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

几乎所有的自然和人为过程在不同的时间尺度上表现不同。例如，如果我们绘制考文垂一小时内每分钟的温度图，我们会看到微小的平滑变化，没有明显的趋势。另一方面，我们预计一年内的周气温会出现较大的波动和季节性趋势。为了捕捉所有尺度上温度变化的行为，我们需要使用复杂的高维动力系统。然而，这样的系统可能效率极低，这就是为什么经常使用描述长期动态的粗粒度模型的原因。我们的目标是解决将粗粒度模型拟合到数据中的问题。主要的挑战是，粗粒度模型虽然成功地提供了长期动态的良好近似，但往往不能捕获系统的精细尺度属性。通常，粗粒度模型比完整的复杂系统表现出更粗糙的行为（例如，相当于布朗运动），后者在非常精细的尺度上通常是有界变化。因此，直接使用标准估计器可能会导致错误的结果，除非仔细处理模型和数据之间的不匹配。当前方法的主要限制是它依赖于尺度分离参数的显式知识，这允许我们使用与粗粒度模型兼容的尺度数据。然而，这些信息通常是不可用的。我们将基于一个快速发展的工具构建一个新的估计器，称为粗糙路径特征，这是一个专门为具有多尺度行为的随机模型构建的工具。粗糙路径签名是一个序列，其中第一项描述模型在平滑尺度上的行为，而第二项看到更精细的布朗尺度，以此类推。签名的极限渐近捕获了模型在所有尺度上的行为，并且可以通过适当的归一化在单个尺度上提取行为。我们的目标是通过隐式地利用数据所显示的尺度分离，确定将导致提取布朗尺度的归一化，从而提供扩散系数的估计器。这个估计器的关键理论基础是co-I和他的合作者最近发现的一个公式，用于从签名中提取平滑和布朗尺度上的路径行为。该项目的第二个目标是将这些结果扩展到有界变异尺度。一个根本的困难是如何超越连续导数的假设。然而，合著者和他的合作者最近在一类二维模型中成功地实现了这一点。我们将在这一发现的基础上，在第二个目标中展示从签名中提取有界变化行为的一般公式。拟议中的研究是迈向更大规模研究计划的第一步。我们的方法的主要优点之一是基于签名的估计器应该自然地推广到所有尺度，因此，更一般的模型。为了使签名充分发展成为多尺度建模的标准工具，我们必须将这种“尺度提取”结果扩展到所有尺度。这将需要一种系统的方法来识别适当的归一化常数，无论是在精确模型还是粗粒度模型的上下文中。

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.