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. 2013:3:1236.
doi: 10.1038/srep01236. Epub 2013 Feb 7.

Extracting insights from the shape of complex data using topology

P Y Lum  1 , G Singh, A Lehman, T Ishkanov, M Vejdemo-Johansson, M Alagappan, J Carlsson, G Carlsson
Affiliations

Extracting insights from the shape of complex data using topology

P Y Lum et al. Sci Rep. 2013.

Abstract

This paper applies topological methods to study complex high dimensional data sets by extracting shapes (patterns) and obtaining insights about them. Our method combines the best features of existing standard methodologies such as principal component and cluster analyses to provide a geometric representation of complex data sets. Through this hybrid method, we often find subgroups in data sets that traditional methodologies fail to find. Our method also permits the analysis of individual data sets as well as the analysis of relationships between related data sets. We illustrate the use of our method by applying it to three very different kinds of data, namely gene expression from breast tumors, voting data from the United States House of Representatives and player performance data from the NBA, in each case finding stratifications of the data which are more refined than those produced by standard methods.

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Figures

Figure 1
Figure 1. The approach as applied to a data set in our analysis pipeline.
A) A 3D object (hand) represented as a point cloud. B) A filter value is applied to the point cloud and the object is now colored by the values of the filter function. C) The data set is binned into overlapping groups. D) Each bin is clustered and a network is built.
Figure 2
Figure 2. Networks derived from NKI (panels A and B) and GSE2034 (panels C and D).
Two filter functions, L-Infinity centrality and survival or relapse were used to generate the networks. The top half of panels A and B are the networks of patients who didn't survive, the bottom half are the patients who survived. Panels C and D are similar to panels A and B except that one of the filters is relapse instead of survival. Panels A and C are colored by the average expression of the ESR1 gene. Panels B and D are colored by the average expression of the genes in the KEGG chemokine pathway. Metric: Correlation; Lens: L-Infinity Centrality (Resolution 70, Gain 3.0x, Equalized) and Event Death (Resolution 30, Gain 3.0x). Color bar: red: high values, blue: low values.
Figure 3
Figure 3. Single linkage hierarchical clustering and PCA of the NKI data set.
Highlighted in red are the lowERNS (top panel) and the lowERHS (bottom panel) patient sub-groups.
Figure 4
Figure 4. Top panel is the fragmentation index calculated from the number of sub-networks formed each year per political party.
X-axis: 1990–2011. Y-axis: Fragmentation index. Color bars denote, from top to bottom, party of the President, party for the House, party for the Senate (red: republican; blue: democrat; purple: split). The bottom 3 panels are the actual topological networks for the members. Networks are constructed from voting behavior of the member of the house, with an “aye” vote coded as a 1, “abstain” as zero, and “nay” as a -1. Each node contains sets of members. Each panel labeled with the year contains networks constructed from all the members for all the votes of that year. Note high fragmentation in 2010 in both middle panel and in the Fragmentation Index plot (black bar). The distance metric and filters used in the analysis were Pearson correlation and principal and secondary metric SVD. Metric: Correlation; Lens: Principal SVD Value (Resolution 120, Gain 4.5x, Equalized) and Secondary SVD Value (Resolution 120, Gain 4.5x, Equalized). Color: Red: Republican; Blue: Democrats.
Figure 5
Figure 5. The following map of players was constructed using the principal and secondary SVD filters at two different resolutions.
A) Low resolution map at 20 intervals for each filter B) High resolution map at 30 intervals for each filter. The overlap is such at that each interval overlaps with half of the adjacent intervals, the graphs are colored by points per game, and a variance normalized Euclidean distance metric is applied. Metric: Variance Normalized Euclidean; Lens: Principal SVD Value (Resolution 20, Gain 2.0x, Equalized) and Secondary SVD Value (Resolution 20, Gain 2.0x, Equalized). Color: red: high values, blue: low values.
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