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Assessing Methods for Evaluating the Number of Components in Non-Negative Matrix Factorization
Non-negative matrix factorization is a relatively new method of matrix decomposition which factors an m×n data matrix X into an m×k matrix W and a k×n matrix H, so that X≈W×H. Importantly, all values in X, W, and H are constrained to be non-negative. NMF can be used for dimensionality reduction, sin...
Autores principales: | , , , , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
2021
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9181460/ https://www.ncbi.nlm.nih.gov/pubmed/35694180 http://dx.doi.org/10.3390/math9222840 |
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author | Maisog, José M. DeMarco, Andrew T. Devarajan, Karthik Young, S. Stanley Fogel, Paul Luta, George |
author_facet | Maisog, José M. DeMarco, Andrew T. Devarajan, Karthik Young, S. Stanley Fogel, Paul Luta, George |
author_sort | Maisog, José M. |
collection | PubMed |
description | Non-negative matrix factorization is a relatively new method of matrix decomposition which factors an m×n data matrix X into an m×k matrix W and a k×n matrix H, so that X≈W×H. Importantly, all values in X, W, and H are constrained to be non-negative. NMF can be used for dimensionality reduction, since the k columns of W can be considered components into which X has been decomposed. The question arises: how does one choose k? In this paper, we first assess methods for estimating k in the context of NMF in synthetic data. Second, we examine the effect of normalization on this estimate’s accuracy in empirical data. In synthetic data with orthogonal underlying components, methods based on PCA and Brunet’s Cophenetic Correlation Coefficient achieved the highest accuracy. When evaluated on a well-known real dataset, normalization had an unpredictable effect on the estimate. For any given normalization method, the methods for estimating k gave widely varying results. We conclude that when estimating k, it is best not to apply normalization. If underlying components are known to be orthogonal, then Velicer’s MAP or Minka’s Laplace-PCA method might be best. However, when orthogonality of the underlying components is unknown, none of the methods seemed preferable. |
format | Online Article Text |
id | pubmed-9181460 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2021 |
record_format | MEDLINE/PubMed |
spelling | pubmed-91814602022-06-09 Assessing Methods for Evaluating the Number of Components in Non-Negative Matrix Factorization Maisog, José M. DeMarco, Andrew T. Devarajan, Karthik Young, S. Stanley Fogel, Paul Luta, George Mathematics (Basel) Article Non-negative matrix factorization is a relatively new method of matrix decomposition which factors an m×n data matrix X into an m×k matrix W and a k×n matrix H, so that X≈W×H. Importantly, all values in X, W, and H are constrained to be non-negative. NMF can be used for dimensionality reduction, since the k columns of W can be considered components into which X has been decomposed. The question arises: how does one choose k? In this paper, we first assess methods for estimating k in the context of NMF in synthetic data. Second, we examine the effect of normalization on this estimate’s accuracy in empirical data. In synthetic data with orthogonal underlying components, methods based on PCA and Brunet’s Cophenetic Correlation Coefficient achieved the highest accuracy. When evaluated on a well-known real dataset, normalization had an unpredictable effect on the estimate. For any given normalization method, the methods for estimating k gave widely varying results. We conclude that when estimating k, it is best not to apply normalization. If underlying components are known to be orthogonal, then Velicer’s MAP or Minka’s Laplace-PCA method might be best. However, when orthogonality of the underlying components is unknown, none of the methods seemed preferable. 2021-11-02 2021-11-09 /pmc/articles/PMC9181460/ /pubmed/35694180 http://dx.doi.org/10.3390/math9222840 Text en https://creativecommons.org/licenses/by/4.0/Submitted for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). |
spellingShingle | Article Maisog, José M. DeMarco, Andrew T. Devarajan, Karthik Young, S. Stanley Fogel, Paul Luta, George Assessing Methods for Evaluating the Number of Components in Non-Negative Matrix Factorization |
title | Assessing Methods for Evaluating the Number of Components in Non-Negative Matrix Factorization |
title_full | Assessing Methods for Evaluating the Number of Components in Non-Negative Matrix Factorization |
title_fullStr | Assessing Methods for Evaluating the Number of Components in Non-Negative Matrix Factorization |
title_full_unstemmed | Assessing Methods for Evaluating the Number of Components in Non-Negative Matrix Factorization |
title_short | Assessing Methods for Evaluating the Number of Components in Non-Negative Matrix Factorization |
title_sort | assessing methods for evaluating the number of components in non-negative matrix factorization |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9181460/ https://www.ncbi.nlm.nih.gov/pubmed/35694180 http://dx.doi.org/10.3390/math9222840 |
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