1. CHAPTER 17 Bray-Curtis (Polar) Ordination 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. Table 17.1. Development and implementation of the most important refinements of Bray-Curtis ordination (from McCune & Beals 1993). Stage of DevelopmentImplementation Basic method (Bray's thesis 1955, Bray & Curtis 1957) Ordination scores found mechanically (with compass) Algebraic method for finding ordination scores (Beals 1960) BCORD, the Wisconsin computer program for Bray-Curtis ordination, ORDIFLEX (Gauch 1977), and several less widely used programs developed by various individuals Calculation of matrix of residual distances (since 1970 at Wisconsin; published by Beals 1973), which also perpendicularizes the axes; given this step, the methods for perpendicularizing axes by Beals (1965) and Orloci (1966) are unnecessary. BCORD Variance-regression method of reference point selection (in use since 1973, first published in Beals 1984) BCORD

3. How it works 1. Select a distance measure (usually Sørensen distance) and calculate a matrix of distances (D) between all pairs of N points. 2. Calculate sum of squares of distances for later use in calculating variance represented by each axis.

4. 3. Select two points, A and B, as reference points for first axis. 4. Calculate position (x gi ) of each point i on the axis g. Point i is projected onto axis g between two reference points A and B (Fig. 17.1). The equation for projection onto the axis is: Eqn. 1

5. The basis for the above equation can be seen as follows. By definition, By the law of cosines, Then substitute cos(A) from Equation 2 into Equation 3. Eqn. 3 Eqn. 2

6. 5. Calculate residual distances R gih (Fig. 17.2) between points i and h where f indexes the g preceding axes.

7. 6. Calculate variance represented by axis k as a percentage of the original variance (V k %). The residual sum of squares has the same form as the original sum of squares and represents the amount of variation from the original distance matrix that remains.

8. 7. Substitute the matrix R for matrix D to construct successive axes. 8. Repeat steps 3, 4, 5, and 6 for successive axes (generally 2-3 axes total).

9. Figure 17.3. Example of the geometry of variance- regression endpoint selection in a two-dimensional species space.

10. Table 17.2. Basis for the regression used in the variance-regression technique. Distances are tabulated between each point i and the first endpoint D 1i and between each point and the trial second endpoint D 2i *. point iD1iD1i D 2i * 10.340.88 20.550.63...... n0.280.83

11. Figure 17.4. Using Bray-Curtis ordination with subjective endpoints to map changes in species composition through time, relative to reference conditions (points A and B). Arrows trace the movement of individual SUs in the ordination space.

12. Figure 17.5. Use of Bray- Curtis ordination to describe an outlier (arrow). Radiating lines are species vectors. The alignment of Sp3 and Sp6 with Axis 1 suggests their contribution to the unusual nature of the outlier. SP6

13. Table 17.3. Comparison of Euclidean and city-block methods for calculating ordination scores and residual distances in Bray-Curtis ordination. OperationEuclidean (usual) methodCity-block method Calculate scores x i for item i on new axis between points A and B. Calculate residual distances R ij between points i and j.