Automatic denoising and differentiation based on Savitzky-Golay filtering and Homogeneous Differentiators for attractor reconstruction via differential embedding

Differential embedding methods aim to reconstruct attractors of dynamical systems from noisy measured time series, but require accurate estimates of signal derivatives. We introduce SHADED (Savitzky-Golay and Homogeneous-differentiator based Automatic DEnoising and Differentiation), a novel methodology for denoising and estimation of derivatives up to an arbitrary order, which enables attractor reconstruction via differential embedding from noisy time series data. Homogeneous 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 manual tuning that may lead to inaccurate reconstruction, and can also incorporate prior knowledge, if available, thereby yielding a flexible tool for data-driven numerical differentiation of noisy signals. 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. 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: across all these case studies, 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), without the need for manual parameter tuning.