Inference for dynamical systems

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

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.