Maximum Mean r th-Order Difference: A High-Order Statistical Discrepancy Metric for Fault Transfer Diagnosis

Variable operating conditions often lead to significant distribution shifts in mechanical vibration signals, making it challenging for traditional low-order discrepancy metrics to accurately characterize the underlying distribution differences in fault transfer diagnosis. Existing maximum mean discrepancy (MMD)-based explicit domain-alignment methods mainly rely on low-order statistics, making them insufficient for representing the complex statistical properties of mechanical vibration signals. Therefore, incorporating higher order statistical information is important for accurate distribution characterization. This article bridges the relationship between the MMD and kernel functions. Based on the properties of the reproducing kernel Hilbert space, a high-order discrepancy metric with an adjustable statistical order is proposed. This maximum mean rth-order difference (MMRD) employs the rth-order kernel mean embedding to capture distribution differences at different statistical orders. Considering the limited number of samples available in real scenarios, two empirical MMRD statistics—biased and unbiased—are developed, and the error bounds between them and the true distribution discrepancy are given. ... mehrThe MMRD method is validated by exploring six bearing fault transfer diagnosis tasks across three bearing datasets, covering cross-speed, cross-load, and cross-device conditions. Experimental results on six transfer tasks demonstrate that the proposed framework based on the designed feature extractor and high-order discrepancy alignment metric achieves the improved and stable transfer-diagnosis performance compared with existing methods.