polarord: Polar (Bray-Curtis) Ordination

In jarioksa/natto: An Extreme 'vegan' Package of Experimental Code

Polar (Bray-Curtis) Ordination

Description

Polar or Bray-Curtis ordination is a historic ordination method that could be performed without computers with simple hand calculations (Bray & Curtis 1957). Ordination axis is found by selecting two extreme points and projecting all points between these end points. The current function follows Beals (1984) in selecting the endpoints, projection of points on axis, and defining the residual distances for later axes.

Usage

polarord(d, k = 2, endpoints, metric = c("euclidean", "manhattan"))

## S3 method for class 'polarord'
plot(x, choices = c(1, 2), type = "t", display, ...)

## S3 replacement method for class 'polarord'
sppscores(object) <- value

## S3 method for class 'polarord'
eigenvals(x, ...)

Arguments

d	Dissimilarities or distances: a dist object.
k	Number of dimensions.
endpoints	Indices (not names) of endpoints for axes. This can be a single integer for the first endpoint, and the second endpoint and later axes are found automatically. If this is a vector, its non-zero values are taken as indices of endpoints of sequential axes.
metric	Use either "euclidean" or "manhattan" (city-block) metric in projecting points onto axes and calculating residual distances.
x	polarord result.
choices	Axes shown.
type	Type of graph which may be "t" for text, "p" for points or "n" for none (an empty plot).
display	Items displayed: "sites" are always available, but "species" only if they were added with sppscores.
...	Other arguments to the function (passed to ordiplot).
object	polarord result.
value	Community data to find the species scores.

Details

The implementation follows McCune & Grace (2002, Chapter 17). The endpoints are found using the variance-regression method of Beals (1984). The first endpoint has the highest variance of distances to other points. This guarantees that the point is at the margin of the multivariate cloud but is not an outlier, since outliers have long distances to all points and hence low variance of distances. The second endpoint has the lowest (most negative) regression coefficient between distances from the first and second point to all other points. This selects a point at the margin of the main cloud of points, opposite to the first endpoint. All points are projected on the axis between the endpoints, and this gives the scores on a polar ordination axis. Then the effect of the axis are removed by calculating residual distances. The projection of points and calculation of residuals is based either on Euclidean or Manhattan geometry, depending on the argument metric. Ecological indices are usually semimetric, and negative residual distances can emerge, but these are taken as zero in the current function.

The eigenvalues are estimated from the dispersion of points, and they are not necessarily in descending order. Usually only some first axes are reliable, and too high numbers of dimensions should be avoided. The inertia and eigenvalues give the average dispersion, and they are consistent with dbrda and capscale.

Polar ordination is a historical method that is little used today, but several authors claim that it is a powerful method (see McCune & Grace 2002). Although the basic operations can be easily performed by hand or graphically, the later developments of endpoint selection require more extensive calculations. With modern numerical utilities, the polar ordination is not faster than metric multidimensional scaling (cmdscale, wcmdscale).

It is possible to use predefined endpoints instead of automatic selection. This can be useful for confirmatory analysis (McCune & Grace 2002). If only one endpoint is given, the others are selected automatically.

McCune & Grace (2002) suggest that Polar Ordination can be alternatively performed with Manhattan or City-Block methods for projecting points onto axes and calculating residual ordination, although they say that this is "yet largerly untested". This can be implemented by setting metric = "manhattan". However, two-dimensional configuration cannot be recovered from its Manhattan distances with this method, but it can be exactly recovered from its Euclidean distances and Euclidean metric (see Examples). Ordination is similar to the original two-dimensional configuration only if the endpoints were chosen so that the ordination axes are parallel to the dimensions in the original configuration. Further, ordinations with Manhattan metric are not invariant when rotated: Manhattan distances are calculated with respect to the axes, and they change if axes are rotated. This also means that fitted vectors cannot be used for environmental variables, and the interpretation of the ordination graph needs special and rare intuition. I warn against use of metric = "manhattan".

Function sppscores can be used to add species scores to the ordination result.

Value

The function returns an object of class "polarord" with the following elements:

• points: The ordination scores.

• inertia: Total inertia of the input dissimilarities.

• eig: Eigenvalues of axes. These do not usually add up to total inertia and may not be in strictly descending order.

• endpoints: The indices (not the names) of the endpoints for each axis.

• call: Function call.

References

Beals, E. W. (1984) Bray-Curtis ordination: an effective strategy for analysis of ecological multivariate data. Advances in Ecological Research 14, 1–55.

Bray, J. R. & Curtis, J. T. (1957) An ordination of the upland forest communities in southern Wisconsin. Ecological Monographs 27, 325–349.

McCune, B. & Grace, J. B. (2002) Analysis of Ecological Communities. MjM Software Design.

Examples

data(spurn)
dis <- dist(spurn, method = "binary") ## Jaccard index
ord <- polarord(dis)
ord
summary(eigenvals(ord))
## add species scores
sppscores(ord) <- spurn
plot(ord)
## Two-dimensional configuration recovered with Euclidean metric
if (require(vegan)) { ## Procrustes analysis
set.seed(1009)
x <- matrix(runif(50), 25, 2) # 2-dim matrix
plot(procrustes(x, polarord(dist(x)))) # Euclidean: exact
plot(procrustes(x, polarord(dist(x, "man"), metric="man"))) # diverges
}

jarioksa/natto documentation built on March 28, 2024, 12:45 a.m.