Building a robust Data Science toolset via Computational Rough Paths: Localised Regression based on the Signature Method

本项目的背景是，粗糙路径理论提供了一种方便有效的方法来描述流数据，同时消除了数据采样过程中产生的噪声。作为一种工具，它似乎特别有价值，可以与数据科学提供的快速发展的工具集结合使用。当底层数据是复杂的、多模式的和不断变化的，但不是静态的或定期抽样的时候，它特别有吸引力。这种方法的主要好处是，由签名提供的变换消除了通常会给学习过程带来严重困难的无限维对称群。签名方法代表了一种从数据中提取特征特征的非参数方法。因此，这种方法允许通过将数据中包含的信息转换为一组基本特征来总结数据中包含的信息，并创建一个有利的和有希望的框架来执行机器学习任务。最大和最强大的银行之一目前正在使用这种方法来重新考虑其衍生品投资组合的定价程序，最近如(Arrias，2018)所示。这种方法也被用于精神病学，分析自我报告的情绪，从而区分诊断小组，如Arrias，Kate，Goodwin，&Lyons，2017所述。此外，手机中使用的将手指动作翻译成汉字的技术是使用签名方法(泽成、增辉、连文、子勇和舒业，2016)开发的。我自己的专业知识将数学基础和强大的计算能力结合在一起。我的目标是发展这些最初的优势，以进一步促进粗糙路径理论在数据科学中的有效使用。在许多问题中，人们想要根据大量的个人历史来预测个人的结果。在许多这样的例子中，没有自然的相似性度量。这种方法的优点之一是可以避免过早引入指标。因此，我将开始这个项目，试图建立稳健的原则方法，以执行本地化回归使用中等维度的签名。如果我能开发出一种稳健的数学原则性方法，并与SCRICKIT-LEARN和TensorFlow的包相平衡，那么这将是一个很好的个人成果。这个项目属于EPSRC数学科学研究领域。

The context of this project is that Rough Paths Theory provides a convenient and effective way to describe streamed data while dropping the noise generated throughout the data sampling process. It seems particularly valuable as a tool to be used in conjunction with the rapidly developing toolset provided by Data Science. It is particularly attractive when the underlying data is complex, multimodal and evolving but not stationary or regularly sampled. The primary benefit of the approach is that the transform provided by the Signature removes an infinite dimensional group of symmetries that would often cause profound difficulties for the learning process. The Signature Method represents a non-parametric way for extracting characteristic features from data. Thus, this approach allows to summarise information contained in the data by transforming it into a set of essential features and create a favourable and promising framework to perform Machine Learning tasks.The applications of this methodology are widespread. One of the biggest and strongest banks is currently using this approach to re-think the pricing procedure of their derivatives portfolio as recently shown in (Arribas, 2018). The method has also been used in psychiatry to analyse self-reported mood and consequently separate diagnostic groups, as reported in (Arribas, Kate, Goodwin, & Lyons, 2017). Furthermore, technology used in mobile phones to translate finger movements into Chinese characters has been developed using the Signature Method ((Zecheng, Zenghui, Lianwen, Ziyong, & Shuye, 2016)).My own expertise combines mathematical foundations with a strong ability to compute. My goal is to develop those initial strengths to further progress the effective use of Rough Paths Theory in Data Science. There are many problems where one would like to predict the outcome for an individual based on a large collection of histories of individuals. In many of these examples there is no natural metric of similarity. One of the advantages of this approach is that one can avoid the introduction of metrics prematurely. Therefore, I will start this project trying to build robust principle ways to perform Localised Regression using Signatures in moderate dimensions. If I can develop a robust mathematically principled approach, balanced with packages for scikit-learn and TensorFlow then this would be a great personal outcome.This project falls within the EPSRC Mathematical sciences research area.