Research Profile:Statistical Inference, reliability, Industrial Statistics, Quality Control
An area of industrial research that has motivated many practical and theoretical statistical procedures is the study of the reliability of products and processes. One of my interests is the development of statistical methods to model and analyze reliability data. Another aspect focuses on the modeling and analysis of discrete data. Specifically, I have studied models and methods for the estimation of population size with application to animal abundance, the use of adaptive rules in clinical trials, and the study of correlated Bernoulli trials.
Currently, I am working on quality control methods, specifically of the multivariate type. Here I study methods to monitor processes where the number of quality indicators varies from small to quite large. In the latter case, dimension-reduction techniques such as principal components are first used, followed by control charts on leading components. The methods have been applied to the monitoring of product profiles.
Research Profile:Harmonic analysis, partial differential equations and function theory
Harmonic analysis, partial differential equations and function theory My research interests in harmonic analysis center on weighted norm inequalities for fractional and singular integrals, with applications to problems in partial differential equations such as regularity of solutions to the oblique derivative problem, sharp estimates for the eigenvalues of Schrodinger operators, regularity for the Monge-Ampere equation and distortion of Hausdorff measure under quasiconformal maps. My interests in function theory include interpolation and corona problems on the unit ball in several complex variables.
Research Profile:Stochastic processes, interacting particle systems
My research interests are in nonlinear stochastic model, large deviation, hydrodynamic limit and stochastic models in genetics. Some concrete problems are the long-time behaviour, the phase structure and metastability of the corresponding system. The method used comes from probability theory, stochastic processes and stochastic analysis.
Research Profile: Statistics, biological and medical applications, mixture distributions, nonlinear estimation, computing New applications demand new statistical methods. Statisticians must ensure that new methodology is available to those who need it in the form of reliable, well-documented computer software, especially when the methods involve intensive computation.
While my interests in statistical applications are very general, much of my research has been in collaboration with fisheries biologists: enumerating migrating populations of juvenile Pacific salmon and inferring the age composition of populations from length-frequency data. For the latter problem I use computer graphics and non-linear estimation methods to resolve age groups from overlapping peaks in the length-frequency distribution. This is called mixture distribution analysis and my software developed at McMaster is now used world-wide for diverse applications. With the assistance of graduate students, I have implemented this methodology as the mixdist package in the open source R environment. Future work will develop the interactive graphical interface and bootstrap calculation of standard errors.
I served on Scientific Advisory Panels for the United States Environmental Protection Agency from 2000 to 2014, evaluating models to estimate human uptake of pesticide residues.
I am now an associate member of the GERAS Centre for Aging Research and collaborate with the GERAS team in study design and analysis and in medical education.
Research Profile: Computational Statistics
Dr. McNicholas’ research focuses on computational statistics, and he is at the cutting edge of international research on mixture model-based clustering and classification. Current research includes work on big data featuring outlying or spurious points, with a focus on classification, clustering, dimension reduction and discriminant analysis. Another important aspect of Dr. McNicholas? current research is work on non-Gaussian mixture models, which present a useful alternative to the Gaussian mixture model. Work on clustering categorical data and data of mixed type is ongoing. Applications of Dr. McNicholas? research are readily found in several fields, including bioinformatics, sensometrics, and psychometrics.
Research Profile:Geometry and Topology
More specifically, I compute topological invariants, such as equivariant cohomology theories, of spaces with such structure. Symplectic geometry is the mathematical framework of classical physics; hyperkahler manifolds are symplectic manifolds wiht extra structure, are of particular recent interest due to their connections to theoretical physics. I am mainly concerned with the theory of symmetries of manifolds with these structures, as encoded by a Hamiltonian Lie group action, i.e. there exists a moment map on M encoding the action by Hamiltonian flows. Such group actions on symplectic and hyperkahler manifolds arise naturally in the context of physics, representation theory, and algebraic geometry. To a Hamiltonian space, one associates a symplectic (hyperkahler) quotient, which inherits a symplectic (hyperkahler) structure from the original manifold. The main theme of my recent research is the study of the topology and equivariant topology of these quotients, in particular the computation of their cohomology and complex K-theory rings.