Mathematical morphology spectrum entropy is a signal feature extraction method based on information entropy and mathematical morphology. The scale of structure element is a critical parameter, whose value determines the accuracy of feature extraction. Existing scale selection methods depend on experiment parameters or external indicators including noise ratio, fault frequencies, etc. In many cases, existing methods obtain fix scale and they are not suitable for quantifying the performance degradation and the fault degree of bearings. There are few researches on scale selection based on the properties of mathematical morphology spectrum. In this study, a scale-adaptive mathematical morphology spectrum entropy (AMMSE) is proposed to improve the scale selection. To support the proposed method, two properties of the mathematical morphology spectrum (MMS), namely non-negativity and monotonic decreasing, are proved. It can be concluded from the two properties that the feature loss of MMS decreases with the increase of scale. Based on the conclusion, two adaptive scale selection strategies are proposed to automatically determine the scale by reducing the feature loss of MMS. AMMSE is the integration of two strategies. Compare to the existing methods, AMMSE is not constrained by the information of the experiment and the signal. The scale of AMMSE changes with the signal characteristics and is no longer fixed by experimental parameters. The parameters of AMMSE are more generalizable as well. The presented method is applied to identify fault degree on CWRU bearing data set and evaluate performance degradation on IMS bearing data set. The experiment result shows that AMMSE has better results in both experiments with the same parameters.

数学形态谱熵是一种基于信息熵和数学形态学的信号特征提取方法。结构元素的尺度是一个关键参数，其值决定了特征提取的准确性。现有的尺度选择方法依赖于实验参数或外部指标，包括噪声比、故障频率等，在很多情况下，现有方法获得固定尺度，不适合量化轴承的性能退化和故障程度。基于数学形态谱特性的尺度选择研究较少。在这项研究中，提出了一种尺度自适应数学形态谱熵（AMMSE）来改进尺度选择。为了支持所提出的方法，证明了数学形态谱（MMS）的两个性质，即非负性和单调递减。从这两个性质可以得出结论，MMS的特征损失随着尺度的增加而减小。基于该结论，提出了两种自适应尺度选择策略，通过减少MMS的特征损失来自动确定尺度。AMMSE 是两种策略的整合。与现有方法相比，AMMSE 不受实验信息和信号的约束。AMMSE 的尺度随信号特性而变化，不再受实验参数固定。AMMSE 的参数也更通用。所提出的方法用于识别CWRU轴承数据集的故障程度和评估IMS轴承数据集的性能下降。