Leung, Kin Pang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 70-71). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Model --- p.4 / Chapter 2.1 --- The Basic Thurstonian Model --- p.4 / Chapter 2.2 --- The Thurstonian Model with Mean Structure in 3-object Ranking Data --- p.6 / Chapter 3 --- Implementation Using the Mx --- p.13 / Chapter 4 --- Simulation Study --- p.21 / Chapter 4.1 --- 2 covariate values --- p.23 / Chapter 4.2 --- 4 covariate values --- p.23 / Chapter 4.3 --- 10 covariate values --- p.23 / Chapter 4.4 --- 50 covariate values --- p.24 / Chapter 5 --- Discussion --- p.37 / Chapter A --- Sample Mx script-2 covariate values --- p.39 / Chapter B --- Sample Mx script-50 covariate values --- p.60 / Bibliography --- p.70

Thesis (M. S.)--Aerospace Engineering, Georgia Institute of Technology, 2010. / Committee Chair: John-Paul Clarke; Committee Member: Ellis Johnson; Committee Member: Eric Feron. Part of the SMARTech Electronic Thesis and Dissertation Collection.

<p> Clustering involves partitioning a given data set into several groups based on some similarity/dissimilarity measurements. Cluster analysis has been widely used in information retrieval, text and web mining, pattern recognition, image segmentation and software reverse engineering.</p> <p> K-means is the most intuitive and popular clustering algorithm and the working horse for clustering. However, the classical K-means suffers from several flaws. First, the algorithm is very sensitive to the initialization method and can be easily trapped at a local minimum regarding to the measurement (the sum of squared errors) used in the model. On the other hand, it has been proved that finding a global minimal sum of the squared errors is NP-hard even when k = 2. In the present model for K-means clustering, all the variables are required to be discrete and the objective is nonlinear and nonconvex.</p> <p> In the first part of the thesis, we consider the issue of how to derive an optimization model to the minimum sum of squared errors for a given data set based on continuous convex optimization. For this, we first transfer the K-means clustering into a novel optimization model, 0-1 semidefinite programming where the eigenvalues of involved matrix argument must be 0 or 1. This provides an unified way for many other clustering approaches such as spectral clustering and normalized cut. Moreover, the new optimization model also allows us to attack the original problem based on the relaxed linear and semidefinite programming.</p> <p> Moreover, we consider the issue of how to get a feasible solution of the original clustering from an approximate solution of the relaxed problem. By using principal component analysis, we construct a rounding procedure to extract a feasible clustering and show that our algorithm can provide a 2-approximation to the global solution of the original problem. The complexity of our rounding procedure is O(n^(k2(k-1)/2)), which improves substantially a similar rounding procedure in the literature with a complexity O(n^k3/2). In particular, when k = 2, our rounding procedure runs in O(n log n) time. To the best of our knowledge, this is the lowest complexity that has been reported in the literature to find a solution to K-means clustering with guaranteed quality.</p> <p> In the second part of the thesis, we consider approximation methods for the so-called balanced bi-clustering. By using a simple heuristics, we prove that we can improve slightly the constrained K-means for bi-clustering. For the special case where the size of each cluster is fixed, we develop a new algorithm, called Q means, to find a 2-approximation solution to the balanced bi-clustering. We prove that the Q-means has a complexity O(n^2).</p> <p> Numerical results based our approaches will be reported in the thesis as well.</p> / Thesis / Master of Science (MSc)

Many models exist in the aerospace industry that attempt to replicate the National Airspace System (NAS). The complexity of the NAS makes it a system that can be modeled in a variety of ways. While some NAS models are very detailed and take many factors into account, runtime of these simulations can be on the magnitude of hours (to simulate a single day). Other models forgo details in order to decrease the runtime of their simulation. Most models are capable of simulating a 24 hour period in the NAS. An analysis of an entire year would mean running the simulation for every day in the year, which would result in a long run time. The following thesis work presents a tool that is capable of giving the user a day that can be used in a simulation and will produce results similar to simulating the entire year. Taking in parameters chosen by the user, the tool outputs a single day, multiple days, or a composite day (based on percentages of days). Statistical methods were then used to compare each day to the overall year. On top of finding a single representative day, the ability to find a composite day was added. After implementing a brute force search technique to find the composite day, the long runtime was deemed inconvenient for the user. To solve this problem, a heuristic search method was created that would search the solution space in a short time and still output a composite day that represented the year. With a short runtime, the user would be able to run the program multiple times. Once the heuristic method was implemented, it was found that it performed well enough to make it an option for the user to choose. The final version of this tool was used to find a representative day and the result was used in comparison with output data from a NAS simulation model. Because the tool found the representative day based on historical data, it could be used to validate the effectiveness of the simulation model. The following thesis will go into detail about how this tool, the Representative Day Finder, was created.