Mathematics (Basel). 2021 Nov 2;9(22):2840. doi:
10.3390/math9222840. Epub 2021 Nov 9.


José M Maisog 1 , Andrew T DeMarco 2 , Karthik Devarajan 3 , S Stanley Young 4 , Paul Fogel 5 , George Luta 6 7 8


  • 1 Blue Health
  • 2 Department of Rehabilitation Medicine, Georgetown University Medical Center.
  • 3 Department of Biostatistics and Bioinformatics, Fox Chase Cancer Center, Temple University Health System, Philadelphia, PA 19111.
  • 4 GCStat, 3401 Caldwell Drive, Raleigh, NC
  • 5 Advestis, 69 Boulevard Haussmann 75008 Paris,
  • 6 Department of Biostatistics, Bioinformatics and Biomathematics, Georgetown University Medical Center.
  • 7 Department of Clinical Epidemiology, Aarhus University, Aarhus,
  • 8 The Parker Institute, Copenhagen University Hospital, Frederiksberg, Denmark.


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.

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