×

Two clustering methods based on the Ward’s method and dendrograms with interval-valued dissimilarities for interval-valued data. (English) Zbl 1497.62155

Summary: Numerous studies have focused on clustering methods for interval-valued data, which is a type of symbolic data. However, limited attention has been awarded to a clustering method employing interval-valued dissimilarity measures. To address this issue, herein, we propose two clustering approaches based on the Ward method using interval-valued dissimilarity for the interval-valued data. Each clustering method has different interval-valued dissimilarities. An interval-valued dissimilarity is generally not used to elucidate the computational result of a hierarchical clustering method by a traditional dendrogram; this is because the nodes of a dendrogram only designate real numbers and not an interval of numbers. We also present a new dendrogram with an arrow symbol, which is named arrow-dendrogram, to demonstrate the results of the clustering methods proposed in this study. In addition, we present the differences between the two clustering methods using numerical examples and numerical experimentation. The results of this study prove that the proposed clustering methods can intuitively provide reasonable and consistent results for our example data, thereby enabling us to completely comprehend the results of the clustering methods using interval-valued dissimilarity, via the arrow-dendrogram.

MSC:

62H30 Classification and discrimination; cluster analysis (statistical aspects)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Billard, L.; Diday, E., Symbolic Data Analysis: Conceptual Statistics and Data Mining (2006), Wiley · Zbl 1117.62002
[2] D’Urso, P.; Leski, J. M., Fuzzy c-ordered medoids clustering for interval-valued data, Pattern Recognit., 58, 49-67 (2016)
[3] Bock, H.-H.; Diday, E., Analysis of Symbolic Data: Exploratory Methods for Extracting Statistical Information from Complex Data (2012), Springer Science & Business Media
[4] de Carvalho, F.d. A.; Brito, P.; Bock, H.-H., Dynamic clustering for interval data based on l 2 distance, Comput. Stat., 21, 2, 231-250 (2006) · Zbl 1114.62070
[5] De Carvalho, F.d. A.; Lechevallier, Y., Partitional clustering algorithms for symbolic interval data based on single adaptive distances, Pattern Recognit., 42, 7, 1223-1236 (2009) · Zbl 1183.68527
[6] Cabanes, G.; Bennani, Y.; Destenay, R.; Hardy, A., A new topological clustering algorithm for interval data, Pattern Recognit., 46, 11, 3030-3039 (2013)
[7] D’Urso, P.; De Giovanni, L., Midpoint radius self-organizing maps for interval-valued data with telecommunications application, Appl. Soft Comput., 11, 5, 3877-3886 (2011)
[8] D’Urso, P.; De Giovanni, L.; Massari, R., Trimmed fuzzy clustering for interval-valued data, Adv. Data Anal. Classif., 9, 1, 21-40 (2015) · Zbl 1414.62242
[9] Sato-Ilic, M., Symbolic clustering with interval-valued data, Proc. Comput. Sci., 6, 358-363 (2011)
[10] D’Urso, P.; Massari, R.; De Giovanni, L.; Cappelli, C., Exponential distance-based fuzzy clustering for interval-valued data, Fuzzy Optim. Decis. Mak., 16, 1, 51-70 (2017) · Zbl 1428.62306
[11] de Carvalho, F.d. A.; Simões, E. C., Fuzzy clustering of interval-valued data with city-block and Hausdorff distances, Neurocomputing, 266, 659-673 (2017)
[12] De Carvalho, F.d. A.; Tenório, C. P., Fuzzy k-means clustering algorithms for interval-valued data based on adaptive quadratic distances, Fuzzy Sets Syst., 161, 23, 2978-2999 (2010) · Zbl 1204.62106
[13] Gowda, K. C.; Diday, E., Symbolic clustering using a new dissimilarity measure, Pattern Recognit., 24, 6, 567-578 (1991)
[14] Gowda, K. C.; Diday, E., Symbolic clustering using a new similarity measure, IEEE Trans. Syst. Man Cybern., 22, 2, 368-378 (1992)
[15] Ichino, M.; Yaguchi, H., Generalized Minkowski metrics for mixed feature-type data analysis, IEEE Trans. Syst. Man Cybern., 24, 4, 698-708 (1994) · Zbl 1371.68235
[16] Denœux, T.; Masson, M.-H., Multidimensional scaling of interval-valued dissimilarity data, Pattern Recognit. Lett., 21, 1, 83-92 (2000)
[17] Masson, M.-H.; Denœux, T., Clustering interval-valued proximity data using belief functions, Pattern Recognit. Lett., 25, 2, 163-171 (2004)
[18] Denœux, T.; Masson, M.-H., Evclus: evidential clustering of proximity data, IEEE Trans. Syst. Man Cybern., Part B, Cybern., 34, 1, 95-109 (2004)
[19] Moore, R. E., Methods and Applications of Interval Analysis (1979), SIAM · Zbl 0417.65022
[20] Moore, R. E.; Kearfott, R. B.; Cloud, M. J., Introduction to Interval Analysis, vol. 110 (2009), SIAM · Zbl 1168.65002
[21] Inuiguchi, M.; Kume, Y., Goal programming problems with interval coefficients and target intervals, Eur. J. Oper. Res., 52, 3, 345-360 (1991) · Zbl 0734.90056
[22] Ogasawara, Y.; Hisano, Y.; Kon, M., Comparing two clustering methods for interval-valued data, (Proceedings of International Conference on Nonlinear Analysis and Convex Analysis-International Conference on Optimization: Techniques and Applications (2019)), submitted for publication
[23] Fowlkes, E. B.; Mallows, C. L., A method for comparing two hierarchical clusterings, J. Am. Stat. Assoc., 78, 383, 553-569 (1983) · Zbl 0545.62042
[24] Ogasawara, Y.; Kon, M., Note on relationships between interval-valued squared Euclidean distance and interval-valued statistics, Metodol. Zv. (2019), submitted for publication
[25] Gan, G.; Ma, C.; Wu, J., Data Clustering: Theory, Algorithms, and Applications, vol. 20 (2007), SIAM · Zbl 1185.68274
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.