Merging the results of soft-clustering algorithm

Question

The established soft-clustering algorithms like Fuzzy-k-means (Wikipedia), Gustafson-Kessel, Gath-Geva for point wise data or the funclust algorithm in functional data context are random-operating algorithms.

The randomness of the initialization is why we get different cluster assignments for the observations at each converged run. If we execute the algorithm more often (let's say $n = 1000$ times) the algorithm will set the initial centroids randomly.

At my point of view that above mentioned is the reason why it is not possible to compare the results (probability of the assignment for an observation to each of $k$ clusters) of the different $n$ runs.

An example just for illustration

We try to cluster an unlabeled dataset of $n_{obs} = 10$ observations (or samples) with the same number of dimensions in $k = 3$ clusters and run, as in my case, the funcust algorithm $n_{run} = 5$ times. (I know that an dataset of $n_{obs}$ is to small for unsupervised clustering, it's only for this example.)

Now we can imagine a matrix $M$ containing the cluster assignments with dimension $n_{obs}$ X $n_{run}$:

M = [1 2 3 2 3 
     3 2 1 2 2 
     1 2 3 3 1 
     2 3 1 1 2 
     1 2 3 2 3 
     1 2 3 2 3 
     3 2 1 3 1 
     2 3 1 1 2 
     3 2 1 3 2 
     1 2 3 2 1]

As each of the 5 runs initialized randomly there is no chance to get an statistical cluster assignment. If that would be possible, it might be possible to calculate the mean (?) for each observation or at least a cluster assignment probability.

I'm very interested in your ideas. Maybe you had the same problem in the past. How did you think about it? Is there a solution for the above mentioned problem?

Answer 1

You need an approach that is insensitive to changing the numbers assigned to clusters, because these are random. The mean is pointless because of this, but there exist other consensus methods. Yet, it is all but trivial, as clusters may be orthogonal concepts.

Also, how would this relate to soft clustering? If you are working with such labels, then you are using hard clustering. In soft clustering, you would have had a vector for each point.