A Fully Multivariate Multifractal Detrended Fluctuation Analysis Method for Fault Diagnosis

We propose a fully multivariate generalization of multifractal detrended fluctuation analysis (MFDFA) and leverage it to develop a fault diagnosis framework for multichannel machine vibration data. We introduce a novel covariance-weighted $L_{pq}$ matrix norm based on Mahalanobis distance to define a fully multivariate fluctuation function that uniquely captures cross-channel dependencies and variance biases in multichannel vibration data. This formulation, termed FM-MFDFA, allows for a more accurate characterization of the multiscale structure of multivariate signals. To enhance feature relevance, the proposed framework integrates multivariate variational mode decomposition (MVMD) to isolate fault-relevant components before applying FM-MFDFA. Results on wind turbine gearbox data demonstrate that the proposed method outperforms conventional MFDFA approaches by effectively distinguishing between healthy and faulty machine states, even under noisy conditions.