Extending Rough Path theory to applications in animal movement modelling

动物行为的变化可能预示着生态系统或环境中潜在的重要变化。例如，可以从运动、位置和其他辅助数据中推断不同动物的行为模式。行为模式可以包括进食、移动和休息，它们的最佳数量强烈依赖于动物。测量技术包括GPS发射器、三轴加速计、身体传感器和辅助环境信息，如风速或温度。在文献中，通常使用机器学习方法将日常运动划分为有限数量的行为状态，然后使用隐马尔可夫模型(HMM)来建模这些状态的交互。一个挑战是确定适当数量的状态，并需要为每个物种手动校准这一数量。在过去的十年中，签名方法已经成功地应用于复杂的时间序列数据，并在中文手写识别、金融数据流和脓毒症预测等应用中取得了成功。有充分的理由怀疑它在生态学上也是有用的；它可以自然地连接不同的数据来源，并以几何方式联合分析它们。这是该项目的一部分，我们的目标是确定在哪些条件下可以使用签名方法从动物运动数据中提取信息和知识，同时将计算时间保持在可接受的水平。接下来，我们希望利用签名作为一种工具，从多种形式的数据中建立动物的行为变化，然后在对实践者有用的概率框架中使用，例如HMM方法。在撰写本文时，这将代表着一种对动物路径进行建模的新方法。该项目的第二部分集中在涉及粗糙路径理论元素的理论问题上。在这里，我们想扩展由粗糙路径驱动的微分方程的统计推断的领域，并用这些方程来回答生态问题。在动物运动模型中，通常假定动物在其原点的速度动力学为Ornstein-Uhlenbeck(OU)型方程，然而我们希望通过使用粗略类型的微分方程，我们可以更直接地描述动物的位置上的动力学。这里的目标是建立一个稳健的参数估计算法，并对现有的求解这些类型的方程的算法进行改进。粗糙路径理论还没有被应用到生态学领域，在那里它可能会产生重要的影响。它的目标是为更不规则的路径建立一个更严格的框架，比如GPS和加速度计数据集中看到的那些路径。该项目由EPSRC资助，该机构的Impact Acceleration账户旨在“提供有益于社会的新创新，形成成功的企业，并提供建立在英国基础工程和物理科学研究基础上的经济和社会回报。”这个项目与这一使命宣言相一致，它的目标是对动物如何运作产生新的理解，这样我们就有希望帮助自然资源保护者做出更清晰、更准确的决定。

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.