Terry Lyons

Utilizing classical results on the structure of Hopf algebras, we develop a novel approach for the construction of cubature formulae on Wiener space based on unshuffle expansions. We demonstrate the effectiveness of this approach by constructing the first degree-7 cubature formula on d-dimensional Wiener space with drift in the sense of Lyons and Victoir (Lyons & Victoir 2004 Stoch. Anal. Appl. Math. Finance460, 169–198) which is explicit as a function of an underlying Gaussian cubature. The support of our degree-7 formula is significantly smaller than that of currently implemented or proposed constructions.

We frame novelty detection on path space as a hypothesis testing problem with signature-based test statistics. Using transportation-cost inequalities of Gasteratos and Jacquier (2023), we obtain tail bounds for false positive rates that extend beyond Gaussian measures to laws of RDE solutions with smooth bounded vector fields, yielding estimates of quantiles and p-values. Exploiting the shuffle product, we derive exact formulae for smooth surrogates of conditional value-at-risk (CVaR) in terms of expected signatures, leading to new one-class SVM algorithms optimising smooth CVaR objectives. We then establish lower bounds on type-$\mathrm{II}$ error for alternatives with finite first moment, giving general power bounds when the reference measure and the alternative are absolutely continuous with respect to each other. Finally, we evaluate numerically the type-$\mathrm{I}$ error and statistical power of signature-based test statistic, using synthetic anomalous diffusion data and real-world molecular biology data.

The DECOVID database contains harmonized pseudonymized electronic health record (EHR) data on all adult (≥18 years old) patients presenting to two large, digitally mature centers in the United Kingdom between 1 January 2020 and 28 February 2021, with follow-up until at least 28 March 2021. The database was originally developed to support the COVID-19 response but is now available via the PIONEER data hub for researchers to explore a wide range of research questions, including exploratory analyses, risk factor assessment, prediction modeling, and comparative effectiveness studies. Raw data were extracted from local EHRs and transformed into a standardized form (Observational Health Data Sciences and Informatics-Common Data Model version 5.3.1). The database includes 165,420 patients across 256,804 hospital presentations. For these patients, highly granular data are available, including patient demographics, longitudinal vital signs, physiology, treatments, laboratory findings, clinical diagnoses, and outcomes. There are 10,030 patients with COVID-19, of whom 1472 died in hospital.

Signature-based techniques give mathematical insight into the interactions between complex streams of evolving data. These insights can be quite naturally translated into numerical approaches to understanding streamed data, and perhaps because of their mathematical precision, have proved useful in analysing streamed data in situations where the data is irregular, and not stationary, and the dimension of the data and the sample sizes are both moderate. Understanding streamed multi-modal data is exponential: a word in n letters from an alphabet of size d can be any one of d^{n} messages. Signatures provide a “lossy compression” of the information contained within such a stream by filtering out the parameterisation noise. More concretely, suppose we have a time series with 3 channels and N samples. There are 1 + 3N + \frac{3N(3N+1)}{2}= 1 + \frac{9}{2}N(N+1) linearly independent quadratic polynomials defined on the time series. But the signature of this time series truncated to depth 2 only consists of 1 + 3 + 3^{2} = 13 components which is, in particular, independent of the number of samples N . However, whilst the independence of the number of samples N has removed an exponential amount of noise, the dependence on the number of channels to the power of the depth ensures that an exponential amount of information remains. This survey aims to stay in the domain where that exponential scaling can be managed directly. Scalability issues are an important challenge in many problems but would require another survey article and further ideas. This survey describes a range of contexts where the data sets are small and the existence of small sets of context free and principled features can be used effectively. The mathematical nature of the tools can make their use intimidating to non-mathematicians. The examples presented in this article are intended to bridge this communication gap and provide tractable working examples drawn from the machine learning context. Notebooks are available online for several of these examples. This survey builds on the earlier paper of Ilya Chevryev and Andrey Kormilitzin which had broadly similar aims at an earlier point in the development of this machinery. This article illustrates how the theoretical insights offered by signatures are simply realised in the analysis of application data in a way that is largely agnostic to the data type. Larger and more complex problems would expect to address scalability issues and draw on a wider range of data science techniques. The article starts with a brief discussion of background material related to machine learning and signatures. This discussion fixes notation and terminology whilst simplifying the dependencies, but these background sections are not a substitute for the extensive literature they draw from. Hopefully, by working some of the examples, the reader will find access to useful and simple to deploy tools; tools that are moderately effective in analysing longitudinal data that is complex and irregular in contexts where massive machine learning is not a possibility.

Rough path theory is a branch of mathematics arising out of stochastic analysis. One of the main tools of rough path analysis is the signature, which captures the evolution of an unparametrised path including the order in which events occur. This turns out to be a useful tool in data science applications involving sequential data. RoughPy is our new Python package that aims change the way we think about sequential streamed data, by viewing it through the lens of rough paths. In RoughPy, data is wrapped in a stream object which can be composed and queried to obtain signatures that can be used in analysis. It also provides a platform for further exploration of the connections between rough path theory and data science.