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 13.1. Timeline of development of some ordination methods and their application in ecology. BC = Bray-Curtis ordination, CA(RA) = correspondence analysis (reciprocal averaging), CCA = canonical correspondence analysis, DCA = detrended correspondence analysis, ISOMAP = isometric feature mapping, LLE = locally linear embedding, NMS = nonmetric multidimensional scaling, and PCA = principal components analysis.

3. Figure 13.2. Reduction of a p-dimensional matrix to a k-dimensional ordination.

4. Figure 13.3. A two-dimensional ordination of sample units in species space. Distances between sample units approximate dissimilarity in species composition.

5. Fig. 13.4. Overlay with different symbols for different topographic classes of sample units (stands).

6. Fig. 13.5. Overlay with related points enclosed by loops or arcs.

7. Figure 13.6. Overlay with sizes of symbols proportional to the magnitude of a variable, in this case abundance of the species Festuca idahoensis.

8. Figure 13.7. Scatterplot of abundance, measured as cover, of one species in relation to two ordination axes. Each point represents a plot. Symbol size is proportional to species abundance. Upper right: 2-D ordination of species composition. Lower right: scatterplot of abundance of “ALSA” (Alectoria sarmentosa) against score on Axis 1 (horizontal axis). Upper left: scatterplot of abundance of ALSA against Axis 2. Superimposed on the two abundance scatterplots are the least squares regression lines and a smoothed envelope. See text for interpretation of r and tau.

9. Figure 13.8. Example of a joint plot. The angles and lengths of the radiating lines indicate the direction and strength of relationships of the variables with the ordination scores.

10. Figure 13.9. Method of calculating a joint plot vector in PC-ORD.

11. Figure 13.10. Finding the angle of the vector using vector fitting as described by Dargie (1984); r 2 = 0.66 between moisture and three ordination axes. Assume we have a k-dimensional ordination to which we are fitting a vector for variable V. Regress V against the ordination scores X: Dargie’s method

12. Figure 13.11. Scaling the length of the vector using vector fitting. Vectors were scaled to unit length in three dimensions then projected onto two dimensions. The numbers are r 2 values in three dimensions.

13. Figure 13.12. Grid overlay on an artificial data set. The grid represents a 2-D environmental gradient underlying noiseless species responses. Inset: the ideal result is a regular grid.

14. Figure 13.13. Successional vectors. Lines connect sequential measurements on permanent sample units.

15. Table 13.1. Coefficients of determination for the correlations between ordination distances and distances in the original 40-dimensional space. For this example, the data matrix was 100 sample units  40 species. The number of entity pairs used in the correlation = 4950.

16. Table 13.2. Suggested methods for calculating proportion of variance represented by ordination axes. The after-the-fact method is the r 2 between Euclidean distances in the ordination space and distances in the original space. The distances in the original space can be calculated with various distance measures, but this choice of distance measure is important. “Built-in” means the method is a part of the ordination method itself.