Event Details

STOR-i Masterclass 4: Michelle Carey, Assistant Professor in Statistics, University College Dublin - Dynamic Data Analysis: Modeling Data with Differential Equations - 15-16 March 2021, 09:00-12:00

Dynamic Data Analysis: Modeling Data with Differential Equations

This course focuses on the use of smoothing methods for developing and estimating differential equations following recent developments in functional data analysis and building on techniques described in Ramsay and Silverman (2005) Functional Data Analysis. Differential equation models are theoretically built mechanistic models for complex systems. A single differential equation model can describe a wide variety of behaviours including oscillations, steady states and exponential growth and decay, with few but readily interpretable parameters. Consequently, differential equation models are routinely used in describing chemical reaction dynamics, predator-prey interactions, heat transfer, economic growth, epidemiological outbreaks, climate and weather prediction, gene regulatory pathways, etc.

There are already many texts on the mathematical properties of differential equations or dynamic models, and there is a large literature distributed over many fields on models for real-world processes consisting of differential equations. This course is interested in fitting such models to data and the statistical properties of differential equations estimated from data. Many statistical models involve three distinct groups of variables: local or nuisance parameters, global or structural parameters, and complexity parameters. We introduce the parameter cascading method to estimate these statistical models, which treats one group of parameters as an explicit or implicit function of other parameters. The dimensionality of the parameter space is reduced, and the optimization surface becomes smoother. The Newton-Raphson algorithm is applied to estimate these three distinct groups of parameters in three levels of optimization, with the gradients and Hessian matrices written out analytically by the Implicit Function Theorem if necessary and allowing for different criteria for each level of optimization. Moreover, variances of global parameters are estimated by the Delta method and include the variation coming from the complexity parameters.