Honours Project Report: Functional Linear Regression with Truncated Path Signatures

This project report explores the concept of incorporating path signatures into functional linear regression to address the limitations faced by traditional regression methods when dealing with high-dimensional, sequential data. Replicating Adeline Fermanian's 2022 work, which utilised the signature linear model as an alternative to basis expansion techniques, this project implements her methodology on the original dataset used in her paper and extends it to a high-dimensional, real-world dataset. The theoretical component covers key concepts, including functional data analysis and the signature method, with a focus on how path signatures capture higher-order interactions within functional data. Empirically, we compare traditional regression models, specifically those using Fourier, B-spline, and functional principal components basis expansions, with the signature linear model applied to two datasets: the Air Quality dataset and the Appliance Energy Prediction dataset. Our results indicate that while the traditional models perform well in lower-dimensional cases, the signature linear model demonstrates superior performance in high-dimensional settings. This project also assesses Fermanian's truncation order estimation method, highlighting its impact on both computational efficiency and performance. Ultimately, the signature linear model presents itself as an effective approach for high-dimensional functional data, offering a compelling alternative to traditional regression techniques.