Signature Moments to Characterize Laws of Stochastic Processes
The paper "Signature Moments to Characterize Laws of Stochastic Processes" by Ilya Chevyrev and Harald Oberhauser explores a sophisticated mathematical framework to tackle a recurring problem in the analysis of path-valued random variables - notably stochastic processes. The authors aim to extend the classical characterization of vector-valued random variables through moments to the domain of path-valued variables by using what they term "robust signature moments."
Key Contributions
The document establishes a metric based on the concept of maximum mean discrepancy (MMD) for the laws of stochastic processes. This metric is aligned with kernel methods, which allow for effective computation thanks to the use of the signature kernel. Notably, the signature kernel is a core feature, enabling the discretization and computational feasibility of these metrics.
Furthermore, this metric serves as the foundation for a non-parametric two-sample hypothesis test for stochastic processes. Therefore, it broadens the applicability of these methods across different stochastic processes, providing a robust statistical tool.
Robust Signature Moments
The authors aim to provide a robust feature space for path-valued data, allowing for effective characterization of such data through its probabilistic laws. The paper addresses two primary tasks: function inference on spaces of paths and inference about the probability measures from which data samples are drawn. They leverage a generalization of the Stone-Weierstrass theorem and propose a strategy for handling data's non-compactness nature. The approach serves also to address robustness concerns due to outliers, thus aligning with the notion of "robust statistics."
Theoretical Underpinnings
Central to the discussion is the signature of a path, which generalizes polynomial feature maps for paths. This paper extends to general function spaces, where infinite dimensionality necessitates a thoughtful approach to function approximation and measure characterization. Through the use of universal and characteristic feature maps, the authors claim to provide an adequate framework for this task. For example, a tensor normalization is introduced to bound the feature map while maintaining its algebraic properties.
Implications and Applications
The implications of this research are both theoretical and practical. From a theoretical standpoint, the results furnish a computational framework allowing for the characterization of a wide class of stochastic processes. Practically, the method results in measurable improvements over classical methods, as shown in empirical tests involving multivariate time series datasets across various domains.
The authors conclude that their robust signature kernel provides a valuable tool for non-parametric inference in high-dimensional contexts. They appeal for further exploration into parametric approaches, particularly within the scope of specific types of stochastic differential equations, potentially allowing semi-explicit expressions of signature moments.
Conclusion
Chevyrev and Oberhauser provide a compelling framework that advances the characteristic understanding of stochastic processes through their proposed robust signature moments and kernel methods. This paper is a significant contribution to the field of mathematical statistics and analysis, providing robust methodologies for characterizing and analyzing complex data structures without overly restrictive assumptions.