Extending Rough Path theory to applications in animal movement modelling

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Changes in animal behaviour can be indicative of potentially important alterations in the ecosystem or in the environment. For example, the behavioural patterns of different animals can be inferred, from movement, location, and other auxiliary data. Modes of behaviour can include feeding, moving, and resting, and their optimal number strongly depends on the animal. Measurement technologies include GPS transmitters, tri-axial accelerometers, body sensors and auxiliary environmental information like wind speed, or temperature. It is common in the literature to use machine learning approaches to partition daily movement into a finite number of behavioural states, and then use Hidden Markov Models (HMMs) to model the interactions of these states. One challenge is the identification of an appropriate number of states and the need to manually calibrate this for each species. During the last decade, the signature method has been applied successfully to complex time series data and has been successful in applications such as Chinese handwriting recognition, financial data streams, and sepsis prognosis. There is good reason to suspect that it could be useful in ecology too; it can naturally concatenate different sources of data and analyse them jointly in a geometric fashion. This is one strand of the project, we aim to determine under which conditions the signature methods can be used to extract information and knowledge from animal movement data, while keeping computational times to acceptable levels. Following on from this, we would wish to utilise the signature as a tool to establish behavioural changes in animals, from multiple forms of data, then use in a probabilistic framework useful for practitioners, such as a HMM approach. At the time of writing, this would represent a novel approach for the modelling of animal paths. A second strand to the project focuses on a theoretical problem involving elements of rough path theory. Here, we would like to extend the field of statistical inference for differential equations driven by a rough path and use these equations to answer ecological problems. It is common in animal movement modelling to assume an Ornstein-Uhlenbeck (OU) type equation for the dynamics of an animal's velocity in its' home location, however we hope that by using a rough-type differential equation, we can specify the dynamics more directly on the animal's position. The objective here would be to build a robust algorithm for parameter estimation and make improvements to algorithms already existing for solving these types of equations. Rough path theory has not yet been applied to the field of ecology, where it could have an important impact. It aims to build a more rigorous framework for more irregular paths, such as those seen in GPS and accelerometer datasets. This studentship is funded by EPSRC, who's Impact Acceleration Accounts aims to "deliver new innovations that benefit society, form successful businesses, and offer economic and social returns built on the UK's fundamental engineering and physical sciences research base." This project aligns with this mission statement, its objective being to generate new understanding about how animals operate, so that we can hopefully aid conservationists to make clearer and more accurate decisions.