Abstract：An improved algorithm for phase space reconstruction is proposed for the problem of embedding dimension and delay time uncertainty in chaotic singular spectrum analysis. The joint criterion is evaluated by the supplementary criterion, and the Cao algorithm is improved. An improved embedding dimension stability criterion embedding dimension using the improved Cao algorithm, can quickly and accurately determine the value of the embedding dimension, with accuracy and efficiency; The method of obtaining the delay time based on the maximum joint entropy based on symbol analysis can reduce the amount of calculation and reduce the error. The superiority of the proposed method is verified by numerical comparison experiments. The method is applied in the early fault identification of rolling bearings. The results show that the chaotic singular spectrum can clearly see the pattern distribution of different fault signals and realize the feature extraction of mechanical fault signals. Provide a new and effective way for early diagnosis of mechanical failure.
［1］Bhavsar R, Davey N, Helian N, et al. Time Series Analysis using Embedding Dimension on Heart Rate Variability［J］. Procedia computer science, 2018, 145: 89-96.
［2］Khazraei S M, Amiri-Simkooei A R. On the application of Monte Carlo singular spectrum analysis to GPS position time series［J］. Journal of Geodesy, 2019, 93: 1401-1418.
［3］Muruganatham B, Sanjith M A, Krishnakumar B, et al. Roller element bearing fault diagnosis using singular spectrum analysis［J］. Mechanical systems and signal processing, 2013, 35(1-2): 150-166.
［4］张淑清, 贺朋, 左一格, 等. 混沌奇异谱特性研究及在滚动轴承故障诊断中的应用［J］. 中国机械工程, 2018, 29(12): 1398-1404.
Zhang S Q, He P, Zuo Y G, et al. Study on Characteristics of Chaotic Singular Spectrum and Applications in Rolling Bearing Fault Diagnosis［J］. China Mechanical Engineering, 2018, 29(12): 1398-1404.
［5］Tamma A, Khubchandani B L. Accurate determination of time delay and embedding dimension for state space reconstruction from a scalar time series［EB］. http://www.arXiv preprint arXiv: 1605. 01571, 2016.
［6］余斌, 杨少敏. 基于改进 Cao 算法的奇异谱分析法及其在北斗多路径去噪中的应用［J］. 大地测量与地球动力学, 2019, 39(1): 25-30.
Yu B, Yang SM. Singular Spectrum Analysis Based on Improved Cao Algorithm and Its Application in Beidou Multipath Filtering［J］. Journal of Geodesy and Geodynamics, 2019, 39(1): 25-30.
［7］邹瑞. Lyapunov指数的逼近性质及其应用［D］.苏州:苏州大学,2018.
［8］柏林, 唐滔, 刘小峰, 等. 基于Lyapunov指数的非线性Lamb波的微裂纹检测［J］. 振动, 测试与诊断, 2019,(1): 9-14.
Bo L, Tang T, Liu X F, et al. Micro-crack Detection of Nonlinear Lamb Waves Based on Lyapunov Exponents［J］. Vibration, testing and diagnosis, 2019,(1): 9-14.
［9］泥立丽, 张艳兰. 基于 GP 法和 Cao 法的桥梁变形时间序列最佳嵌入维数的确定［J］. 北方交通, 2017,(5): 5-7.
Ni L L, Zhang Y L. Determination of the best embedding dimension of bridge deformation time series based on GP method and Cao method［J］. Northern traffic, 2017,(5): 5-7.
［10］岳顺, 李小奇, 翟长治. 基于改进Cao 算法确定奇异谱嵌入维数及应用［J］. 测绘工程, 2015, 24(3): 64-68.
Yue S, Li X Q, Zhai C Z. Determining the singular spectrum embedding dimension based on an improved Cao algorithm［J］. Enginnering of Surveying and Mapping, 2015, 24(3): 64-68.
［11］张淑清, 李新新, 张立国, 等. 基于符号分析的极大联合熵延迟时间求取方法［J］. 物理学报, 2013, 62(11): 110506-110506.
Zhang S Q, Li X X, Zhang L G, et al. Maximum joint entropy delay time method based on symbol analysis［J］. Journal of Physics, 2013, 62(11): 110506-110506.
［12］Fraser A M and Swinney H L. Independent coordinates for strange attractors from mutual information［J］. Phys Rev A, 1986, 33(2): 1134-1140.
［13］Ji W, Wu J, Zhang M, et al. Blind Image Quality Assessment With Joint Entropy Degradation［J］. IEEE Access, 2019, 7: 30925-30936.
［14］Pisarchik A N, Huerta-Cuellar G, Kulp C W. Statistical analysis of symbolic dynamics in weakly coupled chaotic oscillators［J］. Communications in Nonlinear Science and Numerical Simulation, 2018, 62: 134-145.
［15］He S, Li C, Sun K, et al. Multivariate multiscale complexity analysis of self-reproducing chaotic systems［J］. Entropy, 2018, 20(8): 556.
［16］李梅红, 连威. 基于变分模态分解和符号熵的齿轮故障诊断方法［J］. 机械传动, 2019, 43(3): 167-171.
Li M H, Lian W. Gear Fault Diagnosis Based on Variational Mode Decomposition and Symbol Entropy［J］. Mechanical Transmission, 2019, 43(3): 167-171.
［17］李磊, 高永明, 吴止锾. 基于混沌吸引子的飞轮故障检测［J］. 北京航空航天大学学报, 2018, 44(9): 111-119.
Li L, Gao Y M, Wu Z Y. Detection of flywheel fault based on chaotic attractor［J］. Journal of Beijing University of Aeronautics and Astronautics, 2018, 44(9): 111-119.
2021, Vol. 42 
