用签名核改进分布对比，提升路径数据的统计检验能力
Signature Maximum Mean Discrepancy Two-Sample Statistical Tests
- 引入签名核扩展MMD，用于路径空间分布对比
- 发现小样本下易出现第二类错误，误判路径来自同一过程
- 提出缓解策略，增强实际应用中的检验可靠性
最大均值差异（MMD）是机器学习中广泛使用的分布比较工具，近年来在有限维分布对比中表现优异。通过引入签名核，MMD可拓展至路径空间分布的比较，形成签名MMD（sig-MMD），从而定义路径分布间的度量。与原始MMD作为两样本检验统计量的应用类似，sig-MMD可用于判断两组路径是否来自同一随机过程。本文系统探讨了sig-MMD在实际应用中的潜力与挑战，介绍了其理论基础，并提供可复现的实用示例。我们揭示了在数据量有限时可能出现第二类错误——错误地认为样本来自相同过程。随后提出相应技术以降低此类误差，提升检验的有效性。
原文摘要 · Abstract (English)
Maximum Mean Discrepancy (MMD) is a widely used concept in machine learning research which has gained popularity in recent years as a highly effective tool for comparing (finite-dimensional) distributions. Since it is designed as a kernel-based method, the MMD can be extended to path space valued distributions using the signature kernel. The resulting signature MMD (sig-MMD) can be used to define a metric between distributions on path space. Similarly to the original use case of the MMD as a test statistic within a two-sample testing framework, the sig-MMD can be applied to determine if two sets of paths are drawn from the same stochastic process. This work is dedicated to understanding the possibilities and challenges associated with applying the sig-MMD as a statistical tool in practice. We introduce and explain the sig-MMD, and provide easily accessible and verifiable examples for its practical use. We present examples that can lead to Type 2 errors in the hypothesis test, falsely indicating that samples have been drawn from the same underlying process (which generally occurs in a limited data setting). We then present techniques to mitigate the occurrence of this type of error.
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