Inference and Scaling in Stochastic Dynamical Systems

动力系统是我们这个世界的一个普遍特征：从小的(单个原子的运动)到大的(星系动力学)，从具体的(蚂蚁的觅食模式)到抽象的(互联网上的信息流)，从快的(神经冲动)到慢的(不同物种基因组的进化轨迹)。能够理解这样的系统对生物、化学、物理、经济学、工程学以及计算机科学等各个领域都至关重要。我的研究重点是动力系统，特别是随机动力系统的分析和估计的计算方法。目前的建议集中在我的实验室最近取得重要突破的两个问题上：离散状态离散时间随机系统(马尔可夫链)的路径分布的刻画和离散状态连续时间随机系统(连续时间马尔可夫链)的最大概率路径的推断。 关于路径分布问题，我们最近澄清和改进了Mandelbrot的一个已有50年历史的猜想，证明了马尔可夫链生成的路径分布是幂定律的充要条件。我们还证明了有限和拉伸指数分布是可能的，其分布类型仅取决于可能的状态跃迁的结构，而不取决于确切的跃迁概率。我们开发了图论计算来区分这些情况，并开发了有效的特征值/动态规划计算来确定分布参数。在这个提案中，我们将调查我们的理论在模型选择中的使用。在观察到路径分布的经验尺度并且需要从数据估计马尔可夫模型的情况下，如何将尺度信息融入到模型估计过程中？我们还将致力于将我们的结果扩展到更复杂的序列数据模型--特别是与自然语言、音乐、基因组结构和许多其他复杂的真实世界系统的分析相关的随机上下文无关文法。 对于路径推理问题，我们最近发展了一种称为状态序列分析的方法，用于识别连续时间马尔可夫链在一段时间内访问的最可能的状态序列。使用这种方法，我们在许多领域获得了新的见解，包括随机蛋白质折叠、艾滋病毒耐药突变的进化和离子通道动力学。在本提案中，我们将寻求将我们的结果扩展到噪声/部分观测模型，以及系统状态中等待时间分布的更一般形式。这将使状态序列分析能够应用于范围更广的目标区域。

Dynamical systems are a ubiquitous feature of our world: from the small (the motions of single atoms) to the large (galactic dynamics), from the concrete (foraging patterns of ants) to the abstract (flow of information on the internet), from the quick (neural impulses) to the slow (evolutionary trajectories of different species' genomes). Being able to understand such systems is crucial to fields as diverse as biology, chemistry, physics, economics, engineering, and, not least, computer science. My research focuses on computational methods for the analysis and estimation of dynamical systems, particularly stochastic dynamical systems. The current proposal focuses on two recent problems on which my lab has made important breakthroughs: characterizing path distributions for discrete-state discrete-time stochastic systems (Markov chains), and inference of maximum-probability paths for discrete-state continuous-time stochastic systems (continuous-time Markov chains). On the path distribution problem, we recently clarified and improved upon a 50-year old conjecture by Mandelbrot by proving necessary and sufficient conditions for the distribution of paths generated by a Markov chain to be powerlaw. We also showed that finite and stretched-exponential distributions are possible, with the distribution type depending only on the structure of possible state transitions, and not on the exact transition probabilities. We developed graph-theoretic computations to discriminate between the cases, and efficient eigenvalue / dynamic programming computations to determine distribution parameters. In this proposal, we will investigate the uses of our theory for model selection. In the case that empirical scaling of the path distribution is observed and a Markov model needs to be estimated from the data, how can we incorporate scaling information into the model estimation process? We will also work to extend our results to more sophisticated models of sequential data--in particular, stochastic context-free grammars, which are relevant to the analysis of natural language, music, genome structure, and many other complex real-world systems. For the path inference problem, we have recently developed an approach called State Sequence Analysis for identifying the most probable sequences of states visited by a continuous-time Markov chain over some period of time. Using this approach, we obtained novel insights into a number of domains, including stochastic protein folding, the evolution of drug-resistance mutations in HIV, and ion channel dynamics. In the present proposal, we will seek to extend our results to noisy / partial observation models, as well as more general forms of waiting-time distributions in the states of the system. This will allow State Sequence Analysis to be applied to a much wider range of target domains.