• Motivation

• Initial methods

• Graph-based methods

  • walktrap
  • louvain
  • leiden

The data has been QCed and normalized, confounders removed, noise limited, dimensionality reduced.

We can now ask biological questions.

• de novo discovery and annotation of cell-types based on transcription profiles

• unsupervised clustering:

  • identification of groups of cells
  • based on the similarities of the transcriptomes
  • without any prior knowledge of the labels
  • usually using the PCA output

We will introduce three widely used clustering methods:

• hierarchical
• k-means
• graph-based

The first two were developed first and are faster for small data sets.

The third is more recent and better suited for scRNA-seq, especially large data sets.

All three identify non-overlapping clusters.

Hierarchical clustering builds:

• a hierarchy of clusters
• yielding a dendrogram (i.e. tree)
  • that groups together cells with similar expression patterns
  • across the chosen genes.

There are two types of strategies:

• Agglomerative (bottom-up):
  • each observation (cell) starts in its own cluster,
  • pairs of clusters are merged as one moves up the hierarchy.
• Divisive (top-down):
  • all observations (cells) start in one cluster,
  • splits are performed recursively as one moves down the hierarchy.

Set of steps to repeat:

• randomly select k data points to serve as initial cluster centers,
• for each centers, 1) compute distance to centroids, 2) assign to closest cluster,
• calculate the mean of each cluster (the ‘mean’ in ‘k-mean’) to define its centroid,
• for each point compute the distance to these means to choose the closest,
• repeat until the distance between centroids and data points is minimal (ie clusters do not change) or the maximum number of iterations is reached, compute the total variation within clusters

Congruence of clusters may be assessed by computing the sillhouette for each cell.

The larger the value the closer the cell to cells in its cluster than to cells in other clusters.

Cells closer to cells in other clusters have a negative value.

Good cluster separation is indicated by clusters whose cells have large silhouette values.

Example with different numbers of neighbours:

Pros

• fast and memory efficient (no distance matrix for all pairs of cells)
• no assumptions on the shape of the clusters or the distribution of cells within each cluster
• no need to specify a number of clusters to identify

Cons

• loss of information beyond neighboring cells, which can affect community detection in regions with many cells.

Several methods to detect clusters (‘communities’) in networks rely on the ‘modulatrity’ metric.

For a given partition of cells into clusters,

modularity measures how separated clusters are from each other,

based on the difference between the observed and expected weight of edges between nodes.

For the whole graph, the closer to 1 the better.

The walktrap method relies on short random walks (a few steps) through the network.

These walks tend to be ‘trapped’ in highly-connected regions of the network.

Node similarity is measured based on these walks.

• Nodes are first each assigned their own community.
• Pairwise distances are computed and the two closest communities are grouped.
• These steps are repeated a given number of times to produce a dendrogram.
  • Hierarchical clustering is applied to the distance matrix.
• The best partition is that with the highest modularity.

Network example:

Issue with the Louvain method: some communities may be disconnected:

(Traag et al, From Louvain to Leiden: guaranteeing well-connected communities)

Clusters that are well separated mostly comprise intra-cluster edges and harbour a high modularity score on the diagonal and low scores off that diagonal.

Two poorly separated clusters will share edges and the pair will have a high score.

• hierarchical and k-means methods are fast for small data sets

• graph-based methods are better suited for large data sets and cluster detection