Theoretical investigation of convex representation based interpretable time-frequency weight optimization for Machine health monitoring

Abstract

Fault feature representation and extraction using time–frequency spectrograms, such as short-time Fourier transform (STFT) and wavelet transform (WT) have been extensively applied in fault detection and diagnostics. However, the weak fault characteristics present during the early stages of fault initiation are often obscured by noise, posing significant challenges for machine health monitoring, particularly for incipient fault detection and diagnosis. To address this problem, this study proposes an interpretable time–frequency weight optimization methodology based on convex representation. First, compact convex hulls are employed to geometrically represent time–frequency spectrograms at different health conditions. A nearest-point convex hull optimization model is then introduced to implicitly learn interpretable weight matrix, enabling the identification of informative frequency bands within the time–frequency domain. Theoretical exploration of the optimized weight matrix reveals their geometric meanings and their correlation with fault characteristics in the time–frequency domain. Additionally, a condition indicator is constructed using information fusion, facilitating rapid detection of incipient faults. Simulated and experimental results validate the proposed method, demonstrating that the learned time–frequency weights are physics-informed and capable of identifying critical frequency bands for fault diagnostics in agreement with theoretical exploration. With explicit physical interpretability, the proposed methodology surpasses current approaches in its effectiveness for incipient fault detection and diagnosis. This study is the first to establish a theoretical framework for interpreting the physical meanings of the optimized weight matrix in the time–frequency domain by employing convex hull representation in the context of machine health monitoring.