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Personal profile

Research Interests

  • Topological data analysis
  • Time series classification

Biography

Colin Stephen teaches computer science in the School of Computing, Electronics and Mathematics, and researches time series classification methods in the Research Centre for Computational Science and Mathematical Modelling at Coventry University. Before this he developed a successful career in software engineering and managing software development teams.

His teaching expertise includes machine learning, software development, and the theoretical foundations of computer science. Current research focuses on the development and application of novel methods for sensor data analysis, using geometric and topological features.

As part of the EnergyREV Research Consortium, in collaboration with stakeholders from UK government, research funders, and the UK energy industry, Colin recently developed the Catalogue of Projects on Energy Data (CoPED), a web platform that collates, analyses, and visualises energy project metadata from across the UK.

PhD Project

Title

Horizontal Visibility Complexes and Topological Merge Trees

Abstract

This thesis unifies two previously distinct mathematical structures, which are both used to represent and compare nonlinear time series in a growing number of scientific applications: the horizontal visibility graph (HVG) and the persistence diagram (PD) summary of persistent homology. Each has evolved independently as a way to extract features from data using invariants of simplicial structures, and each has its own body of literature. However, for ordered data, we show that HVGs and PDs correspond to different sides of the same coin. They respectively capture combinatorial and metric properties of a new structure we introduce and study: the discrete time series merge tree and its plane dual the horizontal visibility complex.

In particular we show that there exists a metric space HVC of weighted plane graphs, horizontal visibility complexes, which generalise HVGs and which are stable under perturbations to time series. The space has equivalent definitions based on horizontal visibility, on cellular partitions of the 2-sphere, and on rooted plane sublevel set merge trees over simple graphs. We use properties of HVC to derive efficient distributed algorithms for visibility graph construction and to propose an effective test for deterministic chaos using topological persistence diagrams.

The original authors of the horizontal visibility approach in 2010 admitted that finding the core of a formal theory for it would be a challenging research program likely to entangle dynamical systems and graph theory together. This thesis argues that horizontal visibility is best viewed as an entanglement between dynamical systems and metric ℝ-trees describing topological filtrations over path graphs. It shows why this is the case, and begins to explore some of the practical consequences of the connection between horizontal visibility and trees.

Supervisory Team

Matthew England, Vasile Palade, Fei He

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