Abstract
Most clustering algorithms have been designed only for pure numerical or pure categorical data sets while nowadays many applications generate mixed data. It arises the question how to integrate various types of attributes so that one could efficiently group objects without loss of information. It is already well understood that a simple conversion of categorical attributes into a numerical domain is not sufficient since relationships between values such as a certain order are artificially introduced. Leveraging the natural conceptual hierarchy among categorical information, concept trees summarize the categorical attributes. In this paper we propose the algorithm ClicoT (CLustering mixed-type data Including COncept Trees) which is based on the Minimum Description Length (MDL) principle. Profiting of the conceptual hierarchies, ClicoT integrates categorical and numerical attributes by means of a MDL based objective function. The result of ClicoT is well interpretable since concept trees provide insights of categorical data. Extensive experiments on synthetic and real data set illustrate that ClicoT is noise-robust and yields well interpretable results in a short runtime.
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Appendices
Appendix
A Probability Adjustment
To adjust the probabilities for a numerical cluster-specific attribute we can safely use mean and variance corresponding to the cluster. In contrast, learning the cluster-specific concept hierarchy is more challenging since we need to maintain the integrity of a hierarchy. We need to assure that node probabilities of siblings in each level sum up to the probability of the parent node. Moreover node probabilities should sum up to one for each level. ProcessHierarchy() in Algorithm 2 is a recursive function to update the concept tree assuming marked cluster-specific elements. Simultaneously in this function, Propagatedown() tries to preserve the concept tree properties by propagating down the parents probabilities to their children.
B MPG
MPG is a slightly modified version of the data set provided in the StatLib library. The data concerns city-cycle fuel consumption in miles per gallon (MPG) in terms of 3 categorical and 5 numerical attributes consisting of different characteristics of 397 cars. We consider MPG ranging from 10 to 46.6 as the ground truth and divide the range to 7 intervals of the same length. Considering a concept hierarchy for the name of cars we group all the cars so that we have three branches: European, American, Japanese cars. Moreover we divide the range of model year attribute to three intervals: 70–74, 75–80, after 80. We leave the third attribute as a flat concept hierarchy since there is no meaningful hierarchy between variation of cylinders.
C Adult Dataset
Adult data set, extracted from the census bureau database, consists of 48,842 instances of 11 attributes excluding the attributes with missing values (six numerical and 5 categorical). The class attribute Salary indicates whether the salary is over 50K or lower. Categorical attributes consist of different information e.g. work-class, education, occupation and so on. Figure 7 indicates concept hierarchies for three selected categorical attributes, including work-class, relationship and education.
Concept tree for 3 categorical attributes of adult dataset.
D Open Flights Dataset
Clustering results applying various algorithms with a better resolution illustrating is provided here (Figs. 8, 9, 10, 11 and 12).
ClicoT.
KMM.
MDBSCAN.
INCONCO and integrate.
DH.
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Behzadi, S., Müller, N.S., Plant, C., Böhm, C. (2019). Clustering of Mixed-Type Data Considering Concept Hierarchies. In: Yang, Q., Zhou, ZH., Gong, Z., Zhang, ML., Huang, SJ. (eds) Advances in Knowledge Discovery and Data Mining. PAKDD 2019. Lecture Notes in Computer Science(), vol 11439. Springer, Cham. https://doi.org/10.1007/978-3-030-16148-4_43
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