{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T23:01:33Z","timestamp":1772060493041,"version":"3.50.1"},"reference-count":49,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,9,15]],"date-time":"2021-09-15T00:00:00Z","timestamp":1631664000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100005046","name":"Natural Science Foundation of Heilongjiang Province","doi-asserted-by":"publisher","award":["LH2021E021"],"award-info":[{"award-number":["LH2021E021"]}],"id":[{"id":"10.13039\/501100005046","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51505079"],"award-info":[{"award-number":["51505079"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Northeast Petroleum University Youth Foundation","award":["2018ANC-31"],"award-info":[{"award-number":["2018ANC-31"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In practical engineering applications, the vibration signals collected by sensors often contain outliers, resulting in the separation accuracy of source signals from the observed signals being seriously affected. The mixing matrix estimation is crucial to the underdetermined blind source separation (UBSS), determining the accuracy level of the source signals recovery. Therefore, a two-stage clustering method is proposed by combining hierarchical clustering and K-means to improve the reliability of the estimated mixing matrix in this paper. The proposed method is used to solve the two major problems in the K-means algorithm: the random selection of initial cluster centers and the sensitivity of the algorithm to outliers. Firstly, the observed signals are clustered by hierarchical clustering to get the cluster centers. Secondly, the cosine distance is used to eliminate the outliers deviating from cluster centers. Then, the initial cluster centers are obtained by calculating the mean value of each remaining cluster. Finally, the mixing matrix is estimated with the improved K-means, and the sources are recovered using the least square method. Simulation and the reciprocating compressor fault experiments demonstrate the effectiveness of the proposed method.<\/jats:p>","DOI":"10.3390\/e23091217","type":"journal-article","created":{"date-parts":[[2021,9,15]],"date-time":"2021-09-15T21:47:11Z","timestamp":1631742431000},"page":"1217","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Fault Feature Extraction for Reciprocating Compressors Based on Underdetermined Blind Source Separation"],"prefix":"10.3390","volume":"23","author":[{"given":"Jindong","family":"Wang","sequence":"first","affiliation":[{"name":"Mechanical Science and Engineering Institute, Northeast Petroleum University, Daqing 163318, China"},{"name":"Heilongjiang Key Laboratory of Petroleum Machinery Engineering, Daqing 163318, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2670-5530","authenticated-orcid":false,"given":"Xin","family":"Chen","sequence":"additional","affiliation":[{"name":"Mechanical Science and Engineering Institute, Northeast Petroleum University, Daqing 163318, China"}]},{"given":"Haiyang","family":"Zhao","sequence":"additional","affiliation":[{"name":"Mechanical Science and Engineering Institute, Northeast Petroleum University, Daqing 163318, China"},{"name":"Heilongjiang Key Laboratory of Petroleum Machinery Engineering, Daqing 163318, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6534-0618","authenticated-orcid":false,"given":"Yanyang","family":"Li","sequence":"additional","affiliation":[{"name":"Mechanical Science and Engineering Institute, Northeast Petroleum University, Daqing 163318, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8491-5712","authenticated-orcid":false,"given":"Zujian","family":"Liu","sequence":"additional","affiliation":[{"name":"Mechanical Science and Engineering Institute, Northeast Petroleum University, Daqing 163318, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/j.ymssp.2018.03.035","article-title":"A Compound Interpolation Envelope Local Mean Decomposition and Its Application for Fault Diagnosis of Reciprocating Compressors","volume":"110","author":"Haiyang","year":"2018","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/j.ymssp.2018.10.016","article-title":"Underdetermined Blind Separation of Bearing Faults in Hyperplane Space with Variational Mode Decomposition","volume":"120","author":"Li","year":"2019","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.neucom.2019.11.006","article-title":"Bayesian Approach and Time Series Dimensionality Reduction to LSTM-Based Model-Building for Fault Diagnosis of a Reciprocating Compressor","volume":"380","author":"Cabrera","year":"2020","journal-title":"Neurocomputing"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Mondal, D., Zhen, D., Gu, F., and Ball, A.D. (2020). Fault diagnosis of reciprocating compressor using empirical mode decomposition-based Teager energy spectrum of airborne acoustic signal. Advances in Asset Management and Condition Monitoring, Springer.","DOI":"10.1007\/978-3-030-57745-2_77"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/j.ymssp.2018.07.029","article-title":"Performance Evaluation of Decomposition Methods to Diagnose Leakage in a Reciprocating Compressor under Limited Speed Variation","volume":"125","author":"Sharma","year":"2019","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1098\/rspa.1998.0193","article-title":"The Empirical Mode Decomposition and the Hilbert Spectrum for Nonlinear and Non-Stationary Time Series Analysis","volume":"454","author":"Huang","year":"1998","journal-title":"Proc. R. Soc. Lond. Ser. Math. Phys. Eng. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/j.isatra.2018.09.008","article-title":"Quantitative Diagnosis for Bearing Faults by Improving Ensemble Empirical Mode Decomposition","volume":"83","author":"Hoseinzadeh","year":"2018","journal-title":"ISA Trans."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1098\/rsif.2005.0058","article-title":"The Local Mean Decomposition and Its Application to EEG Perception Data","volume":"2","author":"Smith","year":"2005","journal-title":"J. R. Soc. Interface"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Zhang, L., Wang, Z., and Quan, L. (2018). Research on Weak Fault Extraction Method for Alleviating the Mode Mixing of LMD. Entropy, 20.","DOI":"10.3390\/e20050387"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1109\/TSP.2013.2288675","article-title":"Variational Mode Decomposition","volume":"62","author":"Dragomiretskiy","year":"2013","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Liang, T., Lu, H., and Sun, H. (2021). Application of Parameter Optimized Variational Mode Decomposition Method in Fault Feature Extraction of Rolling Bearing. Entropy, 23.","DOI":"10.3390\/e23050520"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"102337","DOI":"10.1016\/j.bspc.2020.102337","article-title":"EEG Signal Denoising Using Hybrid Approach of Variational Mode Decomposition and Wavelets for Depression","volume":"65","author":"Kaur","year":"2021","journal-title":"Biomed. Signal Process. Control"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1109\/MSP.2021.3057051","article-title":"Noninvasive Neural Interfacing With Wearable Muscle Sensors: Combining Convolutive Blind Source Separation Methods and Deep Learning Techniques for Neural Decoding","volume":"38","author":"Holobar","year":"2021","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1109\/LSP.2021.3055463","article-title":"Ray-Space-Based Multichannel Nonnegative Matrix Factorization for Audio Source Separation","volume":"28","author":"Pezzoli","year":"2021","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"107268","DOI":"10.1016\/j.measurement.2019.107268","article-title":"Underdetermined Mixing Matrix Estimation by Exploiting Sparsity of Sources","volume":"152","author":"Zhen","year":"2020","journal-title":"Measurement"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Zhao, X., Qin, Y., He, C., and Jia, L. (2020). Underdetermined Blind Source Extraction of Early Vehicle Bearing Faults Based on EMD and Kernelized Correlation Maximization. J. Intell. Manuf., 1\u201317.","DOI":"10.1007\/s10845-020-01655-1"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ho, S.K., Nedunuri, H.C., Balachandran, W., Kanfoud, J., and Gan, T.-H. (2021). Monitoring of Industrial Machine Using a Novel Blind Feature Extraction Approach. Appl. Sci., 11.","DOI":"10.3390\/app11135792"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"622","DOI":"10.1109\/TIE.2020.2967708","article-title":"Data-Driven Monitoring and Diagnosing of Abnormal Furnace Conditions in Blast Furnace Ironmaking: An Integrated PCA-ICA Method","volume":"68","author":"Zhou","year":"2020","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Miao, F., Zhao, R., Jia, L., and Wang, X. (2021). Multisource Fault Signal Separation of Rotating Machinery Based on Wavelet Packet and Fast Independent Component Analysis. Int. J. Rotating Mach., 2021.","DOI":"10.1155\/2021\/9914724"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1016\/j.measurement.2018.06.047","article-title":"Weak Characteristic Determination for Blade Crack of Centrifugal Compressors Based on Underdetermined Blind Source Separation","volume":"128","author":"He","year":"2018","journal-title":"Measurement"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"107590","DOI":"10.1016\/j.sigpro.2020.107590","article-title":"Time and Frequency Based Sparse Bounded Component Analysis Algorithms for Convolutive Mixtures","volume":"173","author":"Babatas","year":"2020","journal-title":"Signal Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"107750","DOI":"10.1016\/j.sigpro.2020.107750","article-title":"Blind Separation of Coherent Multipath Signals with Impulsive Interference and Gaussian Noise in Time-Frequency Domain","volume":"178","author":"Xiao","year":"2021","journal-title":"Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1264","DOI":"10.1007\/s00034-018-0910-9","article-title":"A Novel Underdetermined Source Recovery Algorithm Based on K-Sparse Component Analysis","volume":"38","author":"Eqlimi","year":"2019","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/j.neucom.2020.06.029","article-title":"Underdetermined Blind Separation of Source Using Lp-Norm Diversity Measures","volume":"411","author":"Xie","year":"2020","journal-title":"Neurocomputing"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.jsv.2019.05.037","article-title":"A Novel Underdetermined Blind Source Separation Method with Noise and Unknown Source Number","volume":"457","author":"Lu","year":"2019","journal-title":"J. Sound Vib."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2353","DOI":"10.1016\/S0165-1684(01)00120-7","article-title":"Underdetermined Blind Source Separation Using Sparse Representations","volume":"81","author":"Bofill","year":"2001","journal-title":"Signal Process."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2477","DOI":"10.1109\/TMECH.2019.2951589","article-title":"Step-by-Step Compound Faults Diagnosis Method for Equipment Based on Majorization-Minimization and Constraint SCA","volume":"24","author":"Hao","year":"2019","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1016\/j.ymssp.2018.07.026","article-title":"Enhanced Sparse Component Analysis for Operational Modal Identification of Real-Life Bridge Structures","volume":"116","author":"Xu","year":"2019","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"095001","DOI":"10.1088\/1361-6501\/ab816f","article-title":"Underdetermined Convolutive Blind Source Separation in the Time\u2013Frequency Domain Based on Single Source Points and Experimental Validation","volume":"31","author":"Cheng","year":"2020","journal-title":"Meas. Sci. Technol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2025","DOI":"10.1109\/TASLP.2020.3003855","article-title":"A Novel Directional Framework for Source Counting and Source Separation in Instantaneous Underdetermined Audio Mixtures","volume":"28","author":"Sgouros","year":"2020","journal-title":"IEEE\/ACM Trans. Audio Speech Lang. Process."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"113856","DOI":"10.1016\/j.eswa.2020.113856","article-title":"Fuzzy C-Means Clustering Algorithm for Data with Unequal Cluster Sizes and Contaminated with Noise and Outliers: Review and Development","volume":"165","author":"Askari","year":"2021","journal-title":"Expert Syst. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"106672","DOI":"10.1016\/j.knosys.2020.106672","article-title":"GBK-Means Clustering Algorithm: An Improvement to the K-Means Algorithm Based on the Bargaining Game","volume":"213","author":"Rezaee","year":"2021","journal-title":"Knowl.-Based Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"6965","DOI":"10.1007\/s00521-020-05471-9","article-title":"An Entropy-Based Initialization Method of K-Means Clustering on the Optimal Number of Clusters","volume":"33","author":"Chowdhury","year":"2021","journal-title":"Neural Comput. Appl."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Lu, J., Cheng, W., and Zi, Y. (2019). A Novel Underdetermined Blind Source Separation Method and Its Application to Source Contribution Quantitative Estimation. Sensors, 19.","DOI":"10.3390\/s19061413"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1049\/iet-rpg.2016.0240","article-title":"Sparse Component Analysis-Based under-Determined Blind Source Separation for Bearing Fault Feature Extraction in Wind Turbine Gearbox","volume":"11","author":"Hu","year":"2017","journal-title":"IET Renew. Power Gener."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"658","DOI":"10.1109\/ACCESS.2017.2773665","article-title":"Blind Source Separation Method for Bearing Vibration Signals","volume":"6","author":"Jun","year":"2017","journal-title":"IEEE Access"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"105763","DOI":"10.1016\/j.asoc.2019.105763","article-title":"Improving K-Means Clustering with Enhanced Firefly Algorithms","volume":"84","author":"Xie","year":"2019","journal-title":"Appl. Soft Comput."},{"key":"ref_38","unstructured":"Arthur, D., and Vassilvitskii, S. (2007, January 7\u20139). K-Means++: The Advantages of Careful Seeding. Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans, LA, USA."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2881","DOI":"10.1007\/s00034-015-0173-7","article-title":"Underdetermined BSS Based on K-Means and AP Clustering","volume":"35","author":"He","year":"2016","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1762","DOI":"10.1016\/j.sigpro.2009.03.017","article-title":"An Algorithm for Mixing Matrix Estimation in Instantaneous Blind Source Separation","volume":"89","author":"Reju","year":"2009","journal-title":"Signal Process."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.jsv.2015.10.028","article-title":"Underdetermined Blind Modal Identification of Structures by Earthquake and Ambient Vibration Measurements via Sparse Component Analysis","volume":"366","author":"Amini","year":"2016","journal-title":"J. Sound Vib."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"2888","DOI":"10.1109\/78.726803","article-title":"Blind Source Separation Based on Time-Frequency Signal Representations","volume":"46","author":"Belouchrani","year":"1998","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"100","DOI":"10.2307\/2346830","article-title":"Algorithm AS 136: A K-Means Clustering Algorithm","volume":"28","author":"Hartigan","year":"1979","journal-title":"Appl. Stat."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"3102","DOI":"10.1109\/TNNLS.2016.2610960","article-title":"Underdetermined Blind Source Separation Using Sparse Coding","volume":"28","author":"Zhen","year":"2016","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_45","first-page":"598","article-title":"Automatic Choice of Dimensionality for PCA","volume":"13","author":"Minka","year":"2000","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1016\/j.neucom.2015.08.008","article-title":"Novel Mixing Matrix Estimation Approach in Underdetermined Blind Source Separation","volume":"173","author":"Sun","year":"2016","journal-title":"Neurocomputing"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2037","DOI":"10.1007\/s11517-017-1647-5","article-title":"Refined Multiscale Fuzzy Entropy Based on Standard Deviation for Biomedical Signal Analysis","volume":"55","author":"Azami","year":"2017","journal-title":"Med. Biol. Eng. Comput."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Rodriguez, N., Alvarez, P., Barba, L., and Cabrera-Guerrero, G. (2019). Combining Multi-Scale Wavelet Entropy and Kernelized Classification for Bearing Multi-Fault Diagnosis. Entropy, 21.","DOI":"10.3390\/e21020152"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"107441","DOI":"10.1016\/j.measurement.2019.107441","article-title":"A Novel Method to Classify Bearing Faults by Integrating Standard Deviation to Refined Composite Multi-Scale Fuzzy Entropy","volume":"154","author":"Minhas","year":"2020","journal-title":"Measurement"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/9\/1217\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:00:23Z","timestamp":1760166023000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/9\/1217"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,15]]},"references-count":49,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["e23091217"],"URL":"https:\/\/doi.org\/10.3390\/e23091217","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,15]]}}}