Estimation, Detection and Control of Multiple Change-point Stochastic Systems with Applications to Economics, Engineering, Biology and Climate Science

具有多个变化点的随机系统的参数估计、突变检测和系统控制问题出现在金融计量学、工业质量控制、基因图谱和气候突变研究中。解决这些问题的一个重要因素是对未知多个变点的分段常量参数进行有效估计。在拟议的研究中，研究人员将开发一种统一的隐马尔可夫滤波方法来进行参数估计，并开发可通过并行递归实现的有限复杂度近似算法。特别是，本研究调查了来自不同领域的四类问题。第一部分发展了计量经济学时间序列模型的估计和预测方法，其中条件方差的广义自回归条件异方差过程与变点未知的分段常数无条件方差交织在一起。第二部分研究了随机系统参数变化前后分布未知的检测问题。第三章研究了序列数据中分段恒定信号的分割，并讨论了它在基因组拷贝数变异分析中的应用。第四，在突如其来的气候变化中，发展了著名的洛伦兹系统中的多重结构变化的估计理论和算法，该系统是混沌的。该系统提供了高度简化的大气环流描述，并已成为气候研究中参数估计方法的试验台。研究人员将展示如何将来自不同领域的这些具有挑战性的问题统一起来，并用隐马尔可夫滤波方法解决多个变化点。复杂随机系统中的参数突然或渐进变化在各种科学和工程实践中经常遇到，包括经济、生物和气候研究。虽然参数渐变系统在历史上一直受到人们的关注，但最近的科学和工程技术进步表明，参数突变但未知的随机系统的重要性与日俱增。在经济研究中，当局和行业从业者对确定经济活动的转折点很感兴趣。在目前的基因组研究中，DNA拷贝数变异(即特定染色体片段的获得或丢失)是包括癌症、HIV获得性疾病以及阿尔茨海默病和帕金森病在内的许多疾病发生和发展的关键遗传事件，而研究这些遗传事件的一个重要步骤是识别变异区域。在气候研究中，地球气候的突变因其对区域和全球生态经济系统的严重影响而日益受到研究人员和政策制定者的关注，而对于非平稳气候或与气候相关的变量，由于现有统计工具的价值有限，假设是逐步变化的，因此需要新的统计技术。然而，未知突变带来的统计和计算复杂性阻碍了解决这些问题的努力。这项拟议的研究提出了一个统一的统计理论来模拟和估计复杂系统中的这种变化，这些应用应该会为解决这些具有挑战性的问题的努力带来新的曙光。特别是，这项研究使我们能够通过将当前最先进的气候建模技术与统计分析相结合，在全球层面对气候系统的突然变化进行建模。此外，将拟议的研究应用于气候科学，将统计理论与高性能计算联系起来，从而为统计方法开辟了新的视野。

The problems of parameter estimation, abrupt change detection, and system control in stochastic systems with multiple change-points arise in financial econometrics, industrial quality control, gene mapping and abrupt climate change studies. An important ingredient in the solution to these problems is efficient estimation of piecewise constant parameters with unknown multiple change-points. In the proposed research, the investigator will develop a unified hidden Markov filtering approach to parameter estimation, and bounded complexity approximation algorithms that can be implemented via parallel recursions. In particular, four types of problems from different fields are investigated in the study. The first develops estimation and forecasting procedures for an econometric time series model, in which a generalized autoregressive conditional heteroskedasticity process for conditional variances is intertwined with piecewise constant unconditional variances for which the change-points are unknown. The second studies detection of parameter changes with unknown pre- and post-change distributions in stochastic systems. The third investigates segmentation of piecewise constant signals in sequence data and discusses its applications to the analysis of genomic copy number variation. The fourth, arising in abrupt climate change, develops estimation theory and algorithms for multiple structural changes in the celebrated Lorenz system, which is chaotic. This system provides a highly simplified description of atmospheric circulation and has served as a testbed for methods of parameter estimation in climate studies. The investigator will show how these challenging problems from different areas are unified and solved by the hidden Markov filtering approach to multiple change-points.Abrupt or gradual parameters changes in complex stochastic systems are often encountered in various scientific and engineering practices including economics, biology and climate studies. While systems with gradually changing parameters have received much attention historically, recent advances in science and engineering show the growing importance of stochastic systems with abrupt, but unknown parameter changes. In economic studies, the authorities and industrial practitioners are interested in dating turning points of economic activities. In current genomic research, DNA copy number variations (i.e., gains or losses of specific chromosomal segments) are key genetic events in the development and progression of numerous diseases including cancer, HIV acquisition, and Alzheimer and Parkinson's disease, and an important step in studying these genetic events is to identify the regions of variations. In climate studies, abrupt changes of Earth's climate have attracted increasing attention from researchers and policy makers due to their projected severe impact on regional and global ecological and economic systems, and new statistical techniques are needed for nonstationary climate or climate-related variables due to the limited values of current statistical tools with a gradually changing assumption. However, the effort of solving these problems is hampered by the statistical and computational complexity caused by unknown abrupt changes. The proposed research presents a unified statistical theory to model and estimate such changes in complex systems, and the applications should shed new light on efforts at tackling these challenging problems. In particular, this study allows us to model abrupt changes of climate systems at the global level by incorporating current state-of-the-art climate modeling techniques with statistical analysis. Furthermore, the application of the proposed research to climate sciences links the statistical theory with high-performance computation, and hence opens up new horizons to statistical methodologies.