Inference for dynamical systems

所提出的研究的出发点是一个新的算法，最近已被证明，使最大似然估计可行的先前棘手的部分观察到的非线性随机动力系统。 该算法是基于一系列的过滤操作，收敛到一个最大似然参数估计，因此被称为迭代过滤。 迭代过滤方法的可用性开辟了许多可能性，为开发新的类的随机动态模型作为数据分析工具。 建议的研究计划的一个组成部分是发展一类新的马尔可夫链模型适合于生物系统，包括相互作用的泊松过程，其速率受到白色噪声。另一个目标是通过增加对迭代滤波的理论理解来拓宽基于似然的推理是实用的动力系统的类别。 具体而言，迭代过滤的一个新的理论框架将开发，确定与以前研究的随机近似技术的关系的基础上。在迭代过程中求平均和在随机方向序列上搜索的技术，对于其他随机逼近方法具有良好的理论和实际性质，预计将适用于迭代滤波。 拟议研究的第三个组成部分是展示新方法在促进疟疾传播的新的科学相关数据分析方面的作用。传染病提出了具有挑战性和重要性的问题，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类进行数据分析将需要处理新的情况下诊断拟合优度，适当的技术将被开发和demonstrated.Nonlinear随机动态模型被广泛用于研究整个科学和工程系统发生。 这样的模型很容易用公式表示，并且可以用数学和数字进行分析。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的重要问题。 最近，新的方法利用不断增加的计算资源已经取得了进展。 持续的进展需要建立成功证明方法的理论理解，开发新的方法，并展示如何利用这些进展来进一步了解感兴趣的动力系统。了解传染病动态的最新动机包括新出现的疾病（艾滋病毒/艾滋病、严重急性呼吸系统综合症、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。 统计领域形成了一个天然的桥梁，使方法的进步提供给更广泛的研究界。

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.