*K-Means and Cluster Models for Cancer Signatures

Mutation Clusters from Cancer Exome

“…The 96 × 32 matrix G is given in Tables A1-A4 is what we pass into the function bio.cl.sigs() in Appendix A of [16] as the input matrix x. We use: iter.max = 100 (this is the maximum number of iterations used in the built-in R function kmeans(); we note that there was not a single instance in our 30 million runs of kmeans() where more iterations were required -the R function kmeans() produces a warning if it does not converge within iter.max); num.try = 1000 (this is the number of individual k-means samplings we aggregate every time); and num.runs = 30,000 (which is the number of aggregated clusterings we use to determine the "ultimate", that is the most frequently occurring, clustering).…”

“…For Clustering-E1, as in [16], we compute the within-cluster weights based on unnormalized regressions (via Equations (13)- (15) in [16]) and normalized regressions (via Equations (14), (16) and (17) in [16]) with exposures calculated based on arithmetic averages (see Section 2.6 of [16] for details). We give the within-cluster weights for Clustering-E1 in Tables A6 and A7 and plot them in Figures A1-A11 for unnormalized regressions and in Tables 2 and 3 and Figures 1-11 for normalized regressions.…”

“…In [16], by extending a prior work [22] in quantitative finance on building statistical industry classifications using clustering algorithms, we developed a clustering method termed *K-means ("star K-means") and applied it to the extraction of cancer signatures from genome data. *K-means is anchored on the standard k-means algorithm (see [23][24][25][26][27][28][29]) as its basic building block.…”

“…(In [16], we applied it to published genome data. In this work, apart from applying *K-means to exome data, we also perform out-of-sample stability analysis of our results here (see Section 4).)…”

“…To circumvent this, in [16], we proposed an alternative approach that bypasses NMF altogether. As we argue in [16], NMF is, at least to a certain degree, clustering in disguise, e.g., many COSMIC cancer signatures [17] obtained via NMF (augmented with additional heuristics based on biologic intuition and empirical observations) exhibit clustering substructure, i.e., in many of these signatures, there are mutation categories with high weights ("peaks" or "tall mountain landscapes") with other mutation categories having small weights likely well within statistical and systematic errors. For all practical purposes, such low weights could be set to zero.…”