无需调参的高阶在线微分器，可精确处理带噪多项式信号
HOUND: High-Order Universal Numerical Differentiator for a Parameter-free Polynomial Online Approximation
- 基于高阶非线性微分方程构建无参数微分系统
- 对带白噪声多项式信号误差趋近于零，一般情况误差有界
- 支持在线实时处理，无需数据积累或系数拟合
本文提出一种标量数值微分器，由任意高阶非线性微分方程组表示。我们推导出该系统的显式解，并证明在合理选择微分阶数时，对于含加性白噪声的多项式信号，误差可收敛至零；在更一般情况下，若最高估计导数有界，则误差也保持有界。该方法显著优势在于无需根据信号特性进行参数调优。我们还提出一种离散化方案，实现时间序列的累积平滑算法，支持在线处理，无需数据积累，且不需对数据拟合任何系数，同时解决插值与外推问题。
原文摘要 · Abstract (English)
This paper introduces a scalar numerical differentiator, represented as a system of nonlinear differential equations of any high order. We derive the explicit solution for this system and demonstrate that, with a suitable choice of differentiator order, the error converges to zero for polynomial signals with additive white noise. In more general cases, the error remains bounded, provided that the highest estimated derivative is also bounded. A notable advantage of this numerical differentiation method is that it does not require tuning parameters based on the specific characteristics of the signal being differentiated. We propose a discretization method for the equations that implements a cumulative smoothing algorithm for time series. This algorithm operates online, without the need for data accumulation, and it solves both interpolation and extrapolation problems without fitting any coefficients to the data.
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