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There is a need for the development of models that are able to account for discreteness in data, along with its time series properties and correlation. A review of the application of thinning operators to adapt the ARMA recursion to the integer-valued case is first discussed. A class of integer-valued ARMA (INARMA) models arises from this application. Our focus falls on INteger-valued AutoRegressive (INAR) type models. The INAR type models can be used in conjunction with existing model-based clustering techniques to cluster discrete valued time series data. This approach is then illustrated with the addition of autocorrelations. With the use of a finite mixture model, several existing techniques such as the selection of the number of clusters, estimation using expectation-maximization and model selection are applicable. The proposed model is then demonstrated on real data to illustrate its clustering applications. / Thesis / Master of Science (MSc)

Model-based clustering is a probabilistic approach that views each cluster as a component in an appropriate mixture model. The Gaussian mixture model is one of the most widely used model-based methods. However, this model tends to perform poorly when clustering high-dimensional data due to the over-parametrized solutions that arise in high-dimensional spaces. This work instead considers the approach of combining dimension reduction techniques with clustering via a mixture of generalized hyperbolic distributions. The dimension reduction techniques, principal component analysis and factor analysis along with their extensions were reviewed. Then the aforementioned dimension reduction techniques were individually paired with the mixture of generalized hyperbolic distributions in order to demonstrate the clustering performance achieved under each method using both simulated and real data sets. For a majority of the data sets, the clustering method utilizing principal component analysis exhibited better classi cation results compared to the clustering method based on the extending the factor analysis model. / Thesis / Master of Science (MSc)

Clustering and classification is the process of finding underlying group structure in heterogenous data. With the rise of the “big data” phenomenon, more complex data structures have made it so traditional clustering methods are oftentimes not advisable or feasible. This thesis presents methodology for analyzing three different examples of these more complex data types. The first is three-way (matrix variate) data, or data that come in the form of matrices. A large emphasis is placed on clustering skewed three-way data, and high dimensional three-way data. The second is click- stream data, which considers a user’s internet search patterns. Finally, co-clustering methodology is discussed for very high-dimensional two-way (multivariate) data. Parameter estimation for all these methods is based on the expectation maximization (EM) algorithm. Both simulated and real data are used for illustration. / Thesis / Doctor of Philosophy (PhD)

Unsupervised classification is a problem often plagued by outliers, yet there is a paucity of work on handling outliers in unsupervised classification. Mixtures of Gaussian distributions are a popular choice in model-based clustering. A single outlier can affect parameters estimation and, as such, must be accounted for. This issue is further complicated by the presence of multiple outliers. Predicting the proportion of outliers correctly is paramount as it minimizes misclassification error. It is proved that, for a finite Gaussian mixture model, the log-likelihoods of the subset models are distributed according to a mixture of beta-type distributions. This relationship is leveraged in two ways. First, an algorithm is proposed that predicts the proportion of outliers by measuring the adherence of a set of subset log-likelihoods to a beta-type mixture reference distribution. This algorithm removes the least likely points, which are deemed outliers, until model assumptions are met. Second, a hypothesis test is developed, which, at a chosen significance level, can test whether a dataset contains a single outlier. / Thesis / Master of Science (MSc)

MASTER OF SCIENCE (2024) (School of Computational Science and Engineering) McMaster University Hamilton, Ontario, Canada TITLE: Vortex Analysis – Clustering and Temporal Tracking of Vortices AUTHOR: Yucheng Feng M.Eng. (Electrical Engineering) Xi’an Jiaotong University, Xi'an, Shaanxi, China B.Eng. (Electrical Engineering) Shandong University, Jinan, Shandong, China SUPERVISOR: Dr. Li Xi NUMBER OF PAGES: xix, 75 / The vortex is a fundamental concept in fluid dynamics, and analyzing it is crucial for explaining and predicting the behavior of fluids in practical applications. In this thesis, two techniques that can lead to a deeper understanding of vortices will be proposed and verified by applying them to Newtonian turbulence and polymer-added flow. The first technique is vortex clustering. By doing dimension reduction and clustering simultaneously, the performance of vortex clustering is notably improved since the hidden features that are immersed in the original input features but can efficiently distinguish different types of vortices can now be extracted objectively. Then, the reliability of the clustering technique is verified in various Newtonian flows. The second technique is vortex tracking based on vortex axis lines, which can efficiently provide complete evolving routines of each vortex over time. With this tracking method, temporal information of vortices, such as their detailed evolving routines and temporal drift positions, can be fully observed and recorded for a future study. The mechanisms and details of this tracking method will first be illustrated and verified using Newtonian flow. Finally, since these two techniques for vortex analysis are solely developed for Newtonian turbulence, a polymer-added flow, where a small amount of polymer can notably modify the behaviour of vortices in Newtonian turbulence, is introduced to check to which level these two techniques are still reliable. Moreover, these two techniques can be compatibly embedded into existing vortex analyzing tools. By doing this, the interested types of vortices can be found and isolated from others, and their specific features and routines can thus be thoroughly studied. / Thesis / Master of Science (MSc) / In turbulence research, efficient clustering and tracking of vortices are appealing. Hence, the fundamental motivation of this research is to investigate vortex clustering techniques and vortex tracking techniques to analyze vortices in turbulent flows automatically and objectively. With the proposed vortex clustering technique, the hidden features immersed in input data space that can efficiently distinguish different types of vortices can be extracted objectively to classify vortices into various groups. With the proposed vortex tracking technique, the temporal behaviours of vortices, such as their detailed developing routines, can be fully tracked, and recorded in a simple but efficient way. With these two techniques, our understanding of the differences between various types of vortices, the ways vortices evolve under different conditions, etc., can be further improved. Besides, embedding these two techniques in existing vortex analyzing tools makes them more powerful.

Wireless sensor network (WSN) technology has gained in importance due to its potential support for a wide range of applications. Most of the WSN applications consist of a large number of distributed nodes that work together to achieve common objectives. Running a large number of nodes requires an efficient mechanism to bring them all together in order to form a multi-hop wireless network that can accomplish specific tasks. Even with the recent developments made in WSN technology, a number of important challenges still create vulnerabilities for WSNs, including: energy waste sources; synchronisation leaks; low network capacity; and self-configuration difficulties. However, energy efficiency perhaps remains both the most challenging and highest priority problem due to the scarce energy resources available in sensor nodes. Synchronization by means of scheduling clusters allows the nodes to cooperate and transmit traffic in a scheduled manner under the duty cycle mechanism. This paper aims to make further advances in this area of work by achieving higher accuracy and precision in time synchronisation through controlling the network topology, self-configuration and estimation of the clock errors between the nodes and finally correcting the nodes’ clock to the estimated value. Furthermore, the target in designing energy efficient protocol relies on synchronized duty cycle mechanism and requires a precise synchronisation algorithm that can schedule a group of nodes to cooperate by communicating together in a scheduled manner. These techniques are considered as parameters in the proposed OLS-MAC algorithm. This algorithm has been designed with the objective of ensuring the schedules of the clusters overlap by introducing a small shift in time between the adjacent clusters’ schedules to compensate for the clock drift. The OLS-MAC algorithm is simulated in NS-2 and compared to some S-MAC derived protocols. The simulation results verified that the proposed algorithm outperforms previous protocols in number of performance criterion.