The analysis of sequential data is important in business, science, and engineering, for tasks such as signal processing, user behavior mining, and commercial transactions analysis. In this dissertation, we build upon the Collective Dynamical Modeling and Clustering (CDMC) framework for discrete time series modeling, by making contributions to clustering initialization, dynamical modeling, and scaling. We first propose a modified Dynamic Time Warping (DTW) approach for clustering initialization within CDMC. The proposed approach provides DTW metrics that penalize deviations of the warping path from the path of constant slope. This reduces over-warping, while retaining the efficiency advantages of global constraint approaches, and without relying on domain dependent constraints. Second, we investigate the use of semi-Markov chains as dynamical models of temporal sequences in which state changes occur infrequently. Semi-Markov chains allow explicitly specifying the distribution of state visit durations. This makes them superior to traditional Markov chains, which implicitly assume an exponential state duration distribution. Third, we consider convergence properties of the CDMC framework. We establish convergence by viewing CDMC from an Expectation Maximization (EM) perspective. We investigate the effect on the time to convergence of our efficient DTW-based initialization technique and selected dynamical models. We also explore the convergence implications of various stopping criteria. Fourth, we consider scaling up CDMC to process big data, using Storm, an open source distributed real-time computation system that supports batch and distributed data processing. We performed experimental evaluation on human sleep data and on user web navigation data. Our results demonstrate the superiority of the strategies introduced in this dissertation over state-of-the-art techniques in terms of modeling quality and efficiency.

Wir betrachten die stochastische Differentialgleichung mit Gedächtnis (SDDE) mit Gedächtnislänge r dX(t) = b(X(u);u in [t-r,t])dt + sigma(X(u);u in [t-r,t])dB(t) mit eindeutiger schwacher Lösung. Dabei ist B eine Brownsche Bewegung, b and sigma sind stetige, lokal beschränkte Funktionen mit Definitionsbereich C[-r,0], und X(u);u in [t-r,t] bezeichnet das Segment der Werte von X(u) für Zeitpunkte u im Intervall [t,t-r]. Unser Ziel ist eine Folge von diskreten Zeitreihen Xh höherer Ordung zu konstruieren, so dass mit h gegen 0 die Zeitreihen Xh schwach gegen die Lösung X der stochastischen Differentialgleichung mit Gedächtnis konvergieren. Desweiteren werden wir Bedingungen angeben, unter denen eine gegeben Folge von Zeitreihen Xh höherer Ordung schwach gegen die Lösung X einer stochastischen Differentialgleichung mit Gedächtnis konvergiert. Als ein Beispiel werden wir den schwachen Grenzwert einer Folge von diskreten GARCH-Prozessen höherer Ordnung ermitteln. Dieser Grenzwert wird sich als schwache Lösung einer stochastischen Differentialgleichung mit Gedächtnis herausstellen. / Consider the stochastic delay differential equation (SDDE) with length of memory r dX(t) = b(X(u);u in [t-r,t])dt + sigma(X(u);u in [t-r,t])dB(t), which has a unique weak solution. Here B is a Brownian motion, b and sigma are continuous, locally bounded functions defined on the space C[-r,0], and X(u);u in [t-r,t] denotes the segment of the values of X(u) for time points u in the interval [t,t-r]. Our aim is to construct a sequence of discrete time series Xh of higher order, such that Xh converges weakly to the solution X of the stochastic differential delay equation as h tends to zero. On the other hand we shall establish under which conditions time series Xh of higher order converge weakly to a weak solution X of a stochastic differential delay equation. As an illustration we shall derive a weak limit of a sequence of GARCH processes of higher order. This limit tends out to be the weak solution of a stochastic differential delay equation.