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.

在此篇論文中，我們提出一個慢性Ｂ型肝炎病毒感染病程之數學模型。因為在病症間的轉移機率(Transition probability)是隨著患者的年齡變動，所以在過去的文獻中，已經有學者提出，在疾病轉移機率模型中，應加入國民生命表(Life table)，藉此讓機率模型更符合Ｂ型肝炎病患的生命歷程。但是過去的文獻中，學者並沒有利用加入國民生命表之後疾病模型做進一步的病程分析。在這篇論文當中，我們假設原始的疾病轉移模型是符合馬可夫鏈的性質，並且提出一種加入國民生命表的方法，賦予疾病有年齡相關特性之模型。根據文獻數據和類馬可夫機率性質，我們使用著名的Chapman-Kolmogorov公式計算Ｂ型肝炎的自然病程機率，並畫出病人的生存機率曲線(Survival curve)。文章最後將會藉由兩個例子來介紹此篇論文提出的模型。實驗數據結果證實，此模型不僅提供了一個更精確的方法去分析在病症與死亡間的轉移機率、平均餘命(Life expectancy)、以及在不同年齡的存活機率(Survival probability)，並且可以更進一步的分析且瞭解病情狀態之間的轉移狀況。 / In this thesis, we propose a new mathematical model extending the natural history of hepatitis B virus (HBV) prognosis progression on chronic HBV infection. Since the actual transition probabilities between symptoms are dependent of ages, it has been proposed that the life table should be accommodated to the HBV prognosis progression model so that it can more properly explain the disease progression of the HBV patients. But in the literature, no further disease analysis and applications of it with the life table are discussed. In this thesis, we assume that the original disease progression is described by a Markov model, and propose a new method to combine the HBV progression with the life table so that the proposed model integrates data from the life table and allows the accommodation of age-dependent properties of the target disease. With clinical data based on annual incidence rates, the entire model is Semi-Markov based in nature. Computation methods similar to the celebrated Chapman-Kolmogorov equation can be applied to study the associated probability of each likely trajectory with desired initial ages and health states under the scenarios of natural history and various treatment policies. This method provides a more accurate way to analyze the transitions between symptoms, such as the mean life expectancy or the survival probabilities at different ages. We will give examples to demonstrate the proposed method in this thesis. Numerical results show the proposed model not only provides a more accurate method to analyze the mean life expectancy, the survival probabilities at different ages, and the transition probabilities from symptoms to death but also helps us to understand the transitions between symptoms.