Note this is part 2 of a series on clustering RNAseq data. Check out part one on hierarcical clustering here and part two on K-means clustering here.

Clustering gene expression is a particularly useful data reduction technique for RNAseq experiments. It allows us to bin genes by expression profile, correlate those bins to external factors like phenotype, and discover groups of co-regulated genes. Two common methods for clustering are hierarchical (agglomerative) clustering and k-means (centroid based) clustering which we discussed in part one and part two of this series. Today we’re going to discuss yet another approach, fuzzy c-means clustering a.k.a. soft K-means clustering.

Fuzzy c-means clustering is useful for RNAseq data since gene expression is inherently noisy and fuzzy clustering is more robust to this noise. Furthermore we can use the membership score, a key feature of soft clustering, to filter genes which don’t have a high membership for any cluster (because they’re probably noise). First let’s talk a bit about what fuzzy c-means is and how it relates to k-means. Feel free to skip down to the part where we apply this algorithm to transcriptomic data.

What is fuzzy c-means clustering?

Fuzzy c-means clustering, oftentimes called soft k-means clustering, is a variant of k-means clustering in which each datapoint simulataneously exists in all clusters with varying degrees of membership which are on a scale of 0 and 1. By convention, we classify the datapoint into a cluster to which it has the highest membership.

How is fuzzy c-means clustering different from regular (hard) K-means?

As a partioning clustering algorithm K-means clustering will assign each and every datapoint to one and only one cluster. One downside to this is it forces in between datapoints into one clsuter or another. Furthermore since the mean of those data dictate the position of the cluster center these in between and outlier points can affect the overall cluster structure.

Fuzzy c-means on the other hand is very similar except it assigns each data-point a cluster membership score, where being closer to the cluster center means a higher score, and these scores are used to position the centroids. Thus, fuzzy clustering is more robust against noise and outliers since low scoring datapoints have a reduced impact on the position of the cluster center.

Consider the two plots below, the first is hard k-means clustering and the second is fuzzy c-means. Note that in the latter the cluster membership score is reported and those closer to the center have higher scores.

It’s subtle because these dummy data are pretty clusterable but the centroids are different on account that the data points with low membership have less effect on the centroid position.

This concept of membership is an important part of fuzzy c-means and we must keep in mind that every data point belongs in all clusters simultaneously with varying degrees of membership. Here’s the same data showing membership scores for each of the three clusters:

The sum of all of the membership scores for one data point is 1 and by convention we assign the data point to the cluster in which it’s membership is highest. As a side note. By applying a threshold of membership we can de-classify data points and eliminate them from downstream analyses.

Finally, fuzzy c-means introduces a fuzzifier, m, to determine how fuzzy the clusters are. m=1 is equivalent to hard clustering (memberships are binary of 0 or 1) and higher m lowers score overall of outlying data points.

Here’s how increasing the m affects the scores (note the plot is animated):

You can see as m increases the outlying datapoints decrease in membership score.

How to use fuzzy c-means with RNAseq data

Fuzzy clustering has several advantages over hard clustering when it comes to RNAseq data. Because the positioning of the centroids relies on data point membership the clustering is more robust to the noise inherent in RNAseq data.

There is a nice package, mFuzz, for performing fuzzy c-means clustering on expression data. Look out for a future tutorial on this package but for now we’ll use the e1071 package which includes several soft-clustering algorithms so we can get a better feel for the workflow.

For this tutorial we’ll use the same dataset we used previously in the post on post on k-means clustering. First we download the data, apply a quick normalization and, for the sake of this tutorial and computation time, we’ll take a subset of the most highly expressed and variable genes. I’m also injecting a random sample of low variance genes to see how the clustering algorithms hold up to this noise.

counts <- assays(rse_gene)$counts
y <- as.matrix((counts))
y <- DGEList(counts = y, group=c(1,2,3,4,5,6,7,8,9,10))
y <- calcNormFactors(y)
z <- cpm(y, normalized.lib.size=TRUE)

#mean/variance calculations
z_var <- apply(z, 1, var)
z_mean <- apply(z, 1, mean)

#take only the most highly express/ variable
#this is just for the sake of this tutorial
test_data <- z[which(z_var > 50 & z_mean > 50), 6:10]

#sample some of the lw variance genes
test_data <- rbind(test_data, z[sample(which(z_var < 5),1000), 6:10])

Should you filter the data?

Gene expression data is notoriously noisy. For k-means clustering I recommend filtering based on mean expression (drop low counts), variance, FC, or anything else to get rid of background genes. Since fuzzy c-means is more noise robust we could perform an a posteriori filter using the membership scores. This has the benefit of not exluding genes which have an interesting profile but don’t make the filtering threshold. It’s up to you. Either way I would exclude noisy genes from downstream analyses.

Before clustering however we need to scale the data. This is so that we can identify clusters of genes that share similar expression profiles rather than similar expression levels.

scaledata <- t(scale(t(test_data))) # Centers and scales data.
scaledata <- scaledata[complete.cases(scaledata),]

Fuzzy c-means: Estimate the fuzzifier

As we described above, fuzzy c-means relies on a fuzzifier parameter to designate how ‘fuzzy’ the clusters are. Schwaemmle and Jensen described a method for estimating the fuzzifier and Mattias Futschik wrapped it into a function which we are using here:

mestimate<- function(df){
  N <-  dim(df)[[1]]
  D <- dim(df)[[2]] <- 1 + (1418/N + 22.05)*D^(-2) + (12.33/N +0.243)*D^(-0.0406*log(N) - 0.1134)

m <- mestimate(scaledata)
## [1] 2.017301

Fuzzy c-means: How many clusters?

Fuzzy c-means, like K-means, clustering requires a priori knowledge of the number of clusters. The number of clusters can really impact the classifications so it’s an important consideration. Check out the post on k-means clustering for a lengthy discussion for choosing a cluster number. Those methods can be applied to fuzzy c-means as well. For simplicity here we will use the within cluster sum of squared error (elbow method):

#helper function for the within sum of squared error
sumsqr <- function(x, clusters){
  sumsqr <- function(x) sum(scale(x, scale = FALSE)^2)
  wss <- sapply(split(, clusters), sumsqr)

#get the wss for repeated clustering
iterate_fcm_WSS <- function(df,m){
  totss <- numeric()
  for (i in 2:20){
    FCMresults <- cmeans(df,centers=i,m=m)
    totss[i] <- sum(sumsqr(df,FCMresults$cluster))
wss_2to20 <- iterate_fcm_WSS(scaledata,m)
plot(1:20, wss_2to20[1:20], type="b", xlab="Number of Clusters", ylab="wss")

It looks like the inflection is between 4 and 6 clusters. In a full analysis I’d recommend comparing several different cluster numbers. Higher cluster numbers can unmask substructures that are hidden in lower cluster numbers. On the other hand with too many clusters can create redundant or highly overlapping clusters.

Here we will proceed with k=6.

k = 5
fcm_results <- cmeans(scaledata,centers=k,m=m)

First we’ll look the centroid profiles:

#import some data manipulation functions

#get the centroids into a long dataframe:
fcm_centroids <- fcm_results$centers
fcm_centroids_df <- data.frame(fcm_centroids)
fcm_centroids_df$cluster <- row.names(fcm_centroids_df)
centroids_long <- tidyr::gather(fcm_centroids_df,"sample",'value',1:5)

ggplot(centroids_long, aes(x=sample,y=value, group=cluster, colour=as.factor(cluster))) + 
  geom_line() +
  xlab("Time") +
  ylab("Expression") +
  labs(title= "Cluster Expression by Time",color = "Cluster")

So we have some interesting patterns! The cluster centroids are well separated although there might be some redundancy as a couple of the profiles look very similar.

We can assess the similarity of the cluster centroids to identify redundancy or high overlap. To do we simply correlate the cluster centroids with each other. If the centroids are too similar then they will have a high correlation. If your K number produces clusters with high correlation (say above 0.85) then consider reducing the number of clusters.

Correlate the centroids to see how similar they are:

##            1          2           3           4          5
## 1  1.0000000  0.2878932  0.51179922 -0.79775983 -0.8967693
## 2  0.2878932  1.0000000 -0.59583228 -0.67375330 -0.2903642
## 3  0.5117992 -0.5958323  1.00000000  0.06802654 -0.5887290
## 4 -0.7977598 -0.6737533  0.06802654  1.00000000  0.5695780
## 5 -0.8967693 -0.2903642 -0.58872899  0.56957800  1.0000000

It looks like we have good separation of the clusters as no cor score is above 0.85.

Now’s the fun part where we can plot the gene profiles by cluster. We simply subset the results and the centroids by cluster and plot it out. We can also set a color scale for the membership score:

#start with the input data
fcm_plotting_df <- data.frame(scaledata)

#add genes
fcm_plotting_df$gene <- row.names(fcm_plotting_df)

#bind cluster assinment
fcm_plotting_df$cluster <- fcm_results$cluster
#fetch the membership for each gene/top scoring cluster
fcm_plotting_df$membership <- sapply(1:length(fcm_plotting_df$cluster),function(row){
  clust <- fcm_plotting_df$cluster[row]

k_to_plot = 1

#subset the dataframe by the cluster and get it into long form
#using a little tidyr action
cluster_plot_df <- dplyr::filter(fcm_plotting_df, cluster == k_to_plot) %>%
  dplyr::select(.,1:5,membership,gene) %>%

#order the dataframe by score
cluster_plot_df <- cluster_plot_df[order(cluster_plot_df$membership),]
#set the order by setting the factors using forcats
cluster_plot_df$gene = forcats::fct_inorder(cluster_plot_df$gene)

#subset the cores by cluster
core <- dplyr::filter(centroids_long, cluster == k_to_plot)

ggplot(cluster_plot_df, aes(x=sample,y=value)) + 
    geom_line(aes(colour=membership, group=gene)) +
    scale_colour_gradientn(colours=c('blue1','red2')) +
    #this adds the core 
    geom_line(data=core, aes(sample,value, group=cluster), color="black",inherit.aes=FALSE) +
    xlab("Time") +
    ylab("Expression") +
    labs(title= paste0("Cluster ",k_to_plot," Expression by Time"),color = "Score")

In this plot, genes with a profile close to the core have a membership score approaching 1 (red) while those with divergent patterns have a score closer to 0 (blue). You can see there is some noise but the genes mostly fit the cluster model. If you observe many genes with low scores consider increasing your K as they’ve been ‘forced’ into a cluster in which they don’t belong. Too much noise in the data can also lead to low scoring genes.

Should you filter the data (redux)?

This is the point where I’d filter the data based on membership score before proceeding to downstream analysis. This will leave out genes that don’t fit well into any cluster, which are probably noise, before proceeding.

Comparing cluster methods:

Let’s see how this compares to hard K-means clustering!

First we perform the kmeans:

#perform the kmeans
kmeans_results <- kmeans(scaledata,centers=5)

Now we will use the WGCNA function matchLabels to line up the cluster assignments. This doesn’t change the structure of the cluster, but attempts to rename the cluster based on the cluster assignment from another dataset. Then we calculate the cluster overlap using the overlapTable function.

#these functions from the WCGNA package are great for this:

#grab the cluster vector for kmeans
kmeans_clusters <- kmeans_results$cluster
#grab the cluster vector for fuzzy
fcm_clusters <- fcm_results$cluster

#grab the cluster vector
kmeans_clusters_matched <- matchLabels(kmeans_clusters,fcm_clusters)

#add a prefix so we can tell them apart:
kmeans_clusters_matched <- paste0('K-',kmeans_clusters_matched)
fcm_clusters <- paste0('FCM-',fcm_clusters)

#calculate the overlap
OT<- overlapTable(kmeans_clusters_matched, fcm_clusters)
#get rid of 0 values...
OT$pTable[OT$pTable == 0] <- 2e-300

textMatrix= paste(signif(OT$countTable, 2), "\n(",
                      signif(OT$pTable, 1), ")", sep= "")
dim(textMatrix)= dim(OT$countTable)
heatmap.2(x= -log(OT$pTable),
          dendrogram = "none",
          Colv =F,
          Rowv = F,
          scale = c("none"),
          cellnote = textMatrix,
          cexRow = 0.8,
          cexCol = 0.8,
          main = "Cluster-Cluster Overlap",
          xlab = "Fuzzy C-means (k=5)",
          ylab = "Kmeans (k=5)")

As you can see there is some discrepancy in how the two algorithms behave but overall they perform similarly. Since we enriched our dataset in the most highly variable genes we would expect the clustering to be fairly robust in both cases. As we noted above, fuzzy c-means performs better in noisier datasets which might be more typical than our practice data here.

What to do from here?

Once we are happy with our clustering we can do lots of analyses on the clustered dataset. Including but not limited to:

  • Correlate phenotypic data with our clusters (can use the centroids for this).
  • Perform gene set enrichmnet analysis on our clusters (GSEA).
  • Test for gene ontology (GO) term enrichment in our clusters.
  • Compare cluster membership between datasets.
  • Analyze the highest scoring genes within clusters (core genes).

Hope this helps! Tune in for part four of this series on clustering when we take the mFuzz package for a test drive!

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##  [8] datasets  methods   base     
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