Abstract
We shall now formulate in general terms the basic or “generic” problem of cluster analysis, and then discuss the consequences of this formulation.
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Notes
- 1.
Actually, given the sense of step i of the procedure, we do not deal with the matrix of distances between individual objects, d, but with the matrix of distances between clusters, D, which is, at the beginning of the procedure, identical with d, and then is transformed.
- 2.
Since we do by no means intend a survey of methods, only some selected, telling references shall be given for the methods considered. In this case—just three generic references will be mentioned: Florek, Łukaszewicz, Perkal, Steinhaus, and Zubrzycki (1956), where the origins of the so-called “single-linkage” algorithm can be found, and Lance and Williams (1966, 1967), who developed a more general theory of the agglomerative clustering procedures.
- 3.
Here, the seminal references are, first of all, Steinhaus (1956)—again(!), see the preceding footnote in order to appreciate the contribution of this Polish mathematician from the Lwów school of mathematics, largely founded by Stefan Banach; then there come Lloyd (1957)—soon afterwards, but similarly not ‘piercing’, and then Forgy (1965), Ball and Hall (1965), and MacQueen (1967). The fuzzy-set based version of the general k-means method, which became enormously popular, was formulated by Bezdek (1981).
- 4.
We stop here, since his is not really a survey, but also because not so many proper clustering methods exist outside of the paradigms mentioned. Thus, for instance, the so-called spectral clustering is actually simply a dimension reduction technique, which is, in practice, coupled with the other, proper clustering methods.
- 5.
- 6.
Note that in this context we refer only to those of the partitioning or clustering criteria alluded to that are called “internal” (see, e.g., Rendón, Abundez, Arizmendi, & Quiroz, 2011), since the ones called “external” actually verify the classification capabilities of the respective methods, and do not address the clustering performance as such.
- 7.
This supposition is, of course, true, when we deal with a definite, very narrow class of data sets, e.g. we can assume all clusters correspond to some Gaussian distribution functions.
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Owsiński, J.W. (2020). The Problem of Cluster Analysis. In: Data Analysis in Bi-partial Perspective: Clustering and Beyond. Studies in Computational Intelligence, vol 818. Springer, Cham. https://doi.org/10.1007/978-3-030-13389-4_2
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