimisation problem.Comment: 14 page

A geometric approach to visualization of variability in functional data

We propose a new method for the construction and visualization of boxplot-type displays for functional data. We use a recent functional data analysis framework, based on a representation of functions called square-root slope functions, to decompose observed variation in functional data into three main components: amplitude, phase, and vertical translation. We then construct separate displays for each component, using the geometry and metric of each representation space, based on a novel definition of the median, the two quartiles, and extreme observations. The outlyingness of functional data is a very complex concept. Thus, we propose to identify outliers based on any of the three main components after decomposition. We provide a variety of visualization tools for the proposed boxplot-type displays including surface plots. We evaluate the proposed method using extensive simulations and then focus our attention on three real data applications including exploratory data analysis of sea surface temperature functions, electrocardiogram functions and growth curves

Differential network analysis of oral microbiome metatranscriptomes identifies community scale metabolic restructuring in dental caries

Advance access publication date: 18 October 2022Dental caries is a microbial disease and the most common chronic health condition, affecting nearly 3.5 billion people worldwide. In this study, we used a multiomics approach to characterize the supragingival plaque microbiome of 91 Australian children, generating 658 bacterial and 189 viral metagenome-assembled genomes with transcriptional profiling and gene-expression network analysis. We developed a reproducible pipeline for clustering sample-specific genomes to integrate metagenomics and metatranscriptomics analyses regardless of biosample overlap. We introduce novel feature engineering and compositionally-aware ensemble network frameworks while demonstrating their utility for investigating regime shifts associated with caries dysbiosis. These methods can be applied when differential abundance modeling does not capture statistical enrichments or the results from such analysis are not adequate for providing deeper insight into disease. We identified which organisms and metabolic pathways were central in a coexpression network as well as how these networks were rewired between caries and caries-free phenotypes. Our findings provide evidence of a core bacterial microbiome that was transcriptionally active in the supragingival plaque of all participants regardless of phenotype, but also show highly diagnostic changes in the ways that organisms interact. Specifically, many organisms exhibit high connectedness with central carbon metabolism to Cardiobacterium and this shift serves a bridge between phenotypes. Our evidence supports the hypothesis that caries is a multifactorial ecological disease.Josh L. Espinoza, Manolito Torralba, Pamela Leong, Richard Saffery, Michelle Bockmann, Claire Kuelbs, Suren Singh, Toby Hughes, Jeffrey M. Craig, Karen E. Nelson and Chris L. Dupon

The invariance principle for Banach space valued random variables

We extend the invariance principle to triangular arrays of Banach space valued random variables, and as an application derive the invariance principle for lattices of random variables. We also point out how the q-dimensional time parameter Yeh-Wiener process is naturally related to a one dimensional time Wiener process with an infinite dimensional Banach space as a state space.Abstract Wiener spaces measurable norm Gaussian measures Brownian motion on a Banach space invariance principle