Time Series Forecasting with Graphical Structure

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Multivariate time series forecasting models are ubiquitous throughout the sciences, particularly in engineering and finance contexts. The most commonly used models consider the data as arbitrary and depend on the learned parameters to model interdependencies. In many real-world scenarios however, the underlying system has additional inherent structure which can be represented graphically and exploited to yield more accurate and interpretable predictions. Examples of this include disease propagation in epidemiology, traffic flow modelling and covariance modelling in finance. Traditional forecasting methods often struggle to effectively handle these types of problems due to the potentially very large number of parameters involved relative to the available data (as in classical autoregressive models) or are extremely computationally expensive to deploy (as in many deep learning solutions). By utilising the underlying structure of the data however, it is often possible to simultaneously avoid overfitting and reduce the computational complexity involved. Developing and deploying forecasting schemes which take advantage of this is a fledging and promising direction of research. Empirical similarity models build forecasts as a direct function of the previously observed data points most closely related to the test input. To do so, they consist of two fundamental components- a model of similarity and a means of combining similar observations to construct a forecast. They have several attractive properties which make them useful in practice: -they do not require any training and new data can be integrated on an ongoing basis at no additional cost -they are highly modular: the components can be freely interchanged even at forecast time -they are fast and highly interpretable in practice Empirical similarity methods have been deployed to great effect in financial contexts, particularly in volatility forecasting. We intend to adapt and develop these methods to better suit network structured data, in particular to the related problem of covariance forecasting. Covariance matrices of financial assets play a central role in modern asset management- accurate forecasting is critical for example to managing portfolio risk and pricing options. Despite this, the body of research on this topic is relatively modest, with simple vector heterogeneous autoregressive (VHAR) models still performing competitively with much more complex and computationally expensive state of the art models. We have seen that by exploiting the network structure of this problem, it is possible to attain superior results to VHAR models using an empirical similarity-based pipeline, while retaining a high degree of interpretability. There remains much to be done in this space: -the correct notion of similarity is central to the success of the pipeline, and this touches on the highly active field of network embedding techniques -similarly, the choice of construction function can have a significant impact on accuracy, a consideration scarcely explored in the literature to date -the non-stationary nature of the data means that the best results are often attained not through a single model but through diverse ensembles of models suited to different regimes; -the creation of dynamic ensembling schemes well-suited to time series data is another promising topic of research We intend to continue to expand on these observations with the aim of producing fast, interpretable forecasts suitable for deployment in practice. Due to its highly flexible modular structure, we hope that the resulting family of pipelines will be applicable not only to covariance forecasting problems but more broadly to any forecasting application with graphical structure. This project falls within the following EPSRC research areas: artificial intelligence technologies, statistics and applied probability.