Bray-Curtis%20(Polar)%20Ordination

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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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Table 17.1. Development and implementation of
the most important refinements of Bray-Curtis
ordination (from McCune Beals 1993).
Stage of Development Implementation
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
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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.
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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
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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.
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5. Calculate residual distances Rgih (Fig. 17.2)
between points i and h where f indexes the g
preceding axes.
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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.
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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).
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Figure 17.3. Example of the geometry of
variance-regression endpoint selection in a
two-dimensional species space.
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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.
point i D1i D2i
1 0.34 0.88
2 0.55 0.63
. . .
. . .
n 0.28 0.83
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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.
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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
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Table 17.3. Comparison of Euclidean and
city-block methods for calculating ordination
scores and residual distances in Bray-Curtis
ordination.
Operation Euclidean (usual) method City-block method
Calculate scores xi for item i on new axis between points A and B.
Calculate residual distances Rij between points i and j.