Adaptive Derivative Estimation via Stein's Unbiased Risk
基于Stein无偏风险的自适应导数估计
Yonathan Murin, Ali Ozer Ercan
AI总结 提出SURDE方法,通过Stein无偏风险评估候选滤波器长度并软组合输出,实现因果FIR导数滤波的噪声-偏差权衡,证明极小极大最优性,在仿真和真实数据上优于ICI和AWVE。
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- Submitted to IEEE Transactions on Signal Processing, 23 pages
从含噪采样数据中估计导数对于控制、人机交互和生物医学工程至关重要。因果FIR导数滤波器为此提供了一种自然方法,但其性能取决于滤波器长度。短滤波器放大噪声,长滤波器引入平滑偏差。我们提出SURDE(SURE导数估计器),通过在一组候选长度上评估基于Stein无偏风险估计(SURE)的数据驱动代价,并利用指数加权软组合它们的输出,在每个时间步解决这一权衡。我们证明了软组合估计器的极小极大最优预言不等式,并据此推导出最优加权温度的闭式解。因此,SURDE唯一的调参参数是噪声方差。通过数值模拟,我们展示了SURDE在一阶导数估计中始终优于替代自适应方法(置信区间交集(ICI)规则和自适应窗口速度估计器(AWVE))。我们进一步表明SURDE对噪声方差误设具有鲁棒性(在4倍范围内性能下降9%),并且在真实数据场景(EuRoC MAV数据集)中也优于ICI和AWVE。SURDE是因果的、计算轻量,且仅需噪声方差的粗略估计。
Estimating derivatives from noisy sampled data is fundamental to control, human--computer interaction, and biomedical engineering. Causal FIR derivative filters offer a natural approach for this challenge, yet their performance depend on their length. While short filters amplify noise, long filters introduce smoothing bias. We present SURDE (SURE Derivative Estimator), which addresses this tradeoff at each time step by evaluating a data-driven cost derived from Stein's Unbiased Risk Estimator (SURE) across a bank of candidate lengths and soft-combining their outputs via exponential weighting. We prove a minimax-optimal oracle inequality for the soft-combined estimator and use it to derive the optimal weighting temperature in closed form. Thus, the only tuning parameter for SURDE is the noise variance. Via numerical simulations we show that SURDE consistently outperforms alternative adaptive methods (the Intersection of Confidence Intervals (ICI) rule and the Adaptive Windowing Velocity Estimator (AWVE)) for first-derivative estimation. We further show that \surede{} is robust to noise-variance misspecification (9\% degradation over a $4\times$ range), and that it is superior to ICI and AWVE also over real data scenarios (the EuRoC MAV dataset). SURDE is causal, computationally light, and requires only a rough estimate of the noise variance.