Fast multi-resolution consensus clustering

Clustering is a key tool for exploring results from large scale data collection efforts. It is especially useful to find clusterings (partitions of data into non-overlapping sets) at different levels of granularity, ranging from detailed to abstract. In order to be consistent such a set of clusterings should be nested, requiring that elements that group together in a given clustering group together in all coarser clusterings. Hierarchical clustering provides a tree for the data and naturally induces nested clusterings, but none of the methods have gained traction to provide such a multi-resolution view. Widely used methods such as Leiden and the Markov Cluster method (MCL) can produce clusterings at different scales controlled by a resolution parameter, but the results are generally not nested. I introduce Restricted Contingency Linkage (RCL), a parameter-free consensus method that uniquely integrates and reconciles a set of flat clusterings with potentially widely varying levels of granularity into a single multi-resolution view. A common starting point in consensus methods is the association matrix A, where Aij tallies the number of co-occurrences of i and j in the same cluster across all clusterings. RCL creates a matrix R where Rij tallies a measure that differentiates i-j pairs with respect to the amount that they co-cluster in pairs of clusterings while taking into account the subset relationship between the clusters involved. The entries in R are fully abstracted from the input data, richly differentiated and uniformly normalised, making it suitable for single linkage clustering. From the resulting tree a limited set of flat clusterings is obtained by varying a resolution criterion along a logarithmic scale. I validate the method using large-scale single-cell transcriptome data from human developing kidneys, along with visualisations and quality control measures of clustering ensembles. Marker gene expression is summarised in a heatmap across the multi-level RCL clusters, providing a coherent view of cell type and cell state hierarchies within the tissue. This reveals clusters corresponding to small populations of mast cells, plasma cells and interstitial cells, and shows segregation of cycling cells in small populations across different cell clades. Given an ensemble of input clusterings, RCL rapidly enables these analyses in a parameter-free manner. An RCL reference implementation is provided for clustering ensembles that are associated with a network G, further restricting the RCL matrix to entries that correspond to edges in G. For a network G with m edges this implementation has complexity [O](m(p2 + log(m))) where p is the number of input clusterings, taking less than a minute on a dataset with N = 27k elements, m = 1.5M edges and p = 24 clusterings. The implementation, including software for QC plots, marker gene heatmaps and other visualisations is available as part of the MCL software package (https://github.com/micans/mcl). O_TEXTBOXA suggestion for a quick guide to RCL is the pseudocode in Listing 1 and the multi-resolution cluster/marker-gene heatmap in Figures 11.1-11.2, supplemented with quality control plots for clustering granularity (Figures 8.1 and 8.1) and clustering discrepancy (Pages 5-6 and Figure 8.3). C_TEXTBOX O_FIG O_LINKSMALLFIG WIDTH=171 HEIGHT=200 SRC="FIGDIR/small/511493v1_fig11_1.gif" ALT="Figure 111"> View larger version (85K): org.highwire.dtl.DTLVardef@121dfecorg.highwire.dtl.DTLVardef@f43c0eorg.highwire.dtl.DTLVardef@92e4a7org.highwire.dtl.DTLVardef@1e1ba4b_HPS_FORMAT_FIGEXP M_FIG O_FLOATNOFig. 11.1:C_FLOATNO ComplexHeatmap [37] of marker genes for RCL clustering of cell transcriptomes from 27203 developing kidney cells. RCL clustering was performed as described in Section 8. Heatmap columns correspond to the rows in the tabular representation of the most fine-grained clusters (Section 10). Coarser levels are visible in the column clustering and the associated dendrogram. Clusters of residual nodes (Section 10) can be recognised as horizontal bars without a vertical descent. (Heatmap continued on the following page). C_FIG O_FIG O_LINKSMALLFIG WIDTH=171 HEIGHT=200 SRC="FIGDIR/small/511493v1_fig11_2.gif" ALT="Figure 112"> View larger version (77K): org.highwire.dtl.DTLVardef@8da768org.highwire.dtl.DTLVardef@1e9ae91org.highwire.dtl.DTLVardef@c3b292org.highwire.dtl.DTLVardef@99e4_HPS_FORMAT_FIGEXP M_FIG O_FLOATNOFig. 11.2:C_FLOATNO Continuation of the heatmap of marker genes for RCL clusters across developing kidney cell transcriptomes in Figure 11.1. C_FIG O_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=131 SRC="FIGDIR/small/511493v1_fig8_1.gif" ALT="Figure 81"> View larger version (34K): org.highwire.dtl.DTLVardef@1776682org.highwire.dtl.DTLVardef@1eb0783org.highwire.dtl.DTLVardef@13a2f91org.highwire.dtl.DTLVardef@10913b0_HPS_FORMAT_FIGEXP M_FIG O_FLOATNOFig. 8.1:C_FLOATNO Each graph corresponds to a clustering of the kidney data (Section 8) for the specified Leiden resolution value (also called gamma). The graphs show the fraction of elements that are part of a cluster of size at most x for varying x. C_FIG O_FIG O_LINKSMALLFIG WIDTH=159 HEIGHT=200 SRC="FIGDIR/small/511493v1_fig8_3.gif" ALT="Figure 83"> View larger version (78K): org.highwire.dtl.DTLVardef@ebc1caorg.highwire.dtl.DTLVardef@1ffba5forg.highwire.dtl.DTLVardef@24f292org.highwire.dtl.DTLVardef@1b1795f_HPS_FORMAT_FIGEXP M_FIG O_FLOATNOFig. 8.3:C_FLOATNO Partition discrepancy (expressed as percentage) for Leiden and mcl clusterings. The i, j value denotes the percentage of elements needed to split off in order to transform [P]i into gcs([P]i, [P]j). Low i, j value compared to j, i implies that [P]i is close to being a subclustering of [P]j. C_FIG