Rational Krylov Methods and Machine Learning Approaches to Time Series Forecasting

Abstract

This thesis presents new results and insights for rational Krylov methods, a popular class of linear algebra algorithms which are currently used with great success for model order reduction, the solution of linear and nonlinear eigenvalue problems, and nonlinear least squares approximation, to name just a few applications. It also makes a connection between block Krylov methods and the autoregressive modelling of multivariate time series. This thesis also introduces a new symbolic time series representation and reports on experiments of its use to improve the time series forecasting performance of recurrent neural networks. We start by deriving formulae for converting between barycentric, Newton and RKFUN representations of rational interpolants. We show applications of these conversion for the solution of nonlinear eigenvalue problems. We then study block rational Arnoldi decompositions associated with the block version of the rational Arnoldi method and prove an implicit Q theorem. We study premature breakdowns of the method and relate them to nonlinear eigenvalue problems. We provide two different deflation strategies to handle an unavoidable breakdown and introduce the RKFUNB format for representing rational matrix-valued functions. We demonstrate how autoregressive models can be fit using the RKFUN format and vector autoregressive models can be fit using the RKFUNB format. We introduce a new symbolic representation of time series called ABBA, and we demonstrate how ABBA is able to better preserve the essential shape information than existing symbolic representations. We list applications of ABBA including anomaly detection and motif discovery. Finally we demonstrate how the ABBA symbolic representation can be combined with long short-term memory networks for time series forecasting. The symbolic representation speeds up training times and reduces the sensitivity to certain hyper parameters.

Date of Award	26 Mar 2020
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Martin Lotz (Co Supervisor) & Stefan Guettel (Main Supervisor)