A Time-Series Feature-Extraction Methodology Based on Multiscale Overlapping Windows, Adaptive KDE, and Continuous Entropic and Information Functionals

We propose a new methodology to transform a time series into an ordered sequence of any entropic and information functionals, providing a novel tool for data analysis. To achieve this, a new algorithm has been designed to optimize the Probability Density Function (PDF) associated with a time signal in the context of non-parametric Kernel Density Estimation (KDE). We illustrate the applicability of this method for anomaly detection in time signals. Specifically, our approach combines a non-parametric kernel density estimator with overlapping windows of various scales. Regarding the parameters involved in the KDE, it is well-known that bandwidth tuning is crucial for the kernel density estimator. To optimize it for time-series data, we introduce an adaptive solution based on Jensen–Shannon divergence, which adjusts the bandwidth for each window length to balance overfitting and underfitting. This solution selects unique bandwidth parameters for each window scale. Furthermore, it is implemented offline, eliminating the need for online optimization for each time-series window. To validate our methodology, we designed a synthetic experiment using a non-stationary signal generated by the composition of two stationary signals and a modulation function that controls the transitions between a normal and an abnormal state, allowing for the arbitrary design of various anomaly transitions. Additionally, we tested the methodology on real scalp-EEG data to detect epileptic crises. The results show our approach effectively detects and characterizes anomaly transitions. The use of overlapping windows at various scales significantly enhances detection ability, allowing for the simultaneous analysis of phenomena at different scales.