Influence of <i>α</i>-Stable Noise on the Effectiveness of Non-Negative Matrix Factorization—Simulations and Real Data Analysis
Non-negative matrix factorization (NMF) has been used in various applications, including local damage detection in rotating machines. Recent studies highlight the limitations of diagnostic techniques in the presence of non-Gaussian noise. The authors examine the impact of non-Gaussianity levels on t...
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MDPI AG
2024-02-01
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| Series: | Electronics |
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| Online Access: | https://www.mdpi.com/2079-9292/13/5/829 |
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| author | Anna Michalak Rafał Zdunek Radosław Zimroz Agnieszka Wyłomańska |
| author_facet | Anna Michalak Rafał Zdunek Radosław Zimroz Agnieszka Wyłomańska |
| author_sort | Anna Michalak |
| collection | DOAJ |
| description | Non-negative matrix factorization (NMF) has been used in various applications, including local damage detection in rotating machines. Recent studies highlight the limitations of diagnostic techniques in the presence of non-Gaussian noise. The authors examine the impact of non-Gaussianity levels on the extraction of the signal of interest (SOI). The simple additive model of the signal is proposed: SOI and non-Gaussian noise. As a model of the random component, i.e., noise, a heavy-tailed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-stable distribution with two important parameters (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>) was proposed. If SOI is masked by noise (controlled by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>), the influence of non-Gaussianity level (controlled by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>) is more critical. We performed an empirical analysis of how these parameters affect SOI extraction effectiveness using NMF. Finally, we applied two NMF algorithms to several (both vibration and acoustic) signals from a machine with faulty bearings at different levels of non-Gaussian disturbances and the obtained results align with the simulations. The main conclusion of this study is that NMF is a very powerful tool for analyzing non-Gaussian data and can provide satisfactory results in a wide range of a non-Gaussian noise levels. |
| first_indexed | 2024-04-25T00:32:45Z |
| format | Article |
| id | doaj.art-e906796088524772a4618b288d0a803c |
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| issn | 2079-9292 |
| language | English |
| last_indexed | 2024-04-25T00:32:45Z |
| publishDate | 2024-02-01 |
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| series | Electronics |
| spelling | doaj.art-e906796088524772a4618b288d0a803c2024-03-12T16:42:17ZengMDPI AGElectronics2079-92922024-02-0113582910.3390/electronics13050829Influence of <i>α</i>-Stable Noise on the Effectiveness of Non-Negative Matrix Factorization—Simulations and Real Data AnalysisAnna Michalak0Rafał Zdunek1Radosław Zimroz2Agnieszka Wyłomańska3Faculty of Geoengineering, Mining and Geology, Wroclaw University of Science and Technology, Na Grobli 15, 50-421 Wroclaw, PolandFaculty of Electronics, Photonics, and Microsystems, Wroclaw University of Science and Technology, Janiszewskiego 11, 50-372 Wroclaw, PolandFaculty of Geoengineering, Mining and Geology, Wroclaw University of Science and Technology, Na Grobli 15, 50-421 Wroclaw, PolandFaculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wroclaw, PolandNon-negative matrix factorization (NMF) has been used in various applications, including local damage detection in rotating machines. Recent studies highlight the limitations of diagnostic techniques in the presence of non-Gaussian noise. The authors examine the impact of non-Gaussianity levels on the extraction of the signal of interest (SOI). The simple additive model of the signal is proposed: SOI and non-Gaussian noise. As a model of the random component, i.e., noise, a heavy-tailed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-stable distribution with two important parameters (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>) was proposed. If SOI is masked by noise (controlled by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>), the influence of non-Gaussianity level (controlled by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>) is more critical. We performed an empirical analysis of how these parameters affect SOI extraction effectiveness using NMF. Finally, we applied two NMF algorithms to several (both vibration and acoustic) signals from a machine with faulty bearings at different levels of non-Gaussian disturbances and the obtained results align with the simulations. The main conclusion of this study is that NMF is a very powerful tool for analyzing non-Gaussian data and can provide satisfactory results in a wide range of a non-Gaussian noise levels.https://www.mdpi.com/2079-9292/13/5/829NMF spectrogramlocal damage detectionheavy-tailed noiseeffectivenessconstraints |
| spellingShingle | Anna Michalak Rafał Zdunek Radosław Zimroz Agnieszka Wyłomańska Influence of <i>α</i>-Stable Noise on the Effectiveness of Non-Negative Matrix Factorization—Simulations and Real Data Analysis Electronics NMF spectrogram local damage detection heavy-tailed noise effectiveness constraints |
| title | Influence of <i>α</i>-Stable Noise on the Effectiveness of Non-Negative Matrix Factorization—Simulations and Real Data Analysis |
| title_full | Influence of <i>α</i>-Stable Noise on the Effectiveness of Non-Negative Matrix Factorization—Simulations and Real Data Analysis |
| title_fullStr | Influence of <i>α</i>-Stable Noise on the Effectiveness of Non-Negative Matrix Factorization—Simulations and Real Data Analysis |
| title_full_unstemmed | Influence of <i>α</i>-Stable Noise on the Effectiveness of Non-Negative Matrix Factorization—Simulations and Real Data Analysis |
| title_short | Influence of <i>α</i>-Stable Noise on the Effectiveness of Non-Negative Matrix Factorization—Simulations and Real Data Analysis |
| title_sort | influence of i α i stable noise on the effectiveness of non negative matrix factorization simulations and real data analysis |
| topic | NMF spectrogram local damage detection heavy-tailed noise effectiveness constraints |
| url | https://www.mdpi.com/2079-9292/13/5/829 |
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