Applications of a Novel Clustering Approach Using Non-Negative Matrix Factorization to Environmental Research in Public Health
2. Materials and Methods
2.1. Data Sets
- The number of emergency hospital admissions for cardiovascular disease (CVD), myocardial infarction (MI), congestive heart failure (CHF), respiratory disease, and diabetes were collected in 26 US communities, for the years 2000–2003 .
- The Compressed Mortality File (CMF)—a county-level national mortality and population database spanning the years 1968–2010. The table contains death counts for 13 age categories.
2.2.1. Non-Negative Factorization of a General Matrix
- Split V into the positive and negative parts : V = V+ − V− where V+ contains the positive entries, other entries being replaced by 0, and V− contains the absolute values of the negative entries, other entries being replaced by 0.
- When the rows are the observations and the columns are the variables, use the horizontally concatenated matrix VPN in order to give equal weight to low and high features while characterizing the clusters of observations.
- Apply the NMF clustering on the concatenated matrix: VPN = WHT (note that writing V = W(H+ − H−)T where H = [H+ H−] corresponds to the semi-NMF model).
- Substract the minimum of each column: V = V0 + baseline .
- Apply the NMF clustering to V0.
2.2.2. NMF Clustering and Reordering
2.2.3. Stability and Specific Clustering Contribution of NMF Clusters
2.2.4. Rank of the NMF Factorization
2.2.5. Normalization of Contingency Tables
- For each cell, the contingency ratio is calculated by forming the ratio of the true count over the expected count—assuming the independence of rows and columns
- Further normalization steps include the subtraction of the expected ratio under the assumption of independence (=1), yielding a mixed signs matrix, and a subsequent scaling of rows and columns to ensure homogeneous cell variances.
- The SVD is applied to the normalized matrix.
- A biplot based on the first two SVD components is then performed, allowing for a simultaneous clustering of the rows and columns of the table, which we will refer to as the SVD clustering. Note that PCA clustering refers to the same approach, since PCA’s eigen vectors are the column singular vectors .
3.1. Hospital Admissions Data
3.1.1. PosNegNMF Clustering
3.1.2. Affine NMF Clustering
3.1.3. Correspondence Analysis
3.1.4. Additional Remarks
3.2. Compressed Mortality File
4.2. Alternative Approaches
Conflicts of Interest
|NMF||Non-Negative Matrix Factorization|
|PMF||Positive Matrix Factorization|
|SVD||Singular Value Decomposition|
|CHF||congestive heart failure|
|SCC||Specific clustering contribution|
Appendix A: Estimation of the NMF Model Components W and H
Appendix B: Calculation of Leverages
- Initialize the robust estimate by the maximum of each component.
- For each vector component q:
- For each row i of W, calculate the probability p (i, q) and the row score (Equations (C2) and (C1) respectively, Appendix C).
- Force the row score to 0 if p (i, q) < 1/k.
- Update Robust Max(q) by the weighted mean of W(i, q), where the mean is taken over all samples satisfying , and the weights are the row scores. The idea is that rows with higher row scores should weigh more on the max estimation.
- Replace all W(i, q) > Robust Max(q) by Robust Max(q).
- Repeat 2. until convergence.
Appendix C: Stability and Specific Clustering Contribution of NMF Clusters
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|Cluster||High Counts||Low Counts|
|1||Respiratory||CVD, CHF, MI|
© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).
Fogel, P.; Gaston-Mathé, Y.; Hawkins, D.; Fogel, F.; Luta, G.; Young, S.S. Applications of a Novel Clustering Approach Using Non-Negative Matrix Factorization to Environmental Research in Public Health. Int. J. Environ. Res. Public Health 2016, 13, 509. https://doi.org/10.3390/ijerph13050509
Fogel P, Gaston-Mathé Y, Hawkins D, Fogel F, Luta G, Young SS. Applications of a Novel Clustering Approach Using Non-Negative Matrix Factorization to Environmental Research in Public Health. International Journal of Environmental Research and Public Health. 2016; 13(5):509. https://doi.org/10.3390/ijerph13050509Chicago/Turabian Style
Fogel, Paul, Yann Gaston-Mathé, Douglas Hawkins, Fajwel Fogel, George Luta, and S. Stanley Young. 2016. "Applications of a Novel Clustering Approach Using Non-Negative Matrix Factorization to Environmental Research in Public Health" International Journal of Environmental Research and Public Health 13, no. 5: 509. https://doi.org/10.3390/ijerph13050509