August 21, 2007
Discovering Clusters of Arbitrary Shapes
Rachsuda Jiamthapthaksin, Jiyeon Choo, Christian Giusti1,
Dan Jiang, Christoph F. Eick, and Ricardo Vilalta
Department of Computer Science, University of Houston
Houston, Texas 77204-3010, U.S.A
{rachsuda, jchoo, djiang, ceick, vilalta}@cs.uh.edu
Clustering is an unsupervised learning technique that is used for exploratory data
analysis, for summary generation, and as a preprocessing step for other data mining
tasks. Our on-going research applies clustering in scientific discovery, such as
identifying pollution hotspots in environmental data, discovering co-location patterns
concerning chemicals on the Martian surface and in the Texas ground water, and for
region discovery in general. Due to the characteristics and diverse nature of the data
used, clusters may be of arbitrary shapes and can be nested within one another.
Examples of such shapes are chain-like patterns that represent active and inactive
volcanoes, as depicted in Figure 1 (a). Generally traditional clustering algorithms,
such as k-means and k-medoids, fail to detect non-spherical shapes, as demonstrated
in Figure 1 (b). Our research aims to find solutions for this challenging problem. We
focus on developing novel techniques that discover clusters of arbitrary shapes
efficiently and effectively. Additionally, the following other criteria are taken into
consideration: speed and scalability, comprehensive parameters tuning, robustness to
noise and outliers, and the capability to detect clusters at different levels of
granularity.
(a) Chain-like patterns
(b) Clusters detected by K-means
Figure 1: Discovering Chain-like Clusters in the Volcano Dataset.
Two approaches are currently explored in order to find clusters of arbitrary
shapes. The first approach is a generic clustering framework that approximates
arbitrary shape clusters using unions of small convex polygons [CJCCGE2007], as
depicted in Figure 2. One of benefits of this framework its flexibility by allowing for
three plug-in components: 1) the formation of large number of sub-clusters, 2) the
construction of neighboring relation of sub-clusters (Fig. 3), and 3) the fitness
function that measures cluster quality. In this framework, fitness functions play an
important role in capturing arbitrary shapes of clusters—unfortunately, the use of
fitness functions in clustering algorithm has not been studied much by past research.
We investigate two approaches for acquiring fitness functions: 1) using shape
1
Department of Mathematics and Computer Science, University of Udine
Via delle Scienze, 33100, Udine, Italy. {giusti@dimi.uniud.it}
signatures [SS06] as summarized information to capture shape characteristic of
clusters, and 2) learning fitness functions based on the domain expert’s or other
feedback.
(a) Input
(b) Output
Figure 2: An illustration of MOSAIC’s Clustering Approach for the Complex9 Dataset
Figure 3: An illustration of the neighboring relations between sub-clusters
The second approach explores the use of density estimation techniques for
discovering arbitrary shape clusters [Jiang2006, JEC2007]. These techniques rely on
influence functions in which a point's influence on another point decreases as the
distance between the two points increases. Based on influence functions, density
functions are constructed, as depicted in Fig. 4, and clusters are formed by using hill
climbing algorithms that associates objects in the dataset with maxima of the sodefined functionobjects that are associated with the same maximum belong to the
same cluster.
The ability to discover clusters with arbitrary shapes is important in many real
applications, 3 of which are highlighted in this paragraph. Finding non-spherical
regions is of critical importance for hotspot and co-location discovery in spatial
datasets [EVJW2006, EDPS2007]; otherwise, arbitrary shape, regional patterns will
remain undetected in spatial data mining. Another application is the use of clustering
as a preprocessing step for classification. In [VAE2003], a decomposition of classes
into subclasses in order to simplify the approximation of class distributions is
proposed. It is clear that such decompositions do not always have spherical shapes,
Figure 4 (a): Binary Complex9 Dataset
Figure 4: (b) A Density Function for the
Binary Complex9 Dataset
and therefore the capability to discover arbitrary shape clusters is important for class
decomposition. Third, in the image processing field an arbitrary shape clustering
technique is important to obtain a spatial clustering of features extracted from an
image. In [GPP2006] a method for discovering objects in images using spatial
clustering of features is presented. The method extracts features from a picture
through a weak segmentation procedure, and uses the displacement of features over
the picture to subdivide the picture in meaningful parts, relying on clustering
algorithms.
References
[CJCCGE2007] J. Choo, R. Jiamthapthaksin, C. Chen, O. Celepcikay, C. Giusti, and C. Eick,
MOSAIC: A Proximity Graph Approach to Agglomerative Clustering, in Proc. 9th
International Conference on Data Warehousing and Knowledge Discovery (DaWaK),
Regensburg, Germany, September 2007.
[EDPS2007] C. Eick, R. Parmar, W. Ding, T.Stepinksi, and J.-P. Nicot, Finding Regional Colocations Patterns for Sets of Continuous Variables, submitted for publication, October 2007.
[EVJW2006] C.Eick, B. Vaezian, D. Jiang, and J. Wang, Discovery of Interesting Regions in
Spatial Datasets Using Supervised Clustering, in Proceedings 10th European Conference on
Principles and Practice of Knowledge Discovery in Databases (PKDD), Berlin, Germany,
September 2006.
[JEC2007] D. Jiang, C. Eick, and C. Chen, On Supervised Density Estimation Techniques
and Their Application to Spatial Data Mining, in Proc. ACM-GIS Conference, Seattle,
Washington, November 2007.
[J2006] Dan Jiang, Design and Implementation of a Density-based Supervised Clustering
Algorithm, Master Thesis, University of Houston, December 2006.
[GPP2006] C. Giusti, G. G. Pieroni, and L. Pieroni, Attention trees and semantic paths, in
Proceedings of The Human Vision and Electronic Imaging XII conferences, San Jose,
California, February 2007.
[SS06] C. Shahabi. and M. Safar, M., An experimental study of alternative shape-based image
retrieval techniques, Springer Science + Business Media, LLC 2006.
[VAE2003] R. Vilalta, M. Achari, and C. Eick, Class Decomposition via Clustering: A New
Framework for Low-Variance Classifiers, in Proceedings Third IEEE International
Conference on Data Mining (ICDM), Melbourne, Florida, November 2003.