Introduction
Cluster analysis is a way of “slicing and dicing” data to allow the
grouping together of similar entities and the separation of dissimilar
ones. Issues arise due to the existence of a diverse number of
clustering algorithms, each with different techniques and inputs, and
with no universally optimal methodology. Thus, a framework for cluster
analysis and validation methods are needed. Our approach is to use
cluster ensembles from a diverse set of algorithms. This ensures that
the data has been considered from several angles and using a variety of
methods. We have currently implemented about 15 clustering algorithms,
and we provide a simple framework to add additional algorithms (see
example("consensus_cluster")). Although results are
dependent on the subset of algorithms chosen for the ensemble, the
intent is that we capture a variety of clustering outputs and select
those that are most consistent with the data.
You can install diceR from CRAN with:
install.packages("diceR")
Or get the latest development version from GitHub:
# install.packages("devtools")
devtools::install_github("AlineTalhouk/diceR")
library(diceR)
library(dplyr)
library(ggplot2)
library(pander)
data(hgsc)
hgsc <- hgsc[1:100, 1:50]
We load an excerpt of an expression data set from TCGA of 100 high
grade serous carcinoma samples measured on 50 genes.
Clustering Algorithms
The list of clustering algorithms available for use are:
"nmf": Nonnegative Matrix Factorization (using
Kullback-Leibler Divergence or Euclidean distance)"hc": Hierarchical Clustering"diana": DIvisive ANAlysis Clustering"km": K-Means Clustering"pam": Partition Around Medoids"ap": Affinity Propagation"sc": Spectral Clustering using Radial-Basis kernel
function"gmm": Gaussian Mixture Model using Bayesian
Information Criterion on EM algorithm"block": Biclustering using a latent block model"som": Self-Organizing Map (SOM) with Hierarchical
Clustering"cmeans": Fuzzy C-Means Clustering"hdbscan": Hierarchical Density-based Spatial
Clustering of Applications with Noise (HDBSCAN)
They are passed as a character vector into the
algorithms parameter of consensus_cluster().
Note that the list continues to evolve as we implement more algorithms
into diceR.
Cluster Models
The clustering algorithms can be categorized under different models
or paradigms based on how the clusters are formed. We provide a brief
overview to guide the initial selection of algorithms since no single
algorithm works for every data model.1
- Connectivity Models:
"hc" and "diana" are
hierarchical models based on connecting objects using a particular
similarity or distance metric.
- Centroid Models:
"km" and "pam" uses
initial cluster centers calculated based on the mean of the objects,
where "pam" restricts the centroid to be an actual object.
These models are good for modeling data that produces spherical
clusters. "ap" is similar to "pam" because it
calculates “exemplars” (like centroids) of the data but is advantageous
because initialization of k (number of clusters) is not
required.2 "cmeans" is similar to
"km" except that points can be assigned to more than one
cluster.
- Distribution Models:
"gmm" assumes a Gaussian mixture
model used by the EM algorithm for clustering.3 The gain in
model complexity is traded with utility since the assumption is quite
strong.
- Density Models:
"hdbscan" runs hierarchical clustering
on the DBSCAN result.4,5 These models assume that each
cluster has high density, so when there are overlapping regions, the
algorithms may not be able to separate objects that well.
- Subspace Models:
"block" clusters both samples and
features, so is useful when you want to associate these pairs of cluster
entities (e.g. are there functional gene groups that associate with
sample subtypes?). Our implementation uses the latent block
model.6
- Neural Models: our implementation of
"som" reduces the
data to a two-dimensional subspace using a neural network model, before
performing hierarchical clustering.7 A lot of the noise is
phased out and cluster stability can be improved from using
"hc" alone.
- Spectral Clustering: Similar to
"som",
"sc" also performs dimensionality reduction before
clustering, but instead uses eigenvectors of a kernel
matrix.8 We use a radial-basis kernel function in the
package.
- Non-negative Matrix Factorization: the computational complexity with
"nmf" is doubled compared to other algorithms because data
manipulations need to be performed to transform the input data into a
non-negative matrix. Otherwise, NMF has a natural clustering result that
can be used to group together samples and/or features.9
Distance Measures
The list of distance measures available for use are those found in
stats::dist and the spearman distance:
"euclidean": see stats::dist"maximum": see stats::dist"manhattan": see stats::dist"canberra": see stats::dist"binary": see stats::dist"minkowski": see stats::dist"spearman": based on
bioDist::spearman.dist
They are passed as a character vector into the distance
parameter of consensus_cluster(). Only the
"hc", "diana", "pam" algorithms
use the distance parameter.
Consensus Clustering
When Monti et al. (2003) first introduced consensus clustering, the
idea was to use one clustering algorithm on bootstrapped subsamples of
the data. We implement some extensions where a consensus is reached
across subsamples and across algorithms. The final cluster
assignment is then computed using statistical transformations on the
ensemble cluster.
The base function of this package is
consensus_cluster(), which outputs cluster assignments
across subsamples and algorithms, for different k (number of clusters).
For example, let’s say we were interested in clustering the
hgsc data into 3 or 4 clusters, using 80% resampling on 5
replicates, for these clustering algorithms: Hierarchical Clustering,
PAM, and DIvisive ANAlysis Clustering (DIANA). Euclidean distance is
used for all algorithms.
CC <- consensus_cluster(hgsc, nk = 3:4, p.item = 0.8, reps = 5,
algorithms = c("hc", "pam", "diana"))
The output is a 4-dimensional array: rows are samples, columns are
different bootstrap subsample replicates, slices are algorithms, and
each “box” (4th dimension) is for a different k. Below are the first few
cluster assignments for each replicate in the DIANA algorithm for k =
3.
co <- capture.output(str(CC))
strwrap(co, width = 80)
#> [1] "int [1:100, 1:5, 1:3, 1:2] 1 1 NA NA NA 1 1 NA 1 NA ..."
#> [2] "- attr(*, \"dimnames\")=List of 4"
#> [3] "..$ : chr [1:100] \"TCGA.04.1331_PRO.C5\" \"TCGA.04.1332_MES.C1\""
#> [4] "\"TCGA.04.1336_DIF.C4\" \"TCGA.04.1337_MES.C1\" ..."
#> [5] "..$ : chr [1:5] \"R1\" \"R2\" \"R3\" \"R4\" ..."
#> [6] "..$ : chr [1:3] \"HC_Euclidean\" \"PAM_Euclidean\" \"DIANA_Euclidean\""
#> [7] "..$ : chr [1:2] \"3\" \"4\""
Cluster Assignments for DIANA, k = 3
| TCGA.04.1331_PRO.C5 |
2 |
2 |
1 |
1 |
1 |
| TCGA.04.1332_MES.C1 |
1 |
2 |
1 |
1 |
1 |
| TCGA.04.1336_DIF.C4 |
NA |
1 |
NA |
2 |
NA |
| TCGA.04.1337_MES.C1 |
NA |
NA |
2 |
2 |
2 |
| TCGA.04.1338_MES.C1 |
NA |
NA |
2 |
2 |
2 |
| TCGA.04.1341_PRO.C5 |
2 |
NA |
1 |
NA |
1 |
Note the unavoidable presence of NAs because we used 80%
subsampling. This can be problematic in certain downstream ensemble
methods, so we can impute missing values using K-Nearest Neighbours
beforehand. There might still be NAs after KNN because of
how the decision threshold was set. The remaining missing values can be
imputed using majority voting.
CC <- apply(CC, 2:4, impute_knn, data = hgsc, seed = 1)
CC_imputed <- impute_missing(CC, hgsc, nk = 4)
sum(is.na(CC))
#> [1] 2
sum(is.na(CC_imputed))
#> [1] 0
We can carry on the analysis using either CC or
CC_imputed.
Consensus Functions
diceR provides functions for retrieving useful summaries
and other results for consensus clustering.
Compute consensus matrix with consensus_matrix()
The consensus matrix is an n by n matrix, where n is the number of
samples. Each element is a real-valued number between 0 and 1 inclusive,
representing the proportion of times that two samples were clustered
together out of the times that the same samples were chosen in the
bootstrap resampling. The diagonal are all one’s. Suppose we wanted to
compute the consensus matrix for PAM, k = 4, and visualize using
graph_heatmap():
pam.4 <- CC[, , "PAM_Euclidean", "4", drop = FALSE]
cm <- consensus_matrix(pam.4)
dim(cm)
#> [1] 100 100
hm <- graph_heatmap(pam.4)
Combine consensus summaries with
consensus_combine()
If we wish to separately extract consensus matrices and consensus
classes for every algorithm, consensus_combine() is a
convenient wrapper to do so. Setting element = "matrix"
returns a list of consensus matrices. On the other hand, setting
element = "class" returns a matrix with rows as samples,
and columns as clustering assignments for each algorithm.
ccomb_matrix <- consensus_combine(CC, element = "matrix")
ccomb_class <- consensus_combine(CC, element = "class")
str(ccomb_matrix, max.level = 2)
#> List of 2
#> $ 3:List of 3
#> ..$ HC_Euclidean : num [1:100, 1:100] 1 0.8 1 0.8 1 1 1 1 1 1 ...
#> ..$ PAM_Euclidean : num [1:100, 1:100] 1 1 0.2 0.8 0 1 1 0.8 0 0.2 ...
#> ..$ DIANA_Euclidean: num [1:100, 1:100] 1 0.8 0 0 0 1 1 0 0 1 ...
#> $ 4:List of 3
#> ..$ HC_Euclidean : num [1:100, 1:100] 1 0.6 1 0.6 1 1 1 1 1 0.8 ...
#> ..$ PAM_Euclidean : num [1:100, 1:100] 1 1 0.2 0.8 0 0.8 0.8 0.6 0 0 ...
#> ..$ DIANA_Euclidean: num [1:100, 1:100] 1 0.8 0 0 0 1 1 0 0 1 ...
Consensus Classes
| 1 |
1 |
1 |
| 2 |
1 |
1 |
| 1 |
2 |
2 |
| 2 |
1 |
2 |
| 1 |
3 |
2 |
| 1 |
1 |
1 |
One can feed in ccomb_class (instead of CC)
into consensus_matrix() to obtain a consensus matrix across
subsamples and algorithms (and k).
# consensus matrix across subsamples and algorithms within k = 3
cm_k3 <- consensus_matrix(ccomb_class$`3`)
# consensus matrix across subsamples and algorithms within k = 4
cm_k4 <- consensus_matrix(ccomb_class$`4`)
# consensus matrix across subsamples and algorithms and k
cm_all <- consensus_matrix(ccomb_class)
A situation might also arise where we initially decided on using 3
clustering algorithms for the ensemble, but later wish to add additional
algorithms for analysis. consensus_combine() takes in any
number of ensemble objects (e.g. CC and CC2)
and combines the results.
CC2 <- consensus_cluster(hgsc, nk = 3:4, p.item = 0.8, reps = 5,
algorithms = "km")
ccomb_class2 <- consensus_combine(CC, CC2, element = "class")
Consensus Classes with KM added
| 1 |
1 |
1 |
1 |
| 2 |
1 |
1 |
1 |
| 1 |
2 |
2 |
2 |
| 2 |
1 |
2 |
2 |
| 1 |
3 |
2 |
2 |
| 1 |
1 |
1 |
3 |
Evaluate, trim, and reweigh algorithms with
consensus_evaluate()
Internal cluster validation indices assess the performance of results
by taking into account the compactness and separability of the clusters.
We choose a variety of indices on which to compare the collection of
clustering algorithms. We use the PAC (Proportion of Ambiguous
Clusters), the proportion of entries in a consensus matrix that are
strictly between lower (defaults to 0) and
upper (defaults to 1), to give a measure of cluster
stability. In addition, if no reference class is provided, we calculate
the average PAC across algorithms within each k, and choose
the k with the greatest average PAC. If there is a
reference class, k is the number of distinct classes in the
reference.
ccomp <- consensus_evaluate(hgsc, CC, CC2, plot = FALSE)
Internal Indices for k = 4 (continued below)
| HC_Euclidean |
HC_Euclidean |
4.24 |
0.4205 |
11.98 |
0.3404 |
0.7049 |
| PAM_Euclidean |
PAM_Euclidean |
15.4 |
0.2849 |
7.175 |
0.3615 |
0.5475 |
| DIANA_Euclidean |
DIANA_Euclidean |
12.47 |
0.28 |
6.425 |
0.3917 |
0.5588 |
| KM |
KM |
17.22 |
0.3162 |
6.495 |
0.3923 |
0.6113 |
Table continues below
| HC_Euclidean |
0.202 |
1.591 |
0.7303 |
0.3595 |
0.954 |
0.03441 |
| PAM_Euclidean |
0.1882 |
2.341 |
0.7881 |
0.6273 |
2.008 |
0.09865 |
| DIANA_Euclidean |
0.1871 |
1.969 |
0.7822 |
0.4838 |
1.654 |
0.1084 |
| KM |
0.1599 |
1.9 |
0.7685 |
0.4727 |
1.247 |
0.08003 |
| HC_Euclidean |
0.1171 |
NA |
7.833 |
19.03 |
| PAM_Euclidean |
0.08106 |
NA |
6.831 |
65.74 |
| DIANA_Euclidean |
0.02616 |
NA |
7.081 |
47.95 |
| KM |
0.112 |
NA |
6.704 |
62.96 |
We see that the biclustering algorithm is the least ambiguous and
also most well-clustered (high compactness and separability).
Some algorithms perform too poorly to deserve membership in the
cluster ensemble. We consider the relative ranks of each algorithm
across all internal indices, and compute their sum. All algorithms below
a certain quantile for the sum rank are trimmed (removed). By default
this quantile is 75%.
After trimming, we can optionally choose to reweigh the algorithms
based on the internal index magnitudes. Of course, we take into account
the direction of optimality (higher is better is lower is better).
Algorithms reweighed are then fed into the consensus functions. This is
done by replicating each algorithm by a scalar factor that is
proportional to its weight. For example, if we have two algorithms A and
B, and A is given a weight of 80% and B is given a weight of 20%, then
we make 4 copies of A and 1 copy of B. To minimize computational time,
the total number of copies out of all algorithms has an upper bound of
100. Without reweighing, each algorithm is given equal weight.
ctrim <- consensus_evaluate(hgsc, CC, CC2, trim = TRUE, reweigh = FALSE, n = 2)
str(ctrim, max.level = 2)
#> List of 5
#> $ k : int 4
#> $ pac :'data.frame': 2 obs. of 5 variables:
#> ..$ k : chr [1:2] "3" "4"
#> ..$ HC_Euclidean : num [1:2] 0.134 0.397
#> ..$ PAM_Euclidean : num [1:2] 0.458 0.451
#> ..$ DIANA_Euclidean: num [1:2] 0.248 0.233
#> ..$ KM : num [1:2] 0.282 0.272
#> $ ii :List of 2
#> ..$ 3:'data.frame': 4 obs. of 16 variables:
#> ..$ 4:'data.frame': 4 obs. of 16 variables:
#> $ ei : NULL
#> $ trim.obj:List of 5
#> ..$ alg.keep : chr [1:2] "HC_Euclidean" "KM"
#> ..$ alg.remove : chr [1:2] "PAM_Euclidean" "DIANA_Euclidean"
#> ..$ rank.matrix:List of 1
#> ..$ top.list :List of 1
#> ..$ E.new :List of 1
The return value shows which algorithms were kept, removed (if any),
and the trimmed (and potentially reweighed) cluster ensemble.
Significance Testing
To test whether there are four statistically distinct clusters (k =
4) versus no clusters (k = 1) using the PAM algorithm, we run
sigclust with 50 simulations and generate a p-value for
this significance test.
set.seed(1)
pam_4 <- ccomb_class2$`4`[, "PAM_Euclidean"]
sig_obj <- sigclust(hgsc, k = 4, nsim = 100, labflag = 0, label = pam_4)
co <- capture.output(str(sig_obj))
strwrap(co, width = 80)
#> [1] "Formal class 'sigclust' [package \"sigclust\"] with 10 slots"
#> [2] "..@ raw.data : num [1:100, 1:50] -0.0107 -0.7107 0.8815 -1.0851 -0.9322 ..."
#> [3] ".. ..- attr(*, \"dimnames\")=List of 2"
#> [4] ".. .. ..$ : chr [1:100] \"TCGA.04.1331_PRO.C5\" \"TCGA.04.1332_MES.C1\""
#> [5] "\"TCGA.04.1336_DIF.C4\" \"TCGA.04.1337_MES.C1\" ..."
#> [6] ".. .. ..$ : chr [1:50] \"ABAT\" \"ABHD2\" \"ACTB\" \"ACTR2\" ..."
#> [7] "..@ veigval : num [1:50] 11.81 4.51 2.66 2.29 1.84 ..."
#> [8] "..@ vsimeigval: num [1:50] 11.81 4.51 2.66 2.29 1.84 ..."
#> [9] "..@ simbackvar: num 0.42"
#> [10] "..@ icovest : num 2"
#> [11] "..@ nsim : num 100"
#> [12] "..@ simcindex : num [1:100] 0.624 0.692 0.664 0.672 0.637 ..."
#> [13] "..@ pval : num 0.23"
#> [14] "..@ pvalnorm : num 0.206"
#> [15] "..@ xcindex : num 0.63"
The p-value is 0.23, indicating we do not have sufficient evidence to
conclude there are four distinct clusters. Note that we did not use the
full hgsc data set in this example, so the underlying
biological mechanisms may not be fully captured.