Inference for dynamical systems

本研究的出发点是一种新的算法，该算法最近被证明可以对先前难以处理的部分观测非线性随机动力系统进行极大似然估计。该算法基于收敛于最大似然参数估计的一系列滤波操作，因此称为迭代滤波。迭代滤波方法的可用性为开发用作数据分析工具的新型随机动态模型开辟了许多可能性。提出的研究计划的一个组成部分是开发适合于生物系统的一类新的马尔可夫链模型，该模型由相互作用的泊松过程组成，其速率受白噪声影响。另一个目标是通过增加对迭代滤波的理论理解，扩大基于似然推理的动力系统的类别。具体来说，一个新的理论框架迭代滤波将开发，基于识别与先前研究的随机逼近技术的关系。迭代平均技术和随机方向序列搜索技术对于其他随机逼近方法具有良好的理论和实践性质，有望应用于迭代滤波。拟议研究的第三个组成部分是证明新方法在促进对疟疾传播进行新颖和科学相关的数据分析方面的作用。传染病提出了具有挑战性和重要的问题，这些问题长期以来一直是动力系统推理方法的试验场。通过新的连续时间动态模型进行数据分析需要处理新的情况来诊断拟合优度，并且需要开发和演示适当的技术。非线性随机动力学模型被广泛用于研究科学和工程中的系统。这样的模型可以很自然地形成，并且可以进行数学和数值分析。尽管经过几十年的研究，对非线性动力学模型进行统计推断仍然是一个具有挑战性和重要的问题。最近，利用不断增加的计算资源的新方法取得了进展。持续的进展需要对成功演示的方法建立理论理解，开发新的方法，并展示如何将这些进展用于进一步了解感兴趣的动力系统的科学知识。了解传染病动态的最近动机包括新出现的疾病（艾滋病毒/艾滋病、非典型肺炎、大流行性流感）、重新出现的疾病（疟疾、结核病）和生物恐怖主义构成的威胁。动力系统的推理出现在许多不同的领域，包括经济学、神经科学、化学工程、信号处理和分子生物化学。统计学领域形成了一个天然的桥梁，使更广泛的研究界可以获得方法上的进步。

The starting point of the proposed research is a new algorithm that has recently been shown to make maximum likelihood estimation feasible for previously intractable partially-observed nonlinear stochastic dynamical systems. The algorithm is based on a sequence of filtering operations which converges to a maximum likelihood parameter estimate, and is therefore termed iterated filtering. The availability of iterated filtering methodology opens up many possibilities for developing new classes of stochastic dynamic models for use as data analysis tools. One component of the proposed research program is development of a new class of Markov chain models appropriate for biological systems, consisting of interacting Poisson processes whose rates are subject to white noise. Another goal is to broaden the class of dynamical systems for which likelihood based inference is practical, via increased theoretical understanding of iterated filtering. Specifically, a new theoretical framework for iterated filtering will be developed, based on identifying a relationship with previously studied stochastic approximation techniques. Techniques of averaging over iterations and searching over a sequence of random directions, which have good theoretical and practical properties for other stochastic approximation methods, are expected to be applicable to iterated filtering. The third component of the proposed research is to demonstrate the role of the new methodology in facilitating a novel and scientifically relevant data analysis of malaria transmission. Infectious diseases pose challenging and important questions which have long been a testing ground for inference methodology for dynamical systems. Carrying out data analysis via new classes of continuous time dynamic models will require handling novel situations for diagnosing goodness of fit, and appropriate techniques will be developed and demonstrated.Nonlinear stochastic dynamical models are widely used to study systems occurring throughout the sciences and engineering. Such models are natural to formulate and can be analyzed mathematically and numerically. Despite decades of work, carrying out statistical inference for nonlinear dynamical models remains a challenging and important problem. Recently, progress has been made possible by new methodology taking advantage of increasing computational resources. Continued progress requires building theoretical understanding of successfully demonstrated methodology, developing new methodologies, and showing how these advances can be used to further scientific knowledge about dynamical systems of interest. Recent motivations for understanding infectious disease dynamics include the threats posed by emerging diseases (HIV/AIDS, SARS, pandemic influenza), re-emerging diseases (malaria, tuberculosis) and bioterrorism. Inference for dynamical systems arises in many diverse fields, including economics, neuroscience, chemical engineering, signal processing, and molecular biochemistry. The field of Statistics forms a natural bridge to make methodological advances available to a wider research community.