Ward method of hierarchical clustering for non-Euclidean similarity measures

The Ward linkage method in agglomerative hierarchical clustering is sometimes used for non-Euclidean similarity, i.e., non-positive definite matrix of similarity, which is not an adequate use of this method, since the square Euclidean distance should be its basis. Nevertheless, this paper shows that the Ward method for non positive-definite similarity can partly be justified. It is shown that the result from the Ward method to a non positive-definite and normalized similarity is almost the same as another result from the Ward method to a positive-definite matrix obtained from the original similarity by adding a positive constant to the diagonal elements. More precisely, the same clusters are generated by the same order from the both data. Only the levels of their generations are different.