# Information Engineering Lab

F. Höppner, Ostfalia University of Applied Sciences

## Clustering / Fuzzy Clustering

### What is (fuzzy) cluster analysis?

Cluster analysis divides data into groups (clusters) such that similar data objects belong to the same cluster and dissimilar data objects to different clusters. The resulting data partition improves data understanding and reveals its internal structure. Partitional clustering algorithms divide up a data set into clusters or classes, where similar data objects are assigned to the same cluster whereas dissimilar data objects should belong to different clusters. In real applications there is very often no sharp boundary between clusters so that fuzzy clustering is often better suited for the data. Membership degrees between zero and one are used in fuzzy clustering instead of crisp assigments of the data to clusters. The most prominent fuzzy clustering algorithm is the fuzzy c-means, a fuzzification of k-Means or ISODATA.

Areas of application of fuzzy cluster analysis include for example data analysis, pattern recognition, and image segmentation. The detection of special geometrical shapes like circles and ellipses can be achieved by so-called shell clustering algorithms. Fuzzy clustering belongs to the group of soft computing techniques (which include neural nets, fuzzy systems, and genetic algorithms).

The family of objective function-based fuzzy clustering algorithms includes, amongst others, the ...

• fuzzy c-means algorithm (FCM): spherical clusters of approximately the same size
• Gustafson-Kessel algorithm (GK): ellipsoidal clusters with approx. the same size; there are also axis-parallel variants of this algorithm; can also be used to detect lines (to some extent)
• Gath-Geva algorithm (GG) / Gaussian mixture decomposition (GMD): ellipsoidal clusters with varying size; there are also axis-parallel variants of this algorithm; can also be used to detect lines (to some extent)
• fuzzy c-varieties algorithm (FCV): detection of linear manifolds (infinite lines in 2D)
• adaptive fuzzy c-varieties algorithm (AFC): detection of line segments in 2D data
• fuzzy c-shells algorithm (FCS): detection of circles (no closed form solution for prototypes)
• fuzzy c-spherical shells algorithm (FCSS): detection of circles
• fuzzy c-rings algorithm (FCR): detection of circles
• fuzzy c-quadric shells algorithm (FCQS): detection of ellipsoids
• fuzzy c-rectangular shells algorithm (FCRS): detection of rectangles (and variants thereof)
• and many more ...

### To probe further...

You can find some publications related to fuzzy clustering below, a few of them are available on-line (g'zipped postscript (.ps.gz) and portable document format (.pdf)).

• Introduction to Fuzzy Clustering: Many popular fuzzy clustering and shell clustering algorithms are discussed in this book.
• Efficient Implementation: For large data sets a simple data organisation can help to reduce the runtime of the fuzzy c-means algorithm significantly.
• Convergence of Fuzzy Clustering: A paper about the relationship of fixed points and saddle points of FCM and convergence of (probabilistic as well as possibilistic) fuzzy clustering algorithms in general.
• Learning Fuzzy Systems from Data: The advantage of fuzzy systems is their interpretability, which is based on the fact that the membership functions partition the data space appropriately. Many approaches in the literature, however, do not have this property.
• F. Höppner, F. Klawonn, P. Eklund: Learning Indistinguishability from Data. Soft Computing Journal 6(1), pp. 6-13, 2002
• F. Höppner, F. Klawonn: Obtaining Interpretable Fuzzy Models from Fuzzy Clustering and Fuzzy Regression. Proc. of the 4th Int. Conf. on Knowledge-Based Intelligent Engineering Systems and Allied Technologies (KES), Brighton, UK, pp. 162-165, 2000. .ps.gz ] [ .pdf ]
• Shell Clustering: A paper about the detection of rectangles and similar shapes in images.
• Fuzzification: The fuzzification in fuzzy clustering is obtained by introducing a so called "fuzzifier" in the objective function of crisp k-means. Although it is necessary to have such a concept, otherwise the membership degrees will stay crisp, there are also some drawbacks of this way of fuzzification. The following paper shows an alternative approach.