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published_or_final_version / Statistics / Master / Master of Philosophy

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Cluster analysis is the study of how to partition data into homogeneous subsets so that the partitioned data share some common characteristic. In one to three dimensions, the human eye can distinguish well between clusters of data if clearly separated. However, when there are more than three dimensions and/or the data is not clearly separated, an algorithm is required which needs a metric of similarity that quantitatively measures the characteristic of interest. Linear Grouping Analysis (LGA, Van Aelst et al. 2006) is an algorithm for clustering data around hyperplanes, and is most appropriate when: 1) the variables are related/correlated, which results in clusters with an approximately linear structure; and 2) it is not natural to assume that one variable is a “response”, and the remainder the “explanatories”. LGA measures the compactness within each cluster via the sum of squared orthogonal distances to hyperplanes formed from the data. In this dissertation, we extend the scope of problems to which LGA can be applied. The first extension relates to the linearity requirement inherent within LGA, and proposes a new method of non-linearly transforming the data into a Feature Space, using the Kernel Trick, such that in this space the data might then form linear clusters. A possible side effect of this transformation is that the dimension of the transformed space is significantly larger than the number of observations in a given cluster, which causes problems with orthogonal regression. Therefore, we also introduce a new method for calculating the distance of an observation to a cluster when its covariance matrix is rank deficient. The second extension concerns the combinatorial problem for optimizing a LGA objective function, and adapts an existing algorithm, called BIRCH, for use in providing fast, approximate solutions, particularly for the case when data does not fit in memory. We also provide solutions based on BIRCH for two other challenging optimization problems in the field of robust statistics, and demonstrate, via simulation study as well as application on actual data sets, that the BIRCH solution compares favourably to the existing state-of-the-art alternatives, and in many cases finds a more optimal solution.

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The globular cluster system (GCS) of the elliptical galaxy NGC 4649 has been examined using the Gemini Multi-Object Spectrograph (GMOS); spectra for 156 candidate globular clusters (GCs) were obtained, extending to a galactocentric radius of 42 kpc. The system was found to have an even 78 GC candidates per population, using a colour of g-i = 0.92 (Faifer et al. 2011) to split the system into sub-populations. The populations refer to their metalicity; a g-i<0.92 is considered metal-poor (MP), and a g-i>0.92 is metal-rich (MR). Line-of-sight-velocity measurements and subsequent modelling, were used to measure the full GCS rotational velocity as 59+/-28 km/s, with a position angle of 218+/-28 degrees. The MR population was found to have rotational velocity of 81+/-42 km/s with a position angle of 221+/-29 degrees, while the MP population measures a rotational velocity of 30+/-36 km/s with a position angle of 202+/-73 degrees. The average velocity dispersion for the full GCS was calculated at 247+/-61 km/s, the MR population 266+/-94 km/s, and the MP population, 221+/-76 km/s. These findings are consistent (within uncertainties) with previous studies by Hwang et al.(2008), and Bridges et al. (2006). The velocity dispersion profile for all populations is constant with increasing radius, suggesting the presence of a dark matter (DM) halo. A tracer mass estimator was used to measure the mass at 42 kpc as (2.01+/-0.05)X10^{12} solar masses, for an isothermal potential, and (1.21+/-0.05)X10^{12} solar masses if the tracers followed the DM profile. Finally, it was estimated that M/L_{B}=22 - 44, consistent with the presence of considerable amounts of DM for a luminous galaxy. / Thesis (Master, Physics, Engineering Physics and Astronomy) -- Queen's University, 2011-10-03 20:45:36.205

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