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School of Electrical, Computer and Energy Engineering PhD Final Oral Defense Statistical and Dynamical Modeling of Riemannian Trajectories with Application to Human Movement Analysis by Rushil Anirudh Feb 23rd (Tuesday), 2016. 1.45pm, GWC 487. Committee: Dr. Pavan Turaga (chair) Dr. George Runger Dr. Douglas Cochran Dr. Thomas Taylor Abstract The data explosion in the past decade is in part due to the widespread use of rich sensors that measure different physical phenomenon -- such as gyroscopes that measure orientation in phones, the Microsoft Kinect measures depth information, etc. A typical application requires inferring the underlying physical phenomenon from data, which is done using machine learning. A fundamental assumption in training models is that the data is Euclidean, i.e. the metric is the standard Euclidean distance governed by the l2 norm. However in many cases this assumption is violated, such as when data lies on Riemannian manifolds which can be non-Euclidean. For example, gyroscopes measure rotation information, where different orientations result in moving along the 2-sphere, S2. The distance between two points cannot be measured by the Euclidean distance, but rather the length of the shortest arc along the sphere. While the underlying geometry accounts for the non-linearity, accurate analysis of human activity also requires temporal information to be taken into account. Human movement has a natural interpretation as a trajectory on the underlying feature manifold, as it evolves smoothly in time. A commonly occurring theme in many emerging problems is the need to represent, compare, and manipulate such trajectories in a manner that respects the geometric constraints. This dissertation is a comprehensive treatise on modeling Riemannian trajectories to understand and exploit their statistical and dynamical properties. Such properties allow us to formulate novel representations for Riemannian trajectories. For example, the physical constraints on human movement are rarely considered, which results in an unnecessarily large space of features, making search, classification and other applications more complicated. Exploiting statistical properties can help us understand the true space of such trajectories. In applications such as stroke rehabilitation where there is a need to differentiate between very similar kinds of movement, dynamical properties can be much more effective. In this regard, we propose a generalization to the Lyapunov exponent to Riemannian manifolds and show its effectiveness for human activity analysis. The theory developed in this thesis naturally leads to several benefits in areas such as data mining, compression, dimensionality reduction, classification, and regression.