Reviews in Computational Chemistry Volume 18

Clustering Algorithms 13
are equal. Assignment of each compound to the closest-cluster centroid is the
expectation step; recalculation of the cluster centroids (model parameters)
after assignment is the maximization step.
Topographic
Topographic clustering methods attempt to preserve the proximities
between clusters, thus facilitating visualization of the clustering results. For
k-means clustering, the cost function is invariant, whereas in topographic clustering
it is not, and a predefined neighborhood is imposed on the clusters to
preserve the proximities between them. The Kohonen, or self-organizing,
map, 37,38 apart from being one of the most commonly used types of neural
network, is also a topographic clustering method. A Kohonen network uses
an unsupervised learning technique to map higher dimensional spaces of a
data set down to, typically, two or three dimensions (2D or 3D), so that clusters
can be identified from the neurons’ coordinates (topological position); the
values of the output are ignored. Initially, the neurons are assigned weight vectors
with random values (weights). During the self-organization process, the
data vectors of the neuron having the most similar weight vector to each
data vector and its immediately adjacent neurons are updated iteratively to
place them closer to the data vector. The Kohonen mapping thus proceeds
as follows:
1. Initialize each neuron’s weight vector with random values.
2. Assign the next data vector to the neuron having the most similar weight
vector.
3. Update the weight vector of the neuron of step 2 to bring it closer to the
data vector.
4. Update neighboring weight vectors using a given updating function.
5. Repeat steps 2–4 until all data vectors have been processed.
6. Start again with the first data vector, and repeat steps 2–5 for a given
number of cycles.
The iterative adjustment of weight vectors is similar to the iterative refinement
of k-means clustering to derive cluster centroids. The main difference is that
adjustment affects neighboring weight vectors at the same time. Kohonen
mapping requires O(Nmn) time and OðNÞ space, where m is the number of
cycles and n the number of neurons.
Other Nonhierarchical Methods
We have delineated the main categories of clustering methods applicable
to chemical applications above. We have also provided one basic algorithm as
an example of each. Researchers in other disciplines sometimes use variants of
these main categories. The main categories that have been used by those
researchers but omitted here include density-based clustering and graph-based
clustering techniques. These will be mentioned briefly in the next section.