Machine learning applications of controlled differential equations to streamed data

Thesis

The amalgamation of rough path theory and machine learning for sequential data has been a topic of increasing interest over the last ten or so years. The unity of these two subject areas is a natural one: rough path theory provides us with the language to describe the solutions to differential equations driven by multidimensional (and potentially highly irregular) signals, and machine learning provides the tools to learn such solutions from data.

The central aim of rough path theory is to provide a general mathematical framework for answering questions about the effect a stream of data can have on a system. One common example of such data is a time series, pervasive in all areas of life (and will be the type of streams we consider most frequently in this thesis); as such, framing problems in the language of rough paths provides us with models that have genuine utility in the real-world.

The aim of this thesis is to provide an accessible introduction to the field of rough path theory for application in machine learning, and then to provide an account of recent and effective contributions that further connect the two fields.

Topics covered in this thesis include: neural controlled differential equations (neural CDEs) -- extensions of neural ordinary differential equation that can incorporate changes in external data processes; neural rough differential equations (neural RDEs) -- a rough path extension to neural CDEs that leads to benefits for long or high-frequency time series; the generalised signature method -- a collection of feature extraction techniques for multivariate time series; and finally real-world applications of the signature method on sepsis and stress detection.

Authors: Morrill, J

• DOI: 10.5287/ora-xqdwzxq4e
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: neural cdes; neural odes; signatures; sepsis; time series; medical time series; rough paths