| 摘要: | 在現代信息時代，隨著傳感器網絡技術的發展，在科學，工程，金融，經濟學和國土安全應用中 在以前所未有的速度和範圍在生成和探索大量的數據，其廣泛地包括無線傳感器網絡，原位傳感器基 礎設施和遠程傳感器。我們的目標是開發多傳感器網絡系統，不僅收集原始數據，而是幫助學習有價 值知識或快速做出明智決定，例如異常檢測，信號檢測。 為了在固定數據集和顺序數據之間建立聯繫，首先考慮具有固定樣本大小的變化點估計問題。已 知改變點估計的設置等價於確定HMM 隱藏狀態的個數，這稱模型選擇問題。然而，HMM 中模型選 擇問題，尚未出現滿意地解決方式，特別是對具有異質共變異數的高度使用高斯HMM。我們將提出 一種氏模型選擇方法。此外除了使用貝氏模型選擇方法，我們將證明HMM 中貝氏信息準則（BIC） 的一致性。接著，我們將這些結果擴展到馬爾可夫波動切換模型。 第二部分，將研究 HMM 傳感器網絡中最快檢測問題，並提出一些方法，我們有興趣在某些情況 下發展基本的信息邊界，並找到漸近地實現這些邊界的方案。因一般問題的困難度和複雜性，本提案 中，我們將集中一些有用簡單情況，結果將闡明複雜的現實問題。具體地說，我們將研究以下事項： 首先，研究變更前和變更後參數兩者完全指定情況。第二，調查局部變化的幅度指定時的情形，但不 知變化的方向。第三，調查變化的方向和幅度未知的情況，也就是HMM 改變後參數是未知。最後， 探究特殊情況下的穩健最快檢測，當一個隱藏狀態為主導地位，並沒有足夠的觀察值來估計其他非優 勢隱藏狀態下的參數。 本計畫目的是提供理論基礎和許多有效的方法，用於隱馬爾可夫模型中的變化點估計和最快檢測。 貝氏信息準則（BIC），一致性，變化點，累和，相對熵散度，邊際概似，模型選擇，標準化常數，計 分檢定，傳感器網絡，順序檢測 ;In the modern information age, with the advance of sensor network technology, massive volumes of sensor data are being generated and explored at an unprecedented pace and scope in science, bioinformatics, genetics, engineering, finance, economics and homeland security applications, where sensors may broadly include wireless sensor networks, in-situ sensor infrastructures and remote sensors. In many real-world applications such as biosurveillance, quality control, key infrastructure or internet traffic monitoring, we have an ultimate objective to develop multi-sensor network systems that do not simply collect \raw" data, but rather help us to learn valuable knowledge or make intelligent decisions quickly, e.g., anomaly detection, signal detection. To make a linkage between the off line data set and the streaming data set, in this project, we first consider the problem of change point estimation with fixed sample size. It is known that one setting of change point estimation is equivalent to determine the number of hidden states for a HMM. This is also called model selection problem. However, model selection problem in HMM, which has not yet been satisfactorily solved, especially for the highly used Gaussian HMM with heterogeneous covariance. Here we will propose a consistent method for determining the number of hidden states of HMM based on the marginal likelihood, which is obtained by integrating out both the parameters and hidden states. Moreover, other than using Bayesian model selection method, we will try to prove the consistency property of Bayesian information criterion (BIC) in HMM, which is a long-standing open problem in model selection literature. Next, we will extend these results to Markov switching models, both in means and volatilities. In the second part of this research proposal, we will investigate quickest detection problems in HMM sensor networks. Instead of proposing some ad hoc methodologies, we are interested in developing fundamental information bounds in certain scenarios and finding schemes that achieve these bounds asymptotically. Given the difficulty and complexity of the general problem, in this research proposal, we will focus on some simple but useful situations, and the results will shed light on more complicated real-world problems. To be more specific, we will study the following matters: First, we will investigate the case when both pre- and post-change parameters, 0 and 1 , are completely specified. Second, we will investigate the scenario when the magnitude of the local post-change is specified, but we do not know the directions of changes. Third, we will investigate the scenario when both direction and magnitude of the change are unknown. In other words, the post-change parameter 1 of the HMM is unknown. Last, we will investigate the robust quickest detection under the special scenario when one hidden state, say, { 0} n X , is dominant, and we do not have enough observations to estimate the parameters under other non-dominant hidden states. The objective of this proposal is to offer theoretical foundation and a host of efficient scalable methodologies for change point estimation and quickest detection in hidden Markov models. Bayesian Information Criterion (BIC), consistency, change-point, CUSUM, Kullback-Leibler (KL) divergence, marginal likelihood, model selection, normalizing constant, score test, sensor networks, sequential detection. |
