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7) 36 HIERARCHICAL CLUSTERING This definition is equivalent to the calculation of the squared Euclidean distance between the centroids of the two clusters, D (Cl , ( Ci , C j )) = ml − m(ij ) . , 2001; Gower, 1967; Jain and Dubes, 1988). The median linkage is similar to the centroid linkage, except that equal weight is given to the clusters to be merged. Eq. 9) This is a special case when the number of data points in the two merging clusters is the same. , 2001; Jain and Dubes, 1988; Ward, 1963).

Mahalanobis distance tends to form hyperellipsoidal clusters, which are invariant to any nonsingular linear transformation. However, the calculation of the inverse of S may cause some computational burden for large-scale data. When features are not correlated, which leads S to an identity matrix, the squared Mahalanobis distance is equivalent to the squared Euclidean distance (Jain and Dubes, 1988; Mao and Jain, 1996). 23) PROXIMITY MEASURES FOR CONTINUOUS VARIABLES 25 where xr is a reference point, such as the centroid of the cluster, and ||·|| represents the Euclidean norm.

6) x ∈Ci where ni is the number of data points belonging to the cluster. Eq. 1 now is written as D (Cl , ( Ci , C j )) = nj ni D ( Cl , Ci ) + D (C l , C j ) − ni + n j ni + n j ni n j ( ni + n j )2 D (C i , C j ) . 7) 36 HIERARCHICAL CLUSTERING This definition is equivalent to the calculation of the squared Euclidean distance between the centroids of the two clusters, D (Cl , ( Ci , C j )) = ml − m(ij ) . , 2001; Gower, 1967; Jain and Dubes, 1988). The median linkage is similar to the centroid linkage, except that equal weight is given to the clusters to be merged.

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