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Int J Environ Res Public Health. 2016 May 18;13(5). pii: E509. doi: 10.3390/ijerph13050509.

# Applications of a Novel Clustering Approach Using Non-Negative Matrix Factorization to Environmental Research in Public Health.

### Author information

1
Independent Consultant, Paris 75006, France. paul_fogel@hotmail.com.
2
YGM Consult, CEO, Paris 75015, France. yann.gaston.mathe@gmail.com.
3
School of Statistics, University of Minnesota, Minneapolis, MN 55455, USA. dhawkins@umn.edu.
4
Institute Louis Bachelier, Paris 75002, France. fogel@di.ens.fr.
5
Department of Biostatistics, Bioinformatics, and Biomathematics, Georgetown University, Washington, DC 20057, USA. george.luta@georgetown.edu.
6
CGStat, CEO, Raleigh, NC 27607, USA. genetree@bellsouth.net.

### Abstract

Often data can be represented as a matrix, e.g., observations as rows and variables as columns, or as a doubly classified contingency table. Researchers may be interested in clustering the observations, the variables, or both. If the data is non-negative, then Non-negative Matrix Factorization (NMF) can be used to perform the clustering. By its nature, NMF-based clustering is focused on the large values. If the data is normalized by subtracting the row/column means, it becomes of mixed signs and the original NMF cannot be used. Our idea is to split and then concatenate the positive and negative parts of the matrix, after taking the absolute value of the negative elements. NMF applied to the concatenated data, which we call PosNegNMF, offers the advantages of the original NMF approach, while giving equal weight to large and small values. We use two public health datasets to illustrate the new method and compare it with alternative clustering methods, such as K-means and clustering methods based on the Singular Value Decomposition (SVD) or Principal Component Analysis (PCA). With the exception of situations where a reasonably accurate factorization can be achieved using the first SVD component, we recommend that the epidemiologists and environmental scientists use the new method to obtain clusters with improved quality and interpretability.

#### KEYWORDS:

K-means; NMF; PCA; SVD

PMID:
27213413
PMCID:
PMC4881134
DOI:
10.3390/ijerph13050509
[Indexed for MEDLINE]
Free PMC Article