Abstract
Hierarchical clustering algorithms are generally based upon a (dis)similarity that is assumed to be symmetric between object pairs. However, the (dis) similarity used in actual analysis is asymmetric. Therefore, to analyze the asymmetric (dis) similarity data the researcher must perform a somehow symmetrization of his original proximity values in the beginning. On the other hand, the idea that the asymmetry has elemental meaning and the researcher must analyze the data given by using algorithm depending on the asymmetry was suggested Hubert (1973) proposed "min and max clustering" for the asymmetric similarity. He symmetrized the data matrix in the beginning and analyze using the "min and max clustering algorithm". Fujiwara (1980) extend the Hubert's (1973) algorithm. He suggested the researcher should not perform symmetrization and should analyze the original asymmetric data matrix. Algorithms proposed in these two papers were extended the single linkage algorithm and the complete linkage algorithm to the asymmetric clustering algorithm. Okada and Iwamoto (1996) proposed the weighted average algorithm for asymmetric (dis) similarity. In those papers, they defined algorithms by deciding two steps, (i) selects the objects to be combined and (ii) updates the (dis) similarity between the objects, and not proposed uniformly. In this paper, we define an extended updating formula to handle a profusion of asymmetric hierarchical clustering algorithms uniformly in the same manner as the symmetric one by Lance and Williams (1967). Extended dendrogram for representation of the result of analysis for asymmetric data is also proposed.
