Statistical Inference for Complex Dynamical Models

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Dynamical models describe the rate of change of a dynamical system. They are widely used to elucidate complex systems in many areas including neuroscience, biology, medicine, genetics, ecology, engineering, physics, and finance. While the parameters in dynamical models usually have scientific interpretations, their values are often unknown and can sometimes be difficult to estimate directly. This makes it hard to verify dynamical models by showing how well their solutions fit real data. My long-term objective of the proposed research program is to develop statistical methodologies for estimating parameters in dynamical models from measurements of the dynamical system in the presence of measurement errors. I will work broadly in parameter estimation for dynamical models. My research proposal highlights some interesting research problems motivated by important applications. For example, when the dynamical system has replicative measurements over multiple subjects, each subject may have different but correlated values of model parameters. Then it is of great interest to allow both population parameters (fixed effects) and individual parameters (random effects) in the dynamical model, which is then called a mixed-effects dynamical model. Subjects may have missing group information, in which case the measurements for the dynamical system can be reasonably assumed to follow a mixture distribution. I propose a semiparametric method to estimate mixed-effects dynamical models from mixture-distributed data. Another research problem arises when parameters in dynamical models depend on some characters (covariates) of subjects while these parameters have unknown values. It may be desirable to estimate model parameters and their relationships to those covariates simultaneously. I propose an analysis of variance (ANOVA) approach to detect treatment effects on dynamical systems. My proposed research can potentially be applied to many scientific areas such as agriculture, biology, and genetics.