Jonathan Macgregor Keith

Most functional magnetic resonance imaging studies rely on estimates of hierarchically organized functional brain networks whose segregation and integration reflect the cognitive and behavioral changes in humans. However, most existing methods for estimating the community structure of networks from both individual and group-level analysis methods do not account for the variability between subjects. In this paper, we develop a new multilayer community detection method based on Bayesian latent block model (LBM). The method can robustly detect the community structure of weighted functional networks with an unknown number of communities at both individual and group levels and retain the variability of the individual networks. For validation, we propose a new community structure-based multivariate Gaussian generative model to simulate synthetic signal. Our simulation study shows that the community memberships estimated by hierarchical Bayesian inference are consistent with the predefined node labels in the generative model. The method is also tested via split-half reproducibility using working memory task fMRI data of 100 unrelated healthy subjects from the Human Connectome Project. Analyses using both synthetic and real data show that our proposed method is more accurate and reliable compared with the commonly used (multilayer) modularity models. The code of this work is available at:https://github.com/LingbinBian/CommuDetectLBM.

The following document is a translation (from French to English) of: Gabriele H. Greco, Sur la mesurabilité d'une fonction numérique par rapport à une famille d'ensembles, Rendiconti del Seminario Matematico della Università di Padova}, tome 65 (1981), pp. 163--176. Translated by: Jonathan M. Keith, School of Mathematics, Monash University, jonathan.keith@monash.edu. With thanks to: Prof. Andrea D'Agnolo, Editor-in-Chief of the above journal, for permission to publish this translation.

The concept of measurability of functions on a charge space is generalised for functions taking values in a uniform space. Several existing forms of measurability generalise naturally in this context, and new forms of measurability are proposed. Conditions under which the various forms of measurability are logically equivalent are identified. Applying these concepts to real-valued functions, some recent characterisations of measurable functions on a bounded charge space are generalised to the unbounded case.

This work is focused on finding G-optimal designs theoretically for kriging models with two-dimensional inputs and separable exponential covariance structures. For design comparison, the notion of evenness of two-dimensional grid designs is developed. The mathematical relationship between the design and the supremum of the mean squared prediction error ($SMSPE$) function is studied and then optimal designs are explored for both prospective and retrospective design scenarios. In the case of prospective designs, the new design is developed before the experiment is conducted and the regularly spaced grid is shown to be the G-optimal design. The retrospective designs are constructed by adding or deleting points from an already existing design. Deterministic algorithms are developed to find the best possible retrospective designs (which minimizes the $SMSPE$). It is found that a more evenly spread design under the G-optimality criterion leads to the best possible retrospective design. For all the cases of finding the optimal prospective designs and the best possible retrospective designs, both frequentist and Bayesian frameworks have been considered. The proposed methodology for finding retrospective designs is illustrated with a methane flux monitoring design.

A charge space (X, 𝒜, µ) is a generalisation of a measure space, consisting of a sample space X, a field of subsets 𝒜 and a finitely additive measure µ, also known as a charge. Properties a real-valued function on X may possess include T ₁-measurability and integrability. However, these properties are less well studied than their measure-theoretic counterparts. This paper describes new characterisations of T ₁-measurability and integrability for a bounded charge space (µ(X) < ∞). These characterisations are convenient for analytic purposes; for example, they facilitate simple proofs that T ₁-measurability is equivalent to conventional measurability and integrability is equivalent to Lebesgue integrability, if (X, 𝒜, µ) is a complete measure space. New characterisations of equality almost everywhere of two real-valued functions on a bounded charge space are provided. Necessary and sufficient conditions for the function space L ₁(X, 𝒜, µ) to be a Banach space are determined. Lastly, the concept of completion of a measure space is generalised for charge spaces, and it is shown that under certain conditions, completion of a charge space adds no new equivalence classes to the quotient space ℒ _p (X, 𝒜, µ).