Abstract

Mathematical models described by differential equations are a powerful way to describe a real-world system. Parameter estimation allows us to make predictions and quantify uncertainty, like with traditional statistical models, but the parameters also have a more natural or explicit interpretation of the relationships present in these systems. Working with these models therefore improves our understanding of the processes they describe and can offer decision making support. Many of these systems depend on parameters that are not directly measurable or, when they are, it is often either invasive or costly to do so. Consequently, statistical inference is a key factor for dealing with these systems.

There is a barrier, however. These systems are modelled in derivative space, but are observed in solution space i.e. the mathematical models describe the changes in the systems, not the output of them at a given point. There is also usually no closed-form solution to these equations. Parameter inference therefore typically involves repeatedly computing solutions numerically to compare with the observed data, which quickly becomes computationally onerous and impractical for use in practice.

There are a number of ways this can be addressed and this talk will focus on a few. Gradient matching, an approach which avoids numerically solving the equations completely, will be discussed, along with the shortcomings of this method. In short, it is fast and adaptable, but can be temperamental and somewhat frustrating to employ without a lot of experience. Emulation will also be covered, a method that is becoming increasingly popular in recent times. The idea behind this approach is to first numerically solve the system many times ahead of seeing data. These solutions are interpolated, as a function of the parameters, and the interpolant's predictions replace the need for further numerical solving when it comes time to inferring parameters from observed data. The approach requires substantial overhead initially, but once the emulator is trained, prediction is very fast, making it suitable for real-time applications.

This presentation will also cover different interpolation strategies, with a focus on Gaussian processes. Applications consist of recent work from Systems Biology (mainly), as well as from Earth Sciences.