On Engle-Granger Cointegration Using Treed Gaussian Processes

Abstract

Over the last decade or so, cointegration has emerged as arguably the state-of-the-art in terms of removing Environmental and Operational Variations (EOVs) from structural health monitoring (SHM) data. When data channels share common trends which can be removed by linear projection, the Johansen procedure, a maximum-likelihood approach developed within the field of econometrics, is provably optimal. Unfortunately, SHM problems can present where the trends occupy a nonlinear submanifold of the feature space, and in this case, linear cointegration/projection fails. It is still possible to make progress in this case by moving to the older Engle-Granger approach to cointegration, where one linearly regresses one of the feature space variables on the others; nonlinear cointegration is then ‘simply’ the application of an appropriate nonlinear regressor. Over the years, a number of nonlinear regression algorithms have been applied, motivated by machine learning or evolutionary computation; each with pros and cons. The aim of the current paper is to demonstrate an approach based on Treed Gaussian Processes (TGP); the advantage being that the algorithm allows switching between cointegration models in different parts of the feature space. Examination of the switching points can provide insight into the physical processes driving the nonlinearity. The approach is demonstrated here on the well-known Z24 Bridge data set, where the ambient temperature drives EOVs which cannot be removed by linear methods.