Global optimization for structured low rank approximation

In this paper, we investigate the complexity of the numerical construction of the Hankel structured low-rank approximation (HSLRA) problem, and develop a family of algorithms to solve this problem. Briefly, HSLRA is the problem of finding the closest (in some pre-defined norm) rank r approximation of a given Hankel matrix, which is also of Hankel structure. Unlike many other methods described in the literature the family of algorithms we propose has the property of guaranteed convergence

An initial analysis and reflection of the metrics used in the Teaching Excellence Framework in the UK

In this paper, we look at the results from the recent Teaching Excellence Framework (2017), which were made publicly available in June 2017. We offer some initial analysis and commentary, look at the primary reasons for providers being awarded Bronze, Silver and Gold, and look at some providers close to the borderline for their award. We demonstrate that the provider submissions, a narrative document prepared to accompany the submission would have had a significant effect upon the award bestowed

Method of moments estimation in linear regression with errors in both variables

Recently, in this journal, there has been revised attention on estimating the parameters of the errors in variables, linear structural model. For example, O’Driscoll and Ramirez (2011) used a geometric approach to give insight into the performance of various slope estimators for the linear structural model as introduced by the present author. This article aims to provide a unified method of moments approach for estimating the parameters in the linear structural model, concentrating attention on estimators using the higher moments, which to date has received only little attention in the literature

Errors in variables regression: What is the appropriate model?

The fitting of a straight line to bivariate data (x,y) is a common procedure. Standard linear regression theory deals with the situation when there is only error in one variable, either x, or y. A procedure known as y on x regression fits a line where the error is assumed to be associated with the y variable, alternatively, x on y regression fits a line when the error is associated with the x variable. The model to describe the scenario when there are errors in both variables is known as an errors in variables model. Errors in variables modelling is fundamentally different from standard regression techniques. The problems of model fitting and parameter estimation of a straight line errors in variables model cannot be solved by generalising a simple linear regression model. Briefly, this thesis provides a unified framework to the fitting of a straight line errors in variables model using the method of moments. Estimators of the line using a higher moments approach have been detailed, and asymptotic variance covariance matrices of a plethora of slope estimators are provided. Simulations demonstrate that these variance covariance matrices are accurate for even small data sets. The topic of prediction is considered, with an estimator for the latent variable presented, as well as advice on the mean value of y given x via both a parametric and non-parametric approach. The problem of residuals in an errors in variables model is described, and some quick solutions given. Some examples are presented towards the end of this thesis to demonstrate how the ideas provided may be applied to real-life data sets, as well as some areas which may demand further research

Difference-based methods for truncating the singular value decomposition

Given a noisy time series (or signal), one may wish to remove the noise from the observed series. Assuming that the noise-free series lies in some low dimensional subspace of rank r, a common approach is to embed the noisy time series into a Hankel trajectory matrix. The singular value decomposition is then used to deconstruct the Hankel matrix into a sum of rank-one components. We wish to demonstrate that there may be some potential in using difference-based methods of the observed series in order to provide guidance regarding the separation of the noise from the signal, and to estimate the rank of the low dimensional subspace in which the true signal is assumed to lie

Using Bradley-Terry models to analyse test match cricket

In this paper we investigate the use of Bradley–Terry models to analyse test match cricket. Specifically, we develop a new and alternative team ranking and compare our rankings with those produced by the International Cricket Council, we forecast the outcomes of a selected number of test cricket matches and show that our predictions perform well compared to bookmaker predictions. We offer ratings of individual players and use these ratings to predict the results of some recent matches. The general purpose of the paper is to illustrate the potential of Bradley–Terry models, which are effectively models of P(i is preferred to j), and thus can be applied in a number of settings where there are paired comparisons. Popular applications include analysing taste test experiments and modelling sports competitions. More creative examples of applications include statistical modelling of citation exchange among statistics journals, predicting the fighting ability of lizards and estimating driver crash risks

Using singular spectrum analysis to obtain staffing level requirements in emergency units

Many operational research (OR) techniques use historical data to populate model input parameters. Although the majority of these models take into account stochastic variation of the inputs, they do not necessarily take into account seasonal variations and other stochastic effects that might arise. One of the major applications of OR lies within healthcare, where ever increasing pressure on healthcare systems is having major implications on those who plan the provision of such services. Coping with growing demand for healthcare, as well as the volatile nature of the number of arrivals at a healthcare facility makes modelling healthcare provision one of the most challenging fields of OR. This paper proposes the use of a relatively modern time series technique, Singular Spectrum Analysis (SSA), to improve existing algorithms that give required staffing levels. The methodology is demonstrated using data from a large teaching hospital's emergency unit. Using time dependent queueing theory, as well as SSA, staffing levels are obtained. The performance of our technique is analysed using a weighted mean square error measure, introduced in this paper

A wide-ranging computational comparison of high-performance graph colouring algorithms

This paper reviews the current state of the literature surrounding methods for the general graph colouring problem and presents a broad comparison of six high-performance algorithms, each belonging to one of the main algorithmic schemes identified. Unlike many previous computational studies in graph colouring, a large range of both artificially generated and real-world graphs are considered, culminating in over 40,000 individual trials that have consumed more than a decade of computation time in total. The picture painted by the comparison is complex, with each method outperforming all others on at least one occasion; however, general patterns are also observed, particularly with regards to the advantages of hybridising local-search techniques with global-based operators

Rostering staff at a mathematics support service using a finite-source queueing model

We study the problem of staffing university mathematics support services (MSSs) in which students drop in to the service (without appointment) for tutoring support. Our approach seeks to find the minimum sufficient number of tutors (with appropriate specialities) to present by hour and day to cover student demand with tolerable delays. We employ traditional operational research techniques to aid managers and administrators of MSSs to roster their services. The machine interference type queue is adopted to model the number of student queries within a mathematics support session. We define and solve an appropriate integer program to roster the number of tutors needed to run the service efficiently