The primary objective of this thesis is to study change detection problems and their applications in longitudinal and functional data. In particularly, two types of change detection problems for longitudinal data are considered. The first type of problems is on-line change detection for longitudinal data, where we focus on detecting changes within a single longitudinal data stream that arrive into the system sequentially. The other type of change detection problems that will be studied in this thesis concerns about detecting outliers from a set of longitudinal or functional data.
For the first type of change detection problems, we study two novel engineering applications. The first application is studied in Chapter 2, focusing on the on-line steady state detection. The goal is to identify the transition point between transient period and steady state period. We propose a novel on-line steady state detection algorithm based on a multiple change-point state space formulation and the sequential Monte Carlo methods. Compared to other existing methods, the main contribution of this work is its significantly improved computational efficiency by the use of Rao-Blackwellization, making it a much preferred method for many on-line applications where quick processing of the data in real time is critical. Additionally, the proposed method is shown to have more robust detection performance than existing methods when dealing with different types of signals.
In Chapter 3, we study the second application of change detection problems, which focuses on statistical process control for the short-run process. We propose new methods under the Bayesian framework is to track the process mean and detect on-line if the process mean is beyond certain control limits or specification limits. Our model modifies the original model proposed by Tsiamyrtzis and Hawkins (2005) and can be more flexible in handling linear trends of the process. Compared to the method proposed by Tsiamyrtzis and Hawkins (2005), the advantages of our method are two-folds. Firstly, the performance of our method is more robust to parameter misspecification and requires less knowledge of the process to make accurate estimations. Secondly, the resulted posterior inference of the process mean has a significantly reduced number of mixtures, leading to substantially save of computational and memory cost.
The other type of change detection problems studied in this thesis concentrates on analysis of a set of longitudinal or functional data, which is discussed in Chapter 4. In particularly, we focus on the outlier detection for functional data, where the outlier is defined as a curve that is generated from a different process compared to normal curves. Based on the use of data depth, we propose two new depth notions, the weighted band depth and the localized weighted band depth for detecting various outliers. Our main contribution is proposing a new idea called the shape distance, which makes our methods particularly effective in detecting outliers that have different shapes with normal curves.