Path-to-signature isometries with applications to modelling the long-term dynamics of complex systems

Almost all natural and man-made processes behave differently at different time scales. For example, if we plot the temperature in Coventry on a minute-by-minute scale over an hour we would expect to see small smooth changes with no clear trend. On the other hand, we would expect the weekly temperature over a year to exhibit large fluctuations and a seasonal trend. To capture the behaviour of temperature changes on all scales, we would need to use complex, high-dimensional dynamical systems. However, such systems can be extremely inefficient, which is why coarse-grained models, describing the long-term dynamics, are often used instead.

Our aim is to address the problem of fitting coarse-grained models to data. The main challenge is that coarse-grained models, while successful in providing a good approximation of the long-term dynamics, often fail to capture the fine-scale properties of the system. Typically, coarse-grained models exhibit a rougher behaviour (e.g. equivalent to Brownian motion) than the full complex systems, which in the very fine-scale are usually of bounded variation. As a result, direct use of standard estimators can lead to wrong results, unless the mismatch between model and data is carefully addressed. The main limitation of current methodology is that it depends on explicit knowledge of the scale separation parameter, which allows us to use data in a scale compatible with the coarse-grained model. However, this information is usually not available.

We will construct a new estimator based on a rapidly developing tool known as the rough path signature, which is a purpose-built tool for stochastic models with multiscale behaviour. The rough path signature is a sequence where the first term describes the behaviour of the model at a smooth scale, while the second term sees the finer Brownian scale, and so on. The limiting asymptotics of the signature capture the behaviour of the model at all scales, and it is possible to extract the behaviour in a single scale by appropriate normalisation. Our goal will be to identify the normalization that will lead to the extraction of the Brownian scale, thus providing an estimator for the diffusion coefficient, by making implicit use of the scale separation exhibited by the data.

The key theoretical underpinning of this estimator is a recently discovered formula by the co-I and his collaborator for extracting the behaviour of a path at the smooth and Brownian scales from the signature. The second objective of the project is to extend these results to the bounded variation scale. A fundamental difficulty has been how to move beyond the assumption of continuous derivative. However, the co-I and his collaborator have recently managed to achieve this in a class of two-dimensional models. We will build on this discovery to show a general formula for extracting the bounded variation behaviour from the signature in the second objective.

The proposed research is a first step towards a much larger research programme. One of the main advantages of our approach is that signature-based estimators should naturally generalise to all scales and, consequently, more general models. In order to fully develop the signature as a standard tool in multiscale modelling, we must extend this "scale-extraction" result to all scales. This will require a systematic methodology for the identification of the appropriate normalisation constant, both in the context of exact models and coarse-grained models.