- Murphy, Thomas Brendan

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# Murphy, Thomas Brendan

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- PublicationClustering ranked preference data using sociodemographic covariatesRanked preference data arise when a set of judges rank, in order of their preference, a set of objects. Such data arise in preferential voting systems and market research surveys. Covariate data associated with the judges are also often recorded. Such covariate data should be used in conjunction with preference data when drawing inferences about judges. To cluster a population of judges, the population is modelled as a collection of homogeneous groups. The Plackett-Luce model for ranked data is employed to model a judgeâ€™s ranked preferences within a group. A mixture of Plackett-Luce models is employed to model the population of judges, where each component in the mixture represents a group of judges. Mixture of experts models provide a framework in which covariates are included in mixture models. Covariates are included through the mixing proportions and the component density parameters. A mixture of experts model for ranked preference data is developed by combining a mixture of experts model and a mixture of Plackett-Luce models. Particular attention is given to the manner in which covariates enter the model. The mixing proportions and group specific parameters are potentially dependent on covariates. Model selection procedures are employed to choose optimal models. Model parameters are estimated via the â€˜EMM algorithmâ€™, a hybrid of the Expectation-Maximization and the Minorization-Maximization algorithms. Examples are provided through a menu survey and through Irish election data. Results indicate mixture modelling using covariates is insightful when examining a population of judges who express preferences.
603 - PublicationMixed membership models for rank data: Investigating structure in Irish voting dataA mixed membership model is an individual level mixture model where individuals have partial membership of the profiles (or groups) that characterize a population. A mixed membership model for rank data is outlined and illustrated through the analysis of voting in the 2002 Irish general election. This particular election uses a voting system called proportional representation using a single transferable vote (PR-STV) where voters rank some or all of the candidates in order of preference. The data set considered consists of all votes in a constituency from the 2002 Irish general election. Interest lies in highlighting distinct voting profiles within the electorate and studying how voters affiliate themselves to these voting profiles. The mixed membership model for rank data is fitted to the voting data and is shown to give a concise and highly interpretable explanation of voting patterns in this election.
291 - Publication
1204 - PublicationExploring Voting Blocs Within the Irish Electorate: A Mixture Modeling ApproachIrish elections use a voting system called proportion representation by means of a single transferable vote(PR-STV). Under this system, voters express their vote by ranking some (or all) of the candidates in order of preference. Which candidates are elected is determined through a series of counts where candidates are eliminated and surplus votes are distributed.The electorate in any election forms a heterogeneous population: that is voters with different political and ideological persuasions would be expected to have different preferences for the candidates. The purpose of this article is to establish the presence of voting bloes in the Irish electorate, to characterize these blocs and to estimate their size.A mixture modeling approach is used to explore the heterogenecity of the Irish electorate and to establish the existence of clearly defined voting blocs. The voting blocs are characterized by thier voting preferences which are described using a ranking data model. In addition the care with which voters choose lower tier preferences is estimated in the model.The methodology is used to explore data from two Irish election. Data from eight opinion polls taken during the six weeks prior to the 1997 Irish presidential election are analyzed. These data reveal the evolution of the structure of the electorate during the election campaign. In addition data that record the votes from the Dublin West constituency of the 2002 Irish general election are analyzed to reveal distinct voting blocs within the electoate these blocs are characterized by party politics, candidate profile and political ideology.
362ScopusÂ© Citations 64 - PublicationComputational Aspects of Fitting Mixture Models via the Expectation-Maximization AlgorithmThe Expectationâ€“Maximization (EM) algorithm is a popular tool in a wide variety of statistical settings, in particular in the maximum likelihood estimation of parameters when clustering using mixture models. A serious pitfall is that in the case of a multimodal likelihood function the algorithm may become trapped at a local maximum, resulting in an inferior clustering solution. In addition, convergence to an optimal solution can be very slow. Methods are proposed to address these issues: optimizing starting values for the algorithm and targeting maximization steps efficiently. It is demonstrated that these approaches can produce superior outcomes to initialization via random starts or hierarchical clustering and that the rate of convergence to an optimal solution can be greatly improved.
481ScopusÂ© Citations 32 - PublicationClustering with the multivariate normal inverse Gaussian distributionMany model-based clustering methods are based on a finite Gaussian mixture model. The Gaussian mixture model implies that the data scatter within each group is elliptically shaped. Hence non-elliptical groups are often modeled by more than one component, resulting in model over-fitting. An alternative is to use a meanâ€“variance mixture of multivariate normal distributions with an inverse Gaussian mixing distribution (MNIG) in place of the Gaussian distribution, to yield a more flexible family of distributions. Under this model the component distributions may be skewed and have fatter tails than the Gaussian distribution. The MNIG based approach is extended to include a broad range of eigendecomposed covariance structures. Furthermore, MNIG models where the other distributional parameters are constrained is considered. The Bayesian Information Criterion is used to identify the optimal model and number of mixture components. The method is demonstrated on three sample data sets and a novel variation on the univariate Kolmogorovâ€“Smirnov test is used to assess goodness of fit.
17307ScopusÂ© Citations 52 - PublicationA mixture of experts model for rank data with applications in election studiesA voting bloc is defined to be a group of voters who have similar voting preferences. The cleavage of the Irish electorate into voting blocs is of interest. Irish elections employ a 'single transferable vote' electoral system; under this system voters rank some or all of the electoral candidates in order of preference. These rank votes provide a rich source of preference information from which inferences about the composition of the electorate may be drawn. Additionally, the influence of social factors or covariates on the electorate composition is of interest. A mixture of experts model is a mixture model in which the model parameters are functions of covariates. A mixture of experts model for rank data is developed to provide a model-based method to cluster Irish voters into voting blocs, to examine the influence of social factors on this clustering and to examine the characteristic preferences of the voting blocs. The Benter model for rank data is employed as the family of component densities within the mixture of experts model; generalized linear model theory is employed to model the influence of covariates on the mixing proportions. Model fitting is achieved via a hybrid of the EM and MM algorithms. An example of the methodology is illustrated by examining an Irish presidential election. The existence of voting blocs in the electorate is established and it is determined that age and government satisfaction levels are important factors in influencing voting in this election.
284ScopusÂ© Citations 79 - PublicationA Mixture of Experts Latent Position Cluster Model for Social Network DataSocial network data represent the interactions between a group of social actors. Interactions between colleagues and friendship networks are typical examples of such data. The latent space model for social network data locates each actor in a network in a latent (social) space and models the probability of an interaction between two actors as a function of their locations. The latent position cluster model extends the latent space model to deal with network data in which clusters of actors exist â€” actor locations are drawn from a finite mixture model, each component of which represents a cluster of actors. A mixture of experts model builds on the structure of a mixture model by taking account of both observations and associated covariates when modeling a heterogeneous population. Herein, a mixture of experts extension of the latent position cluster model is developed. The mixture of experts framework allows covariates to enter the latent position cluster model in a number of ways, yielding different model interpretations. Estimates of the model parameters are derived in a Bayesian framework using a Markov Chain Monte Carlo algorithm. The algorithm is generally computationally expensive â€” surrogate proposal distributions which shadow the target distributions are derived, reducing the computational burden. The methodology is demonstrated through an illustrative example detailing relationships between a group of lawyers in the USA.
493ScopusÂ© Citations 28 - PublicationInvestigation of parameter uncertainty in clustering using a Gaussian mixture model via jackknife, bootstrap and weighted likelihood bootstrap(Springer Science and Business Media LLC, 2019-05-28)
; ; ; Mixture models with (multivariate) Gaussian components are a popular tool in model-based clustering. Such models are often fitted by a procedure that maximizes the likelihood, such as the EM algorithm. At convergence, the maximum likelihood parameter estimates are typically reported, but in most cases little emphasis is placed on the variability associated with these estimates. In part this may be due to the fact that standard errors are not directly calculated in the model-fitting algorithm, either because they are not required to fit the model, or because they are difficult to compute. The examination of standard errors in model-based clustering is therefore typically neglected. Sampling based methods, such as the jackknife (JK), bootstrap (BS) and parametric bootstrap (PB), are intuitive, generalizable approaches to assessing parameter uncertainty in model-based clustering using a Gaussian mixture model. This paper provides a review and empirical comparison of the jackknife, bootstrap and parametric bootstrap methods for producing standard errors and confidence intervals for mixture parameters. The performance of such sampling methods in the presence of small and/or overlapping clusters requires consideration however; here the weighted likelihood bootstrap (WLBS) approach is demonstrated to be effective in addressing this concern in a model-based clustering framework. The JK, BS, PB and WLBS methods are illustrated and contrasted through simulation studies and through the traditional Old Faithful data set and also the Thyroid data set. The MclustBootstrap function, available in the most recent release of the popular R package mclust, facilitates the implementation of the JK, BS, PB and WLBS approaches to estimating parameter uncertainty in the context of model-based clustering. The JK, WLBS and PB approaches to variance estimation are shown to be robust and provide good coverage across a range of real and simulated data sets when performing model-based clustering; but care is advised when using the BS in such settings. In the case of poor model fit (for example for data with small and/or overlapping clusters), JK and BS are found to suffer from not being able to fit the specified model in many of the sub-samples formed. The PB also suffers when model fit is poor since it is reliant on data sets simulated from the model upon which to base the variance estimation calculations. However the WLBS will generally provide a robust solution, driven by the fact that all observations are represented with some weight in each of the sub-samples formed under this approach.411ScopusÂ© Citations 15 - PublicationAnalysis of Irish third-level college applications dataThe Irish college admissions system involves prospective students listing up to 10 courses in order of preference on their application. Places in third-level educational institutions are subsequently offered to the applicants on the basis of both their preferences and their final second-level examination results. The college applications system is a large area of public debate in Ireland. Detractors suggest that the process creates artificial demand for 'high profile' courses, causing applicants to ignore their vocational callings. Supporters argue that the system is impartial and transparent. The Irish college degree applications data from the year 2000 are analysed by using mixture models based on ranked data models to investigate the types of application behaviour that are exhibited by college applicants. The results of this analysis show that applicants form groups according to both the discipline and the geographical location of their course choices. In addition, there is evidence of the suggested 'points race' for high profile courses. Finally, gender emerges as an influential factor when studying course choice behaviour.
466ScopusÂ© Citations 47