This pipeline calculates clusters based on a consensus nonnegative matrix factorization (NMF) clustering method [1],[2]. This pipeline has the following features:

Convert input data set to a nonnegitive matrix by column rank normalization.

Classify samples into consensus clusters.

Determine differentially expressed marker genes for each subtype.
The most robust consensus NMF clustering of 529 samples using the 1500 most variable genes was identified for k = 8 clusters. We computed the clustering for k = 2 to k = 8 and used the cophenetic correlation coefficient to determine the best solution.
SampleName  cluster  silhouetteValue 

TCGAA1A0SD01A11RA11507  1  0.035 
TCGAA1A0SH01A11RA08407  1  0.011 
TCGAA1A0SM01A11RA08407  1  0.092 
TCGAA2A0SV01A11RA08407  1  0.023 
TCGAA8A06P01A11RA00Z07  1  0.027 
TCGAA8A07I01A11RA00Z07  1  0.015 
TCGAA8A08Z01A21RA00Z07  1  0.11 
TCGAA8A09701A11RA03407  1  0.039 
TCGAANA03X01A21RA00Z07  1  0.078 
TCGAANA0AS01A11RA00Z07  1  0.084 
SampleName  K=3  K=4  K=5  K=6  K=7  K=8  NA 

TCGAA1A0SD01A11RA11507  1  1  1  1  1  1  1 
TCGAA1A0SE01A11RA08407  1  1  2  2  2  2  2 
TCGAA1A0SH01A11RA08407  1  2  2  3  3  3  1 
TCGAA1A0SJ01A11RA08407  1  1  2  2  2  1  3 
TCGAA1A0SM01A11RA08407  1  1  1  3  3  3  1 
TCGAA2A04N01A11RA11507  1  1  1  3  1  3  5 
TCGAA2A04R01A41RA10907  1  1  1  1  5  1  3 
TCGAA2A04V01A21RA03407  1  1  1  1  1  1  3 
TCGAA2A04Y01A21RA03407  1  1  3  1  5  1  3 
TCGAA2A0CP01A11RA03407  1  2  2  2  2  2  2 
Samples most representative of the clusters, hereby called core samples were identified based on positive silhouette width, indicating higher similarity to their own class than to any other class member. Core samples were used to select differentially expressed marker genes for each subtype by comparing the subclass versus the other subclasses, using Student's ttest.
Composite.Element.REF  p  difference  q  subclass 

MMP13  1.6e07  2.7  7.7e06  1 
C3ORF57  0.000095  2.5  0.0014  1 
C20ORF39  9.5e12  2.4  2.3e09  1 
MUM1L1  0.000031  2.4  0.00055  1 
COL11A1  2.4e16  2.3  5.1e13  1 
ZMAT4  0.00022  2.2  0.0027  1 
THBS2  5.2e18  2.2  2.3e14  1 
C1QTNF3  1.1e09  2.2  1.2e07  1 
ITGA11  4.8e12  2.1  1.5e09  1 
HAPLN1  0.000017  2.1  0.00034  1 
Nonnegative matrix factorization (NMF) is an unsupervised learning algorithm that has been shown to identify molecular patterns when applied to gene expression data [1],[2]. Rather than separating gene clusters based on distance computation, NMF detects contextdependent patterns of gene expression in complex biological systems.
We use the cophenetic correlation coefficient [1] to determine the cluster that yields the most robust clustering. The cophenetic correlation coefficient is computed based on the consensus matrix of the CNMF clustering, and measures how reliably the same samples are assigned to the same cluster across many iterations of the clustering lgorithm with random initializations. The cophenetic correlation coefficient lies between 0 and 1, with higher values indicating more stable cluster assignments. We select the number of clusters k based on the largest observed correlation coefficient for all tested values of k.
Silhouette width is defined as the ratio of average distance of each sample to samples in the same cluster to the smallest distance to samples not in the same cluster. If silhouette width is close to 1, it means that sample is well clustered. If silhouette width is close to 1, it means that sample is misclassified [3].
This is an experimental feature. The full results of the analysis summarized in this report can be downloaded from the TCGA Data Coordination Center.