Alberto Alvarez Iglesias, John Newell, John Hinde and Liam Glynn.
Recursive partitioning methods are a popular non-parametric alternative to the classical parametric and non-parametric models in regression, classification and survival problems. They have been recognized as a useful modelling tool as they produce a model that is very easy to interpret. In general, the aim of any statistical model is to explain the complicated interstructure between a large number of explanatory variables and the response, in the simplest way possible, and to use the model to predict the outcome of interest when new observations are considered. Single trees are an excellent way to describe the structure of the learning data but their predictive power can be disappointing. In the last decade, many efforts have been made to overcome this problem. These methods are generally known as "ensemble methods" and they use a set of trees, created by bootstrapping the original data, in order to improve predictability. The price to be paid, however, is the absence of a singular tree.
In this work, a data set of 1586 patients with cardiovascular disease will be analyzed. The primary endpoint was a cardiovascular composite endpoint, which included death from a cardiovascular cause or any of the cardiovascular events of myocardial infarction (MI), heart failure, peripheral vascular disease and stroke. Seventeen factors/covariates will be considered for development of a prognostic model and the results of different methods for growing survival trees will be compared.
Marcus Bishop
We show how a presentation of the descent algebra of the Symmetric Group can be derived from a map from an algebra of binary trees into the free Lie algebra. We then introduce a quiver whose path algebra injects into the algebra of binary trees. Composing these two maps provides a presentation of the descent algebra in terms of a quiver subject to relations which we calculate explicitly.
Jorge Bruno
Given a set X, the lattice LT(X) of all topologies on X (under set inclusion) has been studied extensively and is well understood. In this scenario, it is still interesting to investigate topologies on LT(X) for which a particular family of functions is continuous. In particular, topological lattices are those for which \wedge_x(y) = x \wedge y and \wedge: LT(X) \times LT(X) \rightarrow LT(X) are continuous.
Another less natural, but just as interesting, order on LT(X) is given by the embeddability ordering. Where A \leq B iff A \hookrightarrow B. This order arises naturally within \mathcal{P}(X) to find representations of partial orders within such set. Moreover, many independent results have been encountered within this topic. At this stage, regarding embeddability, the general hypothesis is to investigate the possibility of evolution of topologies (for a fixed set X). By this is meant a way to topologise LT(X), so rendering the discussion of continuous maps f : [0, 1] \rightarrow LT(X) meaningful and useful. One could then track the evolution of topologies from f(0) to f(1), say. Achieving such an aim would enable a host of different research questions to be raised, such as whether there is a continuous function from f(0) to f(1), or whether, given topological properties P and Q, we can determine a continuous function f such that f(0) is P while f(1) is Q.
Hoang Dinh Van
I am working on Vertex Operator Algebras. Here is a short description of my reading including: Formal Distributions, Locality, Vertex Algebras, and Vertex Operator Algebras.Supported by Science Foundation Ireland.
Kevin Doherty
Anaerobic digestion is a process in which biodegradable material is broken down into simpler products by micro-organisms in the absence of oxygen. The process can take place naturally in areas where oxygen is not readily available as the main electron carrier for chemical reactions, such as in landfills, or underwater. We formulated and analysed some models for the digestion process.
We considered one of the simplest feasible models for anaerobic digestion. This model tracks the evolution of two types of substrate (influent and volatile fatty acids) and two populations of microbial species (acidogens and methanogens). We considered the solution to this model subject to conditions appropriate to continuous operation in a reactor. The analysis consists largely of investigating the stability of the equilibrium points. We also performed an asymptotic analysis of the model subject to conditions appropriate to batch operation.
Finally, we considered a much larger model - the Anaerobic Digestion Model No.1 (ADM1) [1]. This model tracks the evolution of twenty four different substrates and seven different microbial populations, and takes account of biochemical processes, physico-chemical processes and various types of inhibition. The model was successfully implemented in both Maple and Matlab.
Cara Dooley, John Hinde, John Newell
The aim of the study was to compare survival of colorectal cancer patients in the whole population against the survival of patients in a sub-population who also had inflammatory bowel disease (IBD). All individuals who suffered from colorectal cancer were drawn from the entire Irish population using data from January 1994 to December 2005 provided by the National Cancer Registry of Ireland (NCRI).
The control group contained many more observations (n>20000) when compared to the IBD group (n=170). Given the number of control patients, there was large diversity in this group. In a conventional designed experiment or trial, patients entering the trial would be taken to be as similar as possible. Usually patients would be similar in age, health etc. As this was an observational study, there was no design prior to collecting the data.
To compensate for this lack of design, each IBD patient is matched to the "closest" control patient. For each pair of IBD and control patients a distance is calculated and those two patients which have the smallest distance between them (and are so are the most similar) are matched. The distance used in this case is a Malanhobis distance based on ranks. The matching is carried out using the Optmatch Package in R.
Orla Doolin
HER2-positive is a breast cancer tumour that tests for the human epidermal growth factor receptor 2, a protein that promotes higher aggressiveness and faster growth of cancer cells. HER2 is overexpessed in 25-30% of breast tumors and has been shown to be associated with poor prognosis and increased disease recurrence. Accurate evaluation of HER2 receptor status in invasive tumors is critical in its prognostic role as well as its ability to predict response to antibody treatments for the patient. This study is part of an increasing effort to create an alternative molecular basis for tumour classification, moving away from the classical morphologic systems based on macro and microscopic histology. There now exist microarray technologies that can simultaneously assess the level of expression of thousands of genes and demonstrate the potential power of expression profiling for classifying tumours. For this cancer cohort, with data from the National Breast Cancer Research Institute Galway, the goal is to quantify the dependence of HER2 status positive or negative across a set of gene markers, ER ALPHA, GRB7, Her2, LASP1, RARA, RPL19, STARD3, TUBG1 and TOP2A. Receptor status on estrogen, progesterone and HER2 divide the cohort of 95 women into four intrinsic molecular subtypes, Luminal A and Basal corresponding to HER2-negative and Luminal B and HER2 Overexpressing corresponding to HER2-positive. Described in particular is the use of recursive partitioning techniques and tree-based methodology in HER2 diagnosis based on microarray gene expression data. Improvements in classification accuracy are obtained by use of the recently published R package party which implements Breiman and Cutler's random forests, using unbiased conditional inference trees from ctree as base learners.
Supervised by Dr.John Newell
Supported by National Breast Cancer Research Institute, Galway.
Paul Geeleher, Cathal Seoige and Aaron Golden
MicroRNAs are small non-coding RNA molecules that regulate the expression level of genes by binding to messenger RNA targets, which prevents these messenger RNA molecules from being translated into mature proteins. We have performed an analysis to investigate the genetic compont involved in the regulatory effect of microRNAs. The Regulatory Effect (RE) score [1] is used to measure the inhibitory effect of a microRNA in a sample by measuring the expression levels of the microRNA's messenger RNA targets, independent of the expression level of the actual microRNA; it is calculated from the the average difference in expression level of the targets versus non-targets, where the targets have been determined previously by target-prediction algorithms such as PITA [2]. The Hapmap project is an international collaboration which aims to catalogue the main sources of variation in the human genome with and between distinct geographic populations based on single nucleotide polymorphisms (SNP). The project members have used SNP microarrays to genotype groups of trios of individuals (two parents and one child) from several worldwide geographic regions. Other affiliated groups have used the cell lines produced by the project to perform gene expression studies that measure messenger RNA levels. Using this data and because these cell line samples are related by genetic lineage we have been able to ascertain that there is a heritable component to the regulatory effect of microRNAs. This seems to suggest that the entire miRNA processing machinery operates at a lower level of activity overall in a sub-population defined by specific genotype SNPs. We have performed a genetical genomics assay but have thus far been unable to identify expression quantitative trait loci (eQTL) that show a statistically significant association with this altered RE-score. We are currently assessing the likely causative effects of such diminished microRNA regulatory efficacy
Fintan Hegarty and Graham Ellis.
Homology is the study of connectivity and ``holes '' in spaces. This thesis is aimed at the homological study of subsets of Rn arising from experimental data. We develop a mathematical machinery for efficiently computing the homology of cellular spaces with a view to applications in image analysis. We discuss and implement in GAP a cubical approach to homology computations, and then introduce and implement a new approach based on permutahedral complexes. We compare the efficiency of these new computations using permutahedral complexes with those using the more traditional cubical and simplicial complexes. Our guiding example is an analysis of computed tomography images used for radiation therapy.Neil Humphries and Petri T.~Piiroinen
Inherent discontinuities in a piecewise-smooth (PWS) dynamical system can cause considerable difficulties in forming a qualitative picture of how the behavior of such a system changes under parameter variation, with an extra range of behaviors not shown by smooth systems, such as period-adding and robust chaos. One type of PWS system is a periodically-forced impact oscillator (PFIO), which has two fundamental components; a smooth dynamical system that models the system in free-flight between impacts and a reset rule that models the impact. If the impact surface is sufficiently far from the natural centre of oscillation, then the only orbit of such a system will be a non-impacting one. Eventually, as the impact surface is brought ever closer to the natural centre, this orbit is destroyed at a grazing bifurcation and impacting orbits will appear. However, under parameter variation there is a wide range of different behaviours that could be manifested around this bifurcation.
This study is exploring a geometric methodology - 'Discontinuity-Geometry' - to assist in both analysing and understanding PFIOs. After a brief introduction to this framework, with definitions of the geometric objects used, we will use the discontinuity-geometry framework to both analyse and explain the differences in behaviour manifested by such a system in two damping scenarios.
Barry Hurley
This thesis is concerned with classification problems for finite linear groups. Our main concern is with groups of prime degree, and groups defined over finite fields; although we obtain results for groups of other degrees and over other fields, some of which are of independent interest.
It is broken down into three parts:
D. Hurley and M. Tuite
We are interested in n-point Correlation functions for these types of Vertex Operator Algebras on Riemann surfaces of genus zero and one. These functions are not manifestly symmetric, however, it can be shown that they are symmetric for z1, ... zn and described using graphical representations with weights attached to the edges in the graph.
For a function over a sphere (genus zero), the n-point function can be defined as an inner product <1,Y(ω,z1)..... Y(ω,zn)1>, a rational function in z1,.....,zn. Using a recursive reduction formula the n-point function can be expressed as a sum of all directed graphs with n vertices and each vertex has exactly 2 edges. It is also possible to describe the n-point function using the set of derangements of the label set Φ={1,....,n}.
For a function over a torus (genus one), the n-point function Gn(z1,.....,zn) can be calculated using Zhu's Reduction Formula. In this case there are terms involving the Eisenstein series E2, modular functions P2(zi-zj) and q-derivative terms. Again we will express this as the sum of all directed graphs with n vertices and now each vertex has 0, 1 or 2 edges. This can be described as the set of all permutations of the label set Φ={1,.....,n}.
Supported by SFI
Computing in Symmetric Groups
Liam Naughton
This research is concerned with the development of new algorithms for the computation of the table of marks of the symmetric group. The table of marks of a finite group G is a matrix who's rows and columns are labelled by the conjugacy classes of subgroups of G and where for any 2 subgroups H1 and H2 of G the (H1, H2)-entry in the table of marks is the number of fixed points of H2 on the right cosets of H1 in G. The table of marks characterises the set of all permutation representations of G.
Anh Thai Nhan and Niall Madden
This work is part of a larger project concerning the modeling of the reaction of ICU patients to certain drug therapies~[1]. Colleagues in Applied Mathematics are working on analysis of existing mathematical models~[2], and the design of new one. Researchers in Information Technology are then incorporating these models into a Dynamic Bayesian Network~[3]. Our role is to design numerical methods that can be used to solve the equations in the model numerically and can integrated efficiently into the DBN. The models present several challenges. Here we focus on problems caused by discontinuities in the data.
Nguyen Trung Thong, Andrew Flaus, and Cathal Seoighe
Nucleosomes are the fundamental organising units of eukaryotic genomes. They control the packaging of DNA as chromatin and play key roles in modulating transcription, replication and repair. We hypothesize that binding sites for different transcription factors have distinct positional preferences relative to the nucleosome which can influence their efficiency. Here, we use public ChIP-seq data of nucleosome occupancy and transcription factor binding sites (TFBSs) and develop a robust methodology for predicting functional TFBSs and inferring the relationship between nucleosome occupancy and TFBSs.
Liam O'Callaghan and Petri T. Piiroinen
The mechanisms by which blood sugar is regulated (by a complex process involving many hormones, including insulin and glucagon) are quite well understood. The dynamics of the system, however, are not. Essentially, a practitioner administering insulin would have an idea of the chain of events that, following the drug's administration, result in lowering a patient's blood sugar level. However, the magnitude of such an effect can at present not be predicted with a significant level of accuracy.
We present an analysis of a ``minimal model'' which was recently developed for glycemic control in the critically ill. A minimal model is essentially the smallest possible model that expresses the features that are required. The analysis consists of looking for blood glucose values to which the model will move towards, or move away from, and also the nature of this motion. Whether or not the predictions made by this model are realistic is obviously of critical importance.
The prospects for future advancement and the development of a full-scale description of the glucose regulatory system will also be presented. In particular, the binding process between insulin and its corresponding receptors needs to be effectively modelled, and the subsequent signalling which results in the lowering of blood glucose explained, if the effect of insulin is to be properly quantified.
Padraig Ó Catháain
In this poster, we show that no Hadamard matrix constructed using the twin prime power method is cocyclic, with the exception of the matrix of order 16. We achieve this by showing that the action of the automorphism group of a cocyclic twin prime power matrix induces a 2-transitive action on the rows of the matrix. We then use Ito's classification of Hadamard matrices with 2-transitive automorphism groups to show that exactly one twin prime power Hadamard matrix is cocyclic. This work answers a research problem posed by K.J. Horadam, and exhibits the first known infinite family of Hadamard matrices which are not cocyclic.
Jessica O'Shaughnessy and Prof. Ted Hurley
An original group ring construction for convolutional codes was proposed in 2009 . This group ring construction uses an isomorphism between group rings and group ring matrices to construct generator and control matrices from units in the group ring. This construction is extended to a new group ring construction. This new construction has several interesting free distance properties. Additionally, it can be used to construct LDPC convolutional codes, systematic convolutional codes, and some other optimal convolutional codes.
Kirsten Pfeiffer
Interested particularly in the students' transition from school to university I explore first year students' behaviour and knowledge when validating and evaluating mathematical proofs. A significant aim of my developmental study is to develop and test a method to describe and analyse students' proof validation and evaluation skills and habits. I will here outline the ideas which led to a new perspective and terminology of description and analysis of transcripts of recently held interviews.
Tobias Rossmann
We describe an algorithm for irreducibility testing of finite nilpotent linear groups over various fields of characteristic zero, including number fields and rational function fields over number fields. For a reducible group, our algorithm constructs a proper submodule. An implementation in Magma is publicly available. Details on the algorithm and its implementation can be found in [1].
We extended our work on irreducibility testing and also obtained an algorithm for primitivity testing of finite nilpotent linear groups over cyclotomic fields. For an imprimitive group, a system of imprimitivity is constructed.
This work is supported by the Research Frontiers Programme of Science Foundation Ireland.
Andrew Simpkin
It is often the case that when analysing data the derivative, or rate of change, of the underlying function describing the observed data is of primary interest. A popular tool for derivative estimation is spline smoothing, with a large number of variants being available. We present a method for derivative estimation which extends the P-spline fitting procedure to include an extra additive penalty term for increased robustness in smoothing. The method is applied to biomedical data and compared with several alternative techniques. A small simulation study is presented to gain further insight about the relative performance of the extended P-spline procedure.VO Thi Ngoc Tuoi and Martin Meere
The effect of binding sites to drug diffusion in the artery tissue is considered. A reversible chemical reaction that explains the prolonged presence of drug in the vascular tissue was used to model the presence of the binding site action. In addition, the effect of polymer swelling for the control of drug release is investigated. One of the useful thermoresponsive polymers is Poly(N-isopropylacrylamide) (PNIPAm). This polymer is hydrophilic below the lower critical solution temperature (LCST) 320C, but hydrophobic above 320C, a property that can be exploited to act as an on/off switch to control drug delivery by varying temperature. Initially, there is a good agreement between numerical solutions of the model and experimental datas.
Nur Fatihah Mat Yusoff
This study investigates dimension-reduction techniques in psychometric testing by using Principal Component Analysis (PCA) and Correspondence Analysis (CA). Psychometric research is one of the fields of social science study that is interested in the theory and techniques of education and psychological measurement. Researchers in this area are frequently concerned with the construction and validation of measurement instruments. Theoretically, PCA is a mathematical algorithm that transforms a number of possibly correlated variables into a smaller number of uncorrelated variables by performing a covariance analysis between variables. The PCA concept is closely related to Factor Analysis (FA) which aims to detect structure in the relationships between variables. It is a common technique that has been used by social science researchers in conducting validity and reliability analysis of their study. The CA can be considered as a factor method for the categorical variables and is often linked with producing a low-dimensional graphical display of variables and units. Simple CA is a technique designed to analyse a two-way table, while Multiple Correspondence Analysis (MCA) is an extension of simple CA in that it is applicable to a large set of variables. The result will provide information which is similar in nature to those produced by principal component analysis, and allows us to explore the structure of the categorical variables included in the table.
This study is concerned with reducing the dimension, or number of variables, in an instrument by using the data from a pilot study on personality traits. The original instrument was developed by Oliver P. John and Sanjay Srivastava from University of California, Berkeley in 1999. The pilot survey was conducted at the University Malaysia Sarawak, Malaysia where 80 students from second year and above were randomly selected as respondents. In the original instrument, there are 44 items to assess five personality traits, or the big five dimensions. We believe that some of the items, or even dimensions are not relevant in the Malaysian context. At the end of this study, our aim is to produce the best instrument that can represent all of the variables that we are interested in for subsequent use in structural equation modelling of student achievement.