SHADED: Automatic denoising and differentiation based on Savitzky-Golay filtering and Homogeneous Differentiators for attractor reconstruction

Differential embedding methods aim to reconstruct attractors of dynamical systems from noisy measured time series, but require accurate estimates of signal derivatives, which are difficult to obtain in the presence of noise. We introduce SHADED ( S avitzky-Golay and H omogeneous-differentiator based A utomatic DE noising and D ifferentiation), a novel methodology for effective denoising and estimation of derivatives up to an arbitrary order, which enables attractor reconstruction via differential embedding from noisy time series data. Homogeneous sliding-mode Differentiators (HD) guarantee finite-time derivative estimates in the presence of noise, while subsequent Savitzky-Golay (SG) filtering attenuates chattering. Crucially, SHADED extracts all parameters required for application of both HD and SG automatically from the data, without requiring time-consuming manual tuning that may lead to inaccurate reconstruction, and can also incorporate prior knowledge, if available, thereby yielding a flexible and effective tool for data-driven numerical differentiation of noisy signals; the obtained derivative estimates are valuable per se and instrumental for attractor reconstruction. The obtained differential embeddings can reveal features of the underlying dynamics that are useful, e.g. , for system identification, pattern recognition and discrimination between dynamic regimes; the latter application is particularly important in biomedical settings, to help distinguish between different physiological and pathological states of a patient. We demonstrate the efficacy of SHADED by testing it on computational neuroscience models, LTspice-simulated chaotic electronic circuits, and photoplethysmography and arterial blood pressure experimental recordings (where the base signal and noise are unknown): across all these case studies, under varying signal-to-noise ratios and different noise types, SHADED produces accurate derivative estimates and accurate attractor reconstructions via differential embedding (whenever a ground truth is available) or geometrically coherent and reproducible reconstructions consistent with the expected dynamics (in the absence of a ground truth), with short computation times, and without the need for manual parameter tuning .