Transcript CHAPTER 17 Bray-Curtis (Polar) Ordination Tables, Figures, and Equations Analysis of

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
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
BCORD
(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.
Variance-regression method of reference
point selection (in use since 1973, first
published in Beals 1984)
BCORD
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.
N-1
SS TOT
=
N

i=1 j=i+1
D
2
ij
3. Select two points, A and B, as
reference points for first axis.
DAi
i
DBi
A
B
xgi
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:
x gi =
D
2
AB
2
Ai
+ D - D
2 D AB
DAB
2
Bi
Eqn. 1
The basis for the above equation can be seen as follows.
By definition,
cos A = x gi / D Ai
Eqn. 2
By the law of cosines,
cos A =
D
2
AB
2
Ai
+ D - D
2 D Ai D AB
2
Bi
Then substitute cos(A) from Equation 2 into
Equation 3.
Eqn. 3
5. Calculate residual distances Rgih (Fig. 17.2) between
points i and h where f indexes the g preceding axes.
g
R gih =
2
ih
D -
 (x
f=1
fi
- x fh )
2
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.
N-1
SS RESID =
N
R
2
ij
i=1 h=i+1
 SS RESID 
Cumulative Vk %  1001 

SSTOT 

Vk % = cumulative Vk %  cumulative Vk 1 %
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).
Figure 17.3. Example of the geometry of varianceregression endpoint selection in a two-dimensional
species space.
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
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.
Axis 2
Figure 17.5. Use of BrayCurtis 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
SP3
Axis 1
SP6
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
Calculate scores xi
for item i on new
axis between points
A and B.
2
2
2
+
D
D
D
AB
Ai
Bi
D AB + D Ai - D Bi
=
xi
xi =
2 D AB
2
Calculate residual
distances Rij between
points i and j.
Rij =
2
D2ij - ( xi - x j )
City-block method
Rij = | Dij - | xi - x j ||