Adaptive Derivative Estimation via Stein's Unbiased Risk

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.