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Gaul et al. 1007/978-3-642-24466-7 4, © Springer-Verlag Berlin Heidelberg 2012 29 30 I. Morlini and S. Zani introduced the so-called cophenetic correlation coefficient (see also Rohlf 1982 and Lapointe and Legendre 1995). Baker (1974) has proposed the rank correlation between stages where pairs of objects combine in the tree for measuring the similarity between two hierarchical clusterings. Reilly et al. (2005) have discussed the use of Cohen’s kappa in studying the agreement between two classifications.
1994) and Faber (1994) analysed real massive data sets obtained by the Landsat system (satellite images). It was a pioneering work, to our knowledge. The main aim was a rigorous reduction of the size of the dataset without loosing the essential information. For the Landsat data in (Faber et al. 1994), a “practical” K D 256 has been chosen because 256 is the greatest number of distinct colours that can be stored as an “inexpensive” byte. Figure 1 (at the left hand side) illustrates such a kind of practical motivated data reduction without looking for a native cluster structure.
2 m21 ... :: :: :: :: : : : : ::: ::: ::: :: : k m1k m2k :: : Total m1: m2: :: : f :: : ... :: : ... :: : mfj :: : ... :: : ... :: : mf: :: : k Total mk1 m:1 ... ::: mkj m:j ::: ::: mkk m:k mk: n 32 I. Morlini and S. g. n pair of partitions in k groups: SD X Vk D X k k P (11) 2/ parts Vk related to each 2Tk P Q k C k k Pk (12) The components Vk , however, are not similarity indices for each k since they assume values < 1 even if the two partitions in k groups are identical. For this reason, we consider the complement to 1 of each Zk in order to obtain a single similarity index for each pair of partitions: Pn Sk D 1 Zk D 1 j D2 Pj P C nj D21 Qj Pk Qk C 2Tk Pn 1 Pn 1 j D2 Pj C j D2 Qj (13) Expression (13) can be written as: P Sk D P Pj C j ¤k Qj C 2Tk P P j Pj C j Qj j ¤k (14) The index suggested by Fowlkes and Mallows (1983) for two partitions in k groups in our notation is given by: Bk D p 2Tk Tk Dp 2Pk 2Qk Pk Qk (15) The statistics Bk and Sk may be thought of as resulting from two different methods of scaling Tk to lie in the unit interval.