Clustering is a central task in big data analyses and clusters are often Gaussian or near Gaussian. However, a flexible Gaussian cluster simulation tool with precise control over the size, variance, and spacing of the clusters in NXN dimensional space does not exist. This is why we created clusterlab. The algorithm first creates X points equally spaced on the circumference of a circle in 2D space. These form the centers of each cluster to be simulated. Additional samples are added by adding Gaussian noise to each cluster center and concatenating the new sample co-ordinates. Then if the feature space is greater than 2D, the generated points are considered principal component scores and projected into N dimensional space using linear combinations using fixed eigenvectors. Through using vector rotations and scalar multiplication clusterlab can generate complex patterns of Gaussian clusters and outliers. Clusterlab is highly customizable and well suited to testing class discovery tools across a range of fields.

Contents

1. Simulating a single cluster
2. Simulating four clusters with equal variances
3. Simulating four clusters with unequal variances
4. Simulating four clusters with one cluster pushed to the outside
5. Simulating four clusters with one small cluster
6. Simulating five clusters with one central cluster
7. Simulating five clusters with ten outliers
8. Generating more complex multi-ringed structures I
9. Generating more complex multi-ringed structures II
10. Keeping track of cluster allocations
11. Closing comments

Simulating a single cluster

Here we simulate a 100 sample cluster with the default number of features (500). The standard deviation is left to default which is 1.

library(clusterlab)
synthetic <- clusterlab(centers=1,numbervec=100)
#> running clusterlab...
#> user has not set standard deviation of clusters, setting automatically...
#> user has not set alphas of clusters, setting automatically...
#> finished.

Simulating four clusters with equal variances

Next, we simulate a 4 cluster dataset with a radius of 8 for the circle on which the centers are placed. Then the standard deviations of the cluster are the same, 2.5. We set the alphas to 1, which is the value the clusters are pushed apart from one another. So there are two ways to seperate the clusters, either by the radius of the circle, or by the alpha parameter.

library(clusterlab)
synthetic <- clusterlab(centers=4,r=8,sdvec=c(2.5,2.5,2.5,2.5),   
                        alphas=c(1,1,1,1),centralcluster=FALSE,   
                        numbervec=c(50,50,50,50))
#> running clusterlab...
#> finished.

Simulating four clusters with unequal variances

The same as above, but 2 clusters have different variances to the other 2.

library(clusterlab)
synthetic <- clusterlab(centers=4,r=8,sdvec=c(1,1,2.5,2.5),   
                        alphas=c(1,1,1,1),centralcluster=FALSE,   
                        numbervec=c(50,50,50,50))
#> running clusterlab...
#> finished.

Simulating four clusters with one cluster pushed to the outside

The alpha parameter allows any number of clusters to be pushed away from the others. Here 1 cluster is pushed away slightly.

library(clusterlab)
synthetic <- clusterlab(centers=4,r=8,sdvec=c(2.5,2.5,2.5,2.5),   
                        alphas=c(1,2,1,1),centralcluster=FALSE,   
                        numbervec=c(50,50,50,50))
#> running clusterlab...
#> finished.

Simulating four clusters with one small cluster

Here we change the number vec entry for 1 cluster to a smaller value, therefore lowering the number of samples in the specified cluster.

library(clusterlab)
synthetic <- clusterlab(centers=4,r=8,sdvec=c(2.5,2.5,2.5,2.5),   
                        alphas=c(1,1,1,1),centralcluster=FALSE,   
                        numbervec=c(15,50,50,50))
#> running clusterlab...
#> finished.

Simulating five clusters with one central cluster

In this case we change the centralcluster parameter to TRUE, in order to make a central cluster as well as those placed on the circumference.

library(clusterlab)
synthetic <- clusterlab(centers=5,r=8,sdvec=c(2.5,2.5,2.5,2.5,2.5),   
                        alphas=c(2,2,2,2,2),centralcluster=TRUE,   
                        numbervec=c(50,50,50,50,50))
#> running clusterlab...
#> finished.

Simulated five clusters with one central cluster

We can add outliers and a distance transform them from their original coordinate. The angle chosen is random for every coordinate.

library(clusterlab)
synthetic <- clusterlab(centers=5,r=7,sdvec=c(2,2,2,2,2),   
                        alphas=c(2,2,2,2,2),centralcluster=FALSE,   
                        numbervec=c(50,50,50,50), seed=123, outliers=10, outlierdist = 20)
#> running clusterlab...
#> user has not set length of numbervec equal to number of clusters, setting automatically...
#> we are generating outliers...
#> finished.

Generating more complex multi-ringed structures I

The package is also capable of generating concentric circles of clusters which allows more complex structures to be generated. The standard parameters we used previously per cluster apply to all clusters. To space the rings out we use the ringalphas parameter. Note, the stepwise number sequence specified below for ringalphas so the clusters do not form on top of each other.

library(clusterlab)
synthetic <- clusterlab(centers=5,r=7,sdvec=c(6,6,6,6,6),   
                        alphas=c(2,2,2,2,2),centralcluster=FALSE,   
                        numbervec=c(50,50,50,50),rings=5,ringalphas=c(2,4,6,8,10,12),
                        seed=123) # for a six cluster solution)
#> running clusterlab...
#> user has not set length of numbervec equal to number of clusters, setting automatically...
#> ring thetas not set, setting automatically...
#> we are generating clusters arranged in rings...
#> finished.

Generating more complex multi-ringed structures II

The ringthetas parameter may be used to rotate each ring individually. Through rotating the clusters complex patterns may be formed.

library(clusterlab)
synthetic <- clusterlab(centers=5,r=7,sdvec=c(6,6,6,6,6),   
                        alphas=c(2,2,2,2,2),centralcluster=FALSE,   
                        numbervec=c(50,50,50,50),rings=5,ringalphas=c(2,4,6,8,10,12), 
                        ringthetas = c(30,90,180,0,0,0), seed=123) # for a six cluster solution)
#> running clusterlab...
#> user has not set length of numbervec equal to number of clusters, setting automatically...
#> we are generating clusters arranged in rings...
#> finished.

Keeping track of cluster allocations

Clusterlab also keeps track of the cluster allocations and gives each sample an unique ID. This may prove useful when scoring class discovery algorithms assignments.

head(synthetic$identity_matrix)
#>   sampleID cluster
#> 1     c1s1       1
#> 2     c1s2       1
#> 3     c1s3       1
#> 4     c1s4       1
#> 5     c1s5       1
#> 6     c1s6       1

Closing comments

We have seen how the clusterlab package may generate NXN Gaussian clusters in a flexible manner. For class discovery of these types of clusters we recommend clusterlab's sister package, M3C which was developed in parallel. M3C is available on the Bioconductor (https://bioconductor.org/packages/devel/bioc/html/M3C.html).

Thanks for using clusterlab.