Order Determination for Hidden Markov and Related Models

隐马尔可夫模型（Hidden Markov Models，简称HMM）是处理时间序列数据的强大工具，广泛应用于语音识别、机器翻译、计算生物学、密码分析和金融等科学和工程领域。HMM的基本组成部分包括噪声观测和相应的隐藏状态。在大多数应用中，隐藏状态的数量（HMM的阶数）是事先不知道的，但传达了有关底层过程的重要信息。例如，在分子生物学中，隐藏状态的总数可以是蛋白质的不同3D构象的数量;在化学中，隐藏状态的总数可以是有机反应中不同化学物质的数量。本计画计划研究Honduras、有限混合模型与阶层Honduras的定阶问题，后两种模型是Honduras的特例与延伸。该项目旨在为订单选择开发一种一致和有竞争力的方法。除了彻底的理论研究，全面的数值研究和生物学和化学中的应用将进行。该项目不仅将大大推进对HSP 70的理论理解，还将为研究人员分析数据提供强大的工具。数据应用将有助于推进分子生物学和生物化学。该项目还旨在支持和培训本科生和研究生，特别注意招收代表性不足群体的学生进入统计和相关领域。本科和研究生教育将纳入研究活动。该项目将建立边际似然法作为一个一致的和有竞争力的顺序选择方法，为障碍，有限的混合模型，分层障碍。将开展五项研究，具体如下。(1)研究HMM的阶数，目标是识别和开发HMM阶数确定的一致方法。(2)研究有限混合模型中的阶数选择问题。有限混合模型可以被重新表述为特殊类型的障碍。我们的目标是开发一种方法，用于一致地估计混合物组分的数量。(3)研究层次化障碍物的顺序确定，其中多个障碍物通过层次化结构链接。这里的目的是识别和开发用于确定分层隐马尔可夫模型顺序的一致方法，并特别努力解决多个隐马尔可夫模型通常具有相当不同的特征（例如长度）的挑战性问题。(4)研究计算挑战，调查和实施有效的计算方法，用于HMM和相关模型的阶数确定，包括实施和发布一个开源的、公开可用的R包。(5)将新方法应用于共翻译蛋白质靶向的离子通道数据和单分子数据。PI还计划开发课程，介绍和指导学生在障碍，混合模型和层次障碍。所提出的研究的成功将开发一个理论基础和相关的方法，为一致的顺序确定的障碍和相关的模型。研究成果和教育部分将广泛影响障碍分析、层次障碍和模型选择，并有助于培养该领域的新一代学者和研究人员。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Hidden Markov models (HMMs) are powerful tools for processing time series data and are widely used in scientific and engineering applications, including speech recognition, machine translation, computational biology, cryptanalysis, and finance. The fundamental components of an HMM include the noisy observations and the corresponding hidden states. In most applications, the number of hidden states (the order of the HMM) is not known beforehand but conveys important information about the underlying process. For example, in molecular biology, the total number of hidden states may be the number of distinct 3D conformations of a protein; in chemistry, the total number of hidden states may be the number of distinct chemical species in an organic reaction. This project plans to investigate order determination for HMMs, finite mixture models, and hierarchical HMMs; the latter two models are special cases and extensions of HMMs. The project aims to develop a consistent and competitive method for order selection. In addition to a thorough theoretical investigation, comprehensive numerical studies and applications in biology and chemistry will be conducted. The project will not only significantly advance the theoretical understanding of HMMs, but also provide powerful tools for researchers to analyze data. The data applications will help advance molecular biology and biochemistry. The project also aims to support and train undergraduate and graduate students, with special attention being given to recruiting students from under-represented groups into statistics and related fields. Education at the undergraduate and graduate levels will be integrated into the research activities. The project will establish the marginal likelihood method as a consistent and competitive order selection method for HMMs, finite mixture models, and hierarchical HMMs. Five research studies will be carried out, enumerated as follows. (1) Investigate the order of HMMs, where the goal is to identify and develop consistent methods for HMM order determination. (2) Investigate order selection issues in finite mixture models. Finite mixture models can be reformulated as special types of HMMs. The goal is to develop a method for consistently estimating the number of mixture components. (3) Investigate order determination of hierarchical HMMs, where multiple HMMs are linked through a hierarchical structure. The aim here is to identify and develop consistent methods for determining the order of hierarchical HMMs, taking special effort to address the challenging issue that multiple HMMs often have quite diverse characteristics, such as lengths. (4) Study computational challenges and investigate and implement efficient computational methods for the order determination of HMM and related models, including the implementation and release of an open source, publicly available R package. (5) Apply the new method to ion channel data and single-molecule data on co-translational protein targeting. The PI also plans to develop courses that introduce and guide students in HMMs, mixture models, and hierarchical HMMs. The success of the proposed research will develop a theoretical basis and associated methodology for consistent order determination of HMMs and related models. The research achievements and the education components will broadly impact the analysis of HMMs, hierarchical HMMs, and model selection and also help train a new generation of scholars and researchers in the field.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.