My research interests generally fall under the following four headings:

1. Matrix low-rank approximation and its interface in statistical methodologies such as time series analysis and signal processing
2. Measurement error models and method comparison studies
3. Development of novel statistical techniques for applied and interdisciplinary research
4. Mathematics education

Matrix low-rank approximation and its interface in statistical methodologies such as time series analysis and signal processing
Finding a lower rank approximation of a given matrix is associated with a number of well established statistical techniques such as regression, principal components analysis and image processing. Further difficulty is established if there is some structure in the original matrix which has to be maintained in the approximation. This is particularly the case in time series and signal processing applications where a one-dimensional series is often mapped to a multidimensional structured matrix to permit analysis via singular value decomposition based algorithms (for example). There is no closed form solution to the so-called structured low rank approximation problem, but methods of finding an approximate can be split into two broad types:

• The lift and project algorithm (or method of alternative projections) iterates between solutions which match the low-rank constraint, and solutions which match the structure constraint. If both of these spaces have a non-empty intersection, then algorithms of this type will eventually converge to some solution. This solution however might not be optimal (in the sense that it might not be the closest approximation to the original matrix), which meets the required constraints. Such algorithms usually work fast for the initial iterations, but then slow-down as a solution gets nearer.
• By formulating the problem in a particular way, the structured total least squares problem can be reduced to minimising some optimization function using a numerical algorithm.

My work to date has primarily concentrated on the former, paying particular attention to Cadzow iterations and singular spectrum analysis.

Measurement error models and method comparison studies
When the independent variables in a regression model are contaminated with measurement error, standard least squares estimators of the regression parameters are biased. Different methodologies to fit such models are resultantly used. Model checking, diagnostics and residual analyses are made more difficult due to the effect of the measurement error on the data.

Such models are often applied to method comparison studies, where two methods of measuring the same quantity are compared. Measurement error models aid in the interpretation of whether the two methods are in agreement, and quantifies the bias in a method related to a gold-standard.

My thesis was mainly on the topic of the linear structural model, which is a specific example of a measurement error model that is commonly used in practise. This work used the method of moments to develop a unified methodology for the estimation and prediction of the model. New asymptotic variance-covariances for the parameters in the model were developed, and the effect of adding measurement error to the independent variable was quantified for some examples.

Development of novel statistical techniques for applied and interdisciplinary research
I am involved in a number of collaborator projects involving the application of statistical techniques in a number of disciplines. Work with Monica Busse (SOHCS) on the management of longitudinal data collected in Huntington’s disease studies, and the reliability and minimal detectable change of measures of participation, functional activities and impairments in individuals with Huntington’s disease is on-going. Such work is closely aligned to the European Huntington’s Disease Network.

Work with Mike Smith (SOHCS) has investigated various normalization approaches for dynamic, surface electromyography. The topic of scapular rotation angles in swimmers is presently under joint study.

I work closely with Kerry Hood and colleagues at SEWTU on a number of projects. Most recently joint work has been undertaken on two studies – MODE ARTS (compares modes of data collection for subjective outcomes), and GRACE (aims to explore the determinants of antibiotic prescribing and resistance, including patient and clinicians beliefs). I am also a statistician on a number of trial steering committees.

Mathematics education
My interests in mathematics education are the scholarly research into the effective provision of mathematics support and measuring the effectiveness of such a service; the early engagement of students and maintaining this engagement throughout the student life-cycle; innovative teaching methods; novel methods of providing feedback and the pedagogy of diagnostic testing.

I have experience of teaching in a wide range of environments and partake in many University based and national programmes concerned with learning and teaching matters. Some examples of my work in this area include:

• Fellow of British Society for Research into Learning Mathematics, and reviewer for education journals “Journal of Learning Development in Higher Education” and “Practise and Evidence of the Scholarship of Teaching and Learning in Higher Education”.
• Involved in the University-wide “FYESTI – First Year Skills Training Initiative” alongside contacts in SOCSI, JOMEC, OPTOM, ENGIN and BIOSI.
• Member of the regional hub of sigma (the centre for excellence in learning and teaching for mathematics, statistics and operational research). Have co-chaired a meeting for this group to disseminate good practices concerning mathematics support services.
• Awarded funding to allow an undergraduate student undertake research over the Summer.

 

Conference Presentations

Smith M and Gillard J W et al.,  “Does scapula position change in the dominant arm of competitive swimmers after a swimming training session?”. WISHES conference, Aberystwyth 2009.

Smith M and Gillard J W et al., “Do competitive swimmers have altered scapular upward rotation angles compared to non-swimmers?”. WISHES conference, Aberystwyth 2009.

Gillard J W, Busse-Morris M, and Williams J, “Management of longitudinal data in Huntington’s disease”. World Congress Meeting on Huntington’s disease, Vancouver, 2009.

Gillard J W, “Structured Total Least Squares and its sensitivity to noise level”. Workshop on Singular Spectrum Analysis, Cardiff, 2008.

Gillard J W, Busse-Morris M, and Williams J, “Management of longitudinal data in Huntington’s disease”. Meeting of the European Huntington Disease Network, Cardiff, 2008.

Wilson R H and Gillard J W, “Some problems associated with running a Maths Support Service”. MSOR-CETL Conference, Lancaster, 2008.

Gillard J W, “A Flavour of Errors in Variables Regression”. Young Statisticians Meeting, UWE, Bristol, 2007. Winner of best paper.

Gillard J W - A Flavour of Errors in Variables Regression, Cardiff Young Statisticians Forum 2006 [106 Kb]

Iles T C and Gillard J W - Should we teach the Method of Moments, 42nd University of Wales Gregynog Colloquium 2006.

Gillard J W and Iles T C - Production and Residuals in a Linear Errors-in-Variables Model, 42nd University of Wales Gregynog Colloquium 2006.

Gillard J W and Iles T C - Errors-in-Variables Regression: a problem solved, South Wales Royal Statistical Society Group 2005.

Gillard J W and Iles T C - Errors-in-Variables Regression - The Final Frontier, 41st University of Wales Gregynog Colloquium 2005.