1. From: McCune, B. & J. B. Grace. 2002. Analysis of Ecological Communities. MjM Software Design, Gleneden Beach, Oregon http://www.pcord.comhttp://www.pcord.com Tables, Figures, and Equations

2. Figure 10.1. Example dendrogram scaled by Wishart’s objective function and percent of information remaining.

3. Hierarchical agglomerative cluster analysis 1.Calculate distance matrix.

4. Hierarchical agglomerative cluster analysis 1.Calculate distance matrix. 2.Merge two groups by a criterion of minimum distance.

5. Hierarchical agglomerative cluster analysis 1.Calculate distance matrix. 2.Merge two groups by a criterion of minimum distance. 3.Combine the attributes of the entities in the two groups that were fused.

6. Hierarchical agglomerative cluster analysis 1.Calculate distance matrix. 2.Merge two groups by a criterion of minimum distance. 3.Combine the attributes of the entities in the two groups that were fused. 4.Merge the next two groups, then go to step 3, until one group remains.

7. Hierarchical agglomerative cluster analysis 1.Calculate distance matrix. 2.Merge two groups by a criterion of minimum distance. 3.Combine the attributes of the entities in the two groups that were fused. 4.Merge the next two groups, then go to step 3, until one group remains. 5.Display the results as a dendrogram.

8. Figure 10.1. Example dendrogram scaled by Wishart’s objective function and percent of information remaining. “R 2” 01

9. The objective function (E) is the sum of the error sum of squares from each centroid to the items in that group: where t indexes the T clusters E t is the error sum of squares for cluster t. Each E t is found by: x ijt is the value of the: jth variable for the ith point of cluster t containing k t points is the mean of the jth variable for cluster t.

10. The objective function can be rescaled from 0% to 100% of information: % information remaining = 100(SST - E)/SST

11. Figure 10.2. Reversal in a dendrogram.

12. Figure 10.3. A dendrogram is an inherently nondimensional representation. Imagine the branches as free to pivot, like a child’s mobile

13. Figure 10.4. Use of average path length to measure percent chaining in cluster analysis. Path length is the number of nodes between the tip of a branch and the trunk. Complete chaining No chaining