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Figure 8.13 In combinatorial clustering methods, the similarity between a cluster hi, resulting from the

Figure 8.13 In combinatorial clustering methods, the similarity between a cluster hi, resulting from the fusion of two previously formed clusters h and i, and an external cluster g is a function of the three similarities between (h and i), (h and g), and (i and g), and of the number of objects in h, i, and g.

When using distances, the combinatorial equation becomes:

Clustering proceeds in the same way for all combinatorial agglomerative methods. As the similarity decreases, a new cluster is obtained by the fusion of the two most similar objects or groups, after which the algorithm proceeds to the fusion of the two corresponding rows and columns in the similarity (or distance) matrix using eq. 8.11 or 8.1k. The matrix is thus reduced by one row and one column at each step. Table 8.8 gives the values of the four parameters for the most commonly used combinatorial agglomerative clustering strategies. Values of the parameters for some other clustering strategies are given by Gordon (1996a).

In the case of equality between two mutually exclusive pairs, the decision may be made on an arbitrary basis (the so-called "right-hand rule" used in most computer programs) or based upon ecological criteria (as, for example, S0rensen's criteria reported at the end of Subsection 8.5.k, or those explained in Subsection 8.9.1).

In several strategies, ah + a- + P = 1, so that the term (1 - ah - a - P) becomes zero and disappears from eq. 8.11. One can show how the values chosen for the four parameters make the general equation correspond to each specific clustering method. For single linkage clustering, for instance, the general equation becomes:

D (hi, g) = ahD (h, g) + aD (i, g) + ßD (h, i) + y|D (h, g) - D (i, g) |

5(hi, g) = 2- [5(h, g) + 5(i, g) + (h, g) -5(i, g)|]

Table 8.8 Values of parameters ah, a,', P, and y in Lance and Williams' general model for combinatorial agglomerative clustering. Modified from Sneath & Sokal (1973) and Jain & Dubes (1988).

Clustering method

ah

ai

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