基于路径签名的异常检测方法，可有效识别复杂动态系统中的异常行为。
Novelty detection on path space
- 用路径签名构造检验统计量，将异常检测转化为假设检验问题。
- 在非高斯路径数据上实现低误报率，给出精确的p值与分位数估计。
- 适用于生物分子轨迹等单类学习场景，对微小异常变化敏感。
我们将路径空间上的异常检测建模为基于签名的假设检验问题。利用Gasteratos与Jacquier（2023）的运输成本不等式，我们获得了在平滑有界向量场下随机微分方程解分布的尾部界，从而扩展了对非高斯测度的误报率控制，并可计算分位数与p值。通过引入交错积，我们推导出条件风险价值（CVaR）光滑近似的精确表达式，该表达式以期望签名表示，由此构建新的单类SVM算法，优化光滑的CVaR目标。进一步，我们在一阶矩有限的备择假设下建立了第二类错误的下界，当参考测度与备择测度绝对连续时，给出通用的检验功效边界。最后，通过合成异常扩散数据和真实分子生物学数据，数值评估了基于签名统计量的类型I误差与统计功效。
原文摘要 · Abstract (English)
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.
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