Exploratory Data Analysis Methods in Functional MRI - Advance Image Processing in Magnetic Resonance Imaging - page 500

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FIGURE 16.4 Two-dimensional objects grouped in three clusters by the k -means algo-
rithm. The cluster centers are the bold circles.

the number of clusters to be chosen based on the evolution of within-class inertia.
In Reference 14 the two approaches are combined in a hierarchical k -means
approach: in this chapter, the combined approach is used to find the number of
clusters and the initial guess of their centers is used in a final k -means algorithm.
The procedure consists of iteratively splitting clusters that are found to have some
structure left into two groups by means of a k -means algorithm. The crucial point
here concerns the identification of the clusters to split, because it is not straight-
forward to define a threshold for a cluster to have some structure. In this paper,
it was proposed to use visual inspection, eigenvalues decomposition of the data
set belonging to each cluster, within-cluster sum of squares, and statistical tests
such as nonparametric Kolmogorov-Smirnov. When the splitting part is finished,
a merging step is performed by comparing the distances between all the centers:
if two clusters show a smaller distance with respect to the other pairs, then they
are merged together. The final centers, as stated before, are the initial guesses of
the final k -means algorithm. The reason is that because of low CNR or SNR,
some objects could have been wrongly assigned to a cluster.

16.3.2.3 Fuzzy Clustering

Besides hard partitioning algorithms, one of the most used approaches for clus-
tering fMRI data is the fuzzy clustering approach [54], which introduces the
concept of fuzziness [55]: each member of the data set may belong to several
clusters, and the degree of belonging is described by a membership index. This
approach differs from hard clustering, which allows each element to belong to
one and only one group. If we define a membership value u ji that describes the