Inference and Scaling in Stochastic Dynamical Systems

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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.