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CHAPTER 17 Bray-Curtis (Polar) Ordination. Tables, Figures, and Equations. From: McCune, B. & J. B. Grace. 2002. Analysis of Ecological Communities . MjM Software Design, Gleneden Beach, Oregon http://www.pcord.com.

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CHAPTER 17 Bray-Curtis (Polar) Ordination

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Chapter 17 bray curtis polar ordination

CHAPTER 17

Bray-Curtis (Polar) Ordination

Tables, Figures, and Equations

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


Chapter 17 bray curtis polar ordination

  • Table 17.1. Development and implementation of the most important refinements of Bray-Curtis ordination (from McCune & Beals 1993).


Chapter 17 bray curtis polar ordination

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.


Chapter 17 bray curtis polar ordination

3. Select two points, A and B, as reference points for first axis.

4. Calculate position (xgi) 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


Chapter 17 bray curtis polar ordination

The basis for the above equation can be seen as follows. By definition,

Eqn. 2

By the law of cosines,

Eqn. 3

Then substitute cos(A) from Equation 2 into Equation 3.


Chapter 17 bray curtis polar ordination

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


Chapter 17 bray curtis polar ordination

6. Calculate variance represented by axis k as a percentage of the original variance (Vk%). 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.


Chapter 17 bray curtis polar ordination

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).


Chapter 17 bray curtis polar ordination

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


Chapter 17 bray curtis polar ordination

  • Table 17.2. Basis for the regression used in the variance-regression technique. Distances are tabulated between each point i and the first endpoint D1i and between each point and the trial second endpoint D2i*.


Chapter 17 bray curtis polar ordination

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.


Chapter 17 bray curtis polar ordination

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


Chapter 17 bray curtis polar ordination

  • Table 17.3. Comparison of Euclidean and city-block methods for calculating ordination scores and residual distances in Bray-Curtis ordination.


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