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.
This project has advanced the theory and practice of iterated filtering methodology for statistical inference on partially observed stochastic dynamic processes. Iterated filtering can be applied to general classes of scientific models due to the plug-and-play property: unlike most previous statistical methods for nonlinear dynamic systems, plug-and-play methods are applicable to any model for which computer simulation code is available whether or not the model is analytically tractable. The first complete theory of iterated filtering was published in Annals of Statistics. Case studies developing novel models for malaria transmission were published in Journal of the American Statistical Association and PLoS Computational Biology. Iterated filtering methodology was fundamental to an investigation of cholera transmission which was published in Nature. A new class of dynamic models prompted by the capabilities of iterated filtering was published in Stochastic Processes and their Applications. The relevance of this new model class to understanding of measles transmission was published in Journal of the Royal Society Interface. Three PhD students have been trained through involvement in this research, and the first to graduate is now starting an Associate Professor position in Statistics at Purdue University. The new statistical methodology has also been propagated through the development of a publicly available software package (http://cran.r-project.org/web/packages/pomp). Further outreach to the scientific community has arisen through my involvement as an investigator for RAPIDD (Research and Policy in Infectious Disease Dynamics), a program of the NIH and DHS. Current scientific interest in iterated filtering methodology has centered on the fields of epidemiology and ecology, though broader applications are anticipated in future.
