NoLAW: A Recursive Non-Linear Adaptive Wiener Filter for Time Series Smoothing

Smoothing is a foundational operation in time series analysis, often used to mitigate the effects of noise and outliers prior to visualization, modeling, or classification. However, traditional filters such as Gaussian smoothing, exponential methods, and the classical Wiener filter typically assume linearity and stationarity, which limits their effectiveness in capturing local non-linearities and abrupt changes. In this work, we introduce NoLAW (Non-Linear Adaptive Wiener), a recursive, higher-order extension of the Wiener filter that models the clean signal as a local polynomial function of the noisy observations. Designed under the assumption of additive, zero-mean Gaussian noise, NoLAW is especially effective in applications where the goal is robust signal reconstruction in the presence of stochastic perturbations. Extensive experiments across 20 real-world datasets show that NoLAW consistently outperforms classical smoothing techniques in terms of denoising accuracy, measured by the Mean Absolute Percentage Error (MAPE). Despite its higher-order formulation, NoLAW retains linear computational complexity with respect to series length, although practical runtime may grow with filter order. We also discuss the trade-offs involved in filtering high-frequency components that may carry structural or predictive information, emphasizing that NoLAW is best suited as a preprocessing tool when the primary objective is signal recovery, feature extraction or classification. These results position NoLAW as a flexible and scalable alternative for denoising in domains such as sensor analysis, biomedical signal processing, and exploratory time series analysis.