On reconstructing high derivatives of noisy time-series with confidence intervals
Mazen Alamir
TL;DR
Reconstructing high-order derivatives from noisy time-series is challenging; the authors introduce a bandwidth- and noise-aware dictionary of linear operators $A_{(j, ell,d)}$ learned by cross-validation, and apply moving-window averaging to produce $oldsymbol{ ilde s}^{[d]}$ with a corresponding confidence interval $oldsymbol{ ilde oldsymbol ext{sigma}}^{[d]}$. The method demonstrates higher accuracy than Kalman-filtering, spectral derivation, Savitzky–Golay, and implicit sliding-mode approaches across derivative orders up to $d_{max}=4$, and remains effective for extensions to higher derivatives. Core components include a bandwidth-limited basis $B$, projection operators $oldsymbol{ ilde s}(oldsymbol{ar ext{Ω}})$ and $oldsymbol{ ilde ext{P}}$, and a design pulsation grid $ar{ ext{Ω}}_{design}$ used to populate the dictionary. The framework is particularly valuable for offline continuous-time identification and normality characterization, while maintaining potential for online use on shorter windows due to favorable computation times.
Abstract
Reconstructing high derivatives of noisy measurements is an important step in many control, identification and diagnosis problems. In this paper, a heuristic is proposed to address this challenging issue. The framework is based on a dictionary of identified models indexed by the bandwidth, the noise level and the required degrees of derivation. Each model in the dictionary is identified via cross-validation using tailored learning data. It is also shown that the proposed approach provides heuristically defined confidence intervals on the resulting estimation. The performance of the framework is compared to the state-of-the-art available algorithms showing noticeably higher accuracy. Although the results are shown for up to the 4-th derivative, higher derivation orders can be used with comparable results.