EPSRC Reference: 
EP/W00707X/1 
Title: 
Pathtosignature isometries with applications to modelling the longterm dynamics of complex systems 
Principal Investigator: 
Papavasiliou, Dr A 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Statistics 
Organisation: 
University of Warwick 
Scheme: 
Standard Research  NR1 
Starts: 
01 April 2022 
Ends: 
31 March 2023 
Value (£): 
51,276

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Almost all natural and manmade processes behave differently at different time scales. For example, if we plot the temperature in Coventry on a minutebyminute 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, highdimensional dynamical systems. However, such systems can be extremely inefficient, which is why coarsegrained models, describing the longterm dynamics, are often used instead.
Our aim is to address the problem of fitting coarsegrained models to data. The main challenge is that coarsegrained models, while successful in providing a good approximation of the longterm dynamics, often fail to capture the finescale properties of the system. Typically, coarsegrained models exhibit a rougher behaviour (e.g. equivalent to Brownian motion) than the full complex systems, which in the very finescale 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 coarsegrained 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 purposebuilt 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 coI 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 coI and his collaborator have recently managed to achieve this in a class of twodimensional 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 signaturebased 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 "scaleextraction" 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 coarsegrained models.

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Organisation Website: 
http://www.warwick.ac.uk 