Cluster analysis (continued)

Complete linkage (furthest neighbor) clustering

Identical to single linkage clustering except that the distance between entities is defined as the point of maximum distance

e.g., distances:

	1	2	3	4	5	6
1	-	3.16	4.47	15.16	11.40	12.32
2		-	7.07	12.04	8.94	9.85

All distances of entities after quadrats 1 and 2 are joined:

	1,2	3	4	5	6
1,2	-	7.07	15.16	11.40	12.32

i.e.,	d(1,3) = 4.47
	d(2,3) = 7.07;
thus,	d[(1,2),3] = 7.07

w/ single linkage clustering, dist = minimum distance - -> d[(1,2),3] = 4.47

Decision rule is still based on smallest distance, but distances are calculated differently

Characteristics of complete linkage clustering:

1. "space-dilating"--as a cluster grows it tends to become more dissimilar to others --> non-chaining

2. group structure is ignored; as w/ single linkage clustering, comparisons are based on indiv. quadrats

3. results often similar to "minimum-variance" clustering

Centroid clustering

Distance between 2 clusters is defined as the euclidean distance between their centroids

Two groups are joined if the distance between their centroids is the smallest of all possible "choices"

e.g., distance between groups:

To calculate centroid:

Quadrat	Species A	Species B
1	15	9
2	12	8
3	17	13
4	0	7
5	8	0
6	3	12

First step is identical to single linkage clustering, since groups are single quadrats. After quadrats 1 and 2 are joined, centroid(1,2) = [(15+12)/2, (9+8)/2] = (13.5,8.5).

centroid (1,2,3) = [(15+12+17)/3, (9+8+13)/3] = (14.333,10)

Then euclidean distance is calculated not between nearest quadrats in group (single linkage) and not between furthest quadrats in group (complete linkage), but between centroids of groups

Thus, group structure is used in determining between- group similarities

A disadvantage of centroid clustering is the potential for reversals

After a fusion, the next fusion occurs at a less dissimilar point (i.e., closer distance)

e.g., consider these 3 quadrats w/ 2 species:

Quadrat	Species A	Species B
1	26	10
2	34	6
3	34	15

d(1,2) = [(26-34)² + (10-6)²] = 8.944

d(1,3) = [(26-34)² + (10-15)²] = 9.434

d(2,3) = [(34-34)² + (6-15)²] = 9.000

Quadrats 1 and 2 are joined --> centroid = [(26+34)/2,(10+6)/2] = (30,8)

d[(1,2),3] = [(30-34)² + (8-15)²] = 9.062

Thus, the second fusion occurs at a smaller distance than the first fusion (i.e., this indicates these entities are more similar than those joined by the first fusion):

Centroid clustering incorporates information about the group when joining groups (vs. single linkage and complete linkage clustering, which do not)

However, reversals create interpretational difficulties, and this has discouraged widespread use of clustering techniques which have potential to show reversals

Comparison studies have shown that single linkage and centroid clustering behave similarly

Minimum-variance clustering (Ward's method) (syn. Orloci's method in ecological literature)

Concept: we can measure the sum of the distances² of the members of a group from the group centroid as an indicator of group heterogeneity or dispersion

Distance (similarity) measure: euclidean distance

Fusion rule: groups are joined only if the increase in d² is less for that pair of groups than for any other pair

Ward's method lends itself to a measure of "classification efficiency":

SS_total = d² of all quadrats from centroid

At any point in the analysis, SS can be calculated for each group (i.e., within-group heterogeneity or dispersion)

Thus, a percentage can be calculated which indicates the proportion of total variability explained by each group: SS_group/SS_total

Characteristics of Ward's method

1. Minimizes dispersion within groups

2. Like complete linkage clustering, it favors the formation of small clusters of approximately equal size

3. Incorporates information about groups, not merely about individual quadrats

4. Computationally complex and time-consuming compared to other methods we've discussed

5. Widely applied in ecology, especially recently (since computers have overcome problems w/ computational complexity)