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&l...

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Autores principales:	José M. Maisog, Andrew T. DeMarco, Karthik Devarajan, Stanley Young, Paul Fogel, George Luta
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	non-negative matrix factorization normalization PCA factorization rank number of factored components high-dimensional data Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/17df6d63695d43868d77feb0c575755d
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spelling	oai:doaj.org-article:17df6d63695d43868d77feb0c575755d2021-11-25T18:16:26ZAssessing Methods for Evaluating the Number of Components in Non-Negative Matrix Factorization10.3390/math92228402227-7390https://doaj.org/article/17df6d63695d43868d77feb0c575755d2021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2840https://doaj.org/toc/2227-7390Non-negative matrix factorization is a relatively new method of matrix decomposition which factors an <i>m</i> × <i>n</i> data matrix <i>X</i> into an <i>m</i> × <i>k</i> matrix <i>W</i> and a <i>k</i> × <i>n</i> matrix <i>H</i>, so that <i>X</i> ≈ <i>W</i> × <i>H</i>. Importantly, all values in <i>X</i>, <i>W</i>, and <i>H</i> are constrained to be non-negative. NMF can be used for dimensionality reduction, since the <i>k</i> columns of <i>W</i> can be considered components into which <i>X</i> has been decomposed. The question arises: how does one choose <i>k</i>? In this paper, we first assess methods for estimating <i>k</i> 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 <i>k</i> gave widely varying results. We conclude that when estimating <i>k</i>, it is best not to apply normalization. If the underlying components are known to be orthogonal, then Velicer’s MAP or Minka’s Laplace-PCA method might be best. However, when the orthogonality of the underlying components is unknown, none of the methods seemed preferable.José M. MaisogAndrew T. DeMarcoKarthik DevarajanStanley YoungPaul FogelGeorge LutaMDPI AGarticlenon-negative matrix factorizationnormalizationPCAfactorization ranknumber of factored componentshigh-dimensional dataMathematicsQA1-939ENMathematics, Vol 9, Iss 2840, p 2840 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	non-negative matrix factorization normalization PCA factorization rank number of factored components high-dimensional data Mathematics QA1-939
spellingShingle	non-negative matrix factorization normalization PCA factorization rank number of factored components high-dimensional data Mathematics QA1-939 José M. Maisog Andrew T. DeMarco Karthik Devarajan Stanley Young Paul Fogel George Luta Assessing Methods for Evaluating the Number of Components in Non-Negative Matrix Factorization
description	Non-negative matrix factorization is a relatively new method of matrix decomposition which factors an <i>m</i> × <i>n</i> data matrix <i>X</i> into an <i>m</i> × <i>k</i> matrix <i>W</i> and a <i>k</i> × <i>n</i> matrix <i>H</i>, so that <i>X</i> ≈ <i>W</i> × <i>H</i>. Importantly, all values in <i>X</i>, <i>W</i>, and <i>H</i> are constrained to be non-negative. NMF can be used for dimensionality reduction, since the <i>k</i> columns of <i>W</i> can be considered components into which <i>X</i> has been decomposed. The question arises: how does one choose <i>k</i>? In this paper, we first assess methods for estimating <i>k</i> 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 <i>k</i> gave widely varying results. We conclude that when estimating <i>k</i>, it is best not to apply normalization. If the underlying components are known to be orthogonal, then Velicer’s MAP or Minka’s Laplace-PCA method might be best. However, when the orthogonality of the underlying components is unknown, none of the methods seemed preferable.
format	article
author	José M. Maisog Andrew T. DeMarco Karthik Devarajan Stanley Young Paul Fogel George Luta
author_facet	José M. Maisog Andrew T. DeMarco Karthik Devarajan Stanley Young Paul Fogel George Luta
author_sort	José M. Maisog
title	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_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_sort	assessing methods for evaluating the number of components in non-negative matrix factorization
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/17df6d63695d43868d77feb0c575755d
work_keys_str_mv	AT josemmaisog assessingmethodsforevaluatingthenumberofcomponentsinnonnegativematrixfactorization AT andrewtdemarco assessingmethodsforevaluatingthenumberofcomponentsinnonnegativematrixfactorization AT karthikdevarajan assessingmethodsforevaluatingthenumberofcomponentsinnonnegativematrixfactorization AT stanleyyoung assessingmethodsforevaluatingthenumberofcomponentsinnonnegativematrixfactorization AT paulfogel assessingmethodsforevaluatingthenumberofcomponentsinnonnegativematrixfactorization AT georgeluta assessingmethodsforevaluatingthenumberofcomponentsinnonnegativematrixfactorization
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Assessing Methods for Evaluating the Number of Components in Non-Negative Matrix Factorization

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