vegan (version 1.6-0)

decorana: Detrended Correspondence Analysis and Basic Reciprocal Averaging

Description

Performs detrended correspondence analysis and basic reciprocal averaging or orthogonal correspondence analysis.

Usage

decorana(veg, iweigh=0, iresc=4, ira=0, mk=26, short=0, before=NULL,
         after=NULL)
## S3 method for class 'decorana':
plot(x, choices=c(1,2), origin=TRUE, display=c("both","sites","species","none"),
     cex = 0.8, cols = c(1,2), type, ...)
## S3 method for class 'decorana':
summary(object, digits=3, origin=TRUE,
     display=c("both", "species","sites","none"), ...)
downweight(veg, fraction = 5)
## S3 method for class 'decorana':
scores(x, display=c("sites","species"), choices =1:4, origin=TRUE, ...)

Arguments

veg

Community data matrix.

iweigh

Downweighting of rare species (0: no).

iresc

Number of rescaling cycles (0: no rescaling).

ira

Type of analysis (0: detrended, 1: basic reciprocal averaging).

mk

Number of segments in rescaling.

short

Shortest gradient to be rescaled.

before

Hill's piecewise transformation: values before transformation.

after

Hill's piecewise transformation: values after transformation -- these must correspond to values in before.

x, object

A decorana result object.

choices

Axes shown.

origin

Use true origin even in detrended correspondence analysis.

display

Display only sites, only species, both or neither.

cex

Plot character size.

cols

Colours used for sites and species.

type

Type of plots, partial match to "text", "points" or "none".

digits

Number of digits in summary output.

fraction

Abundance fraction where downweighting begins.

...

Other parameters for plot function.

Value

• Function returns an object of class decorana, which has print, summary and plot methods.

Details

In late 1970s, correspondence analysis became the method of choice for ordination in vegetation science, since it seemed to be able to cope with non-linear species responses better than principal components analysis. However, even correspondence analysis produced arc-shaped configuration of a single gradient. Mark Hill developed detrended correspondence analysis to correct two assumed `faults' in correspondence analysis: curvature of straight gradients and packing of sites at the ends of the gradient.

The curvature is removed by replacing the orthogonalization of axes with detrending. In orthogonalization the successive axes are made non-correlated, but detrending should remove all systematic dependence between axes. Detrending is made using a five-segment smoothing window with weights (1,2,3,2,1) on mk segments -- which indeed is more robust than the suggested alternative of detrending by polynomials. The packing of sites at the ends of the gradient is undone by rescaling the axes after extraction. After rescaling, the axis is supposed to be scaled by `SD' units, so that the average width of Gaussian species responses is supposed to be one over whole axis. Other innovations were the piecewise linear transformation of species abundances and downweighting of rare species which were regarded to have an unduly high influence on ordination axes.

It seems that detrending works actually by twisting the ordination space, so that the results look non-curved in two-dimensional projections (`lolly paper effect'). As a result, the points have usually an easily recognized triangle or diamond shaped pattern, obviously as a detrending artefact. Rescaling works differently than commonly presented, too. Decorana does not use, or even evaluate, the widths of species responses. Instead, it tries to equalize the weighted variance of species scores on axis segments (parameter mk has only a small effect, since decorana finds the segment number from the current estimate of axis length). This equalizes response widths only for the idealized species packing model, where all species initially have unit width responses and equally spaced modes.

Function summary prints the ordination scores, possible prior weights used in downweighting, and the marginal totals after applying these weights. Function plot plots species and site scores. Classical decorana scaled the axes so that smallest site score was 0 (and smallest species score was negative), but summary, plot and scores use the true origin, unless origin = FALSE.

In addition to proper eigenvalues, the function also reports `decorana values' in detrended analysis. These are the values that the legacy code of decorana returns as `eigenvalues'. They are estimated internally during iteration, and it seems that detrending interferes the estimation so that these values are generally too low and have unclear interpretation. Moreover, `decorana values' are estimated before rescaling which will change the eigenvalues. The proper eigenvalues are estimated after extraction of the axes and they are always the ratio of biased weighted variances of site and species scores even in detrended and rescaled solutions. The `decorana values' are provided only for the the compatibility with legacy software, and they should not be used.

References

Hill, M.O. and Gauch, H.G. (1980). Detrended correspondence analysis: an improved ordination technique. Vegetatio 42, 47--58.

Oksanen, J. and Minchin, P.R. (1997). Instability of ordination results under changes in input data order: explanations and remedies. Journal of Vegetation Science 8, 447--454.

See Also

For unconstrained ordination, non-metric multidimensional scaling in isoMDS may be more robust. Constrained (or `canonical') correspondence analysis can be made with cca. Orthogonal correspondence analysis can be made with ca, or with decorana or cca, but the scaling of results vary.

Examples

data(varespec)
vare.dca <- decorana(varespec)
vare.dca
summary(vare.dca)
plot(vare.dca)
### the detrending rationale:
gaussresp <- function(x,u) exp(-(x-u)^2/2)
x <- seq(0,6,length=15) ## The gradient
u <- seq(-2,8,len=23)   ## The optima
pack <- outer(x,u,gaussresp)
matplot(x, pack, type="l", main="Species packing")
library(mva)    ## prcomp
opar <- par(mfrow=c(2,2))
plot(scores(prcomp(pack)), asp=1, type="b", main="PCA")
plot(scores(decorana(pack, ira=1)), asp=1, type="b", main="CA")
plot(scores(decorana(pack)), asp=1, type="b", main="DCA")
plot(scores(cca(pack ~ x), dis="sites"), asp=1, type="b", main="CCA")
### Let's add some noise:
noisy <- (0.5 + runif(length(pack)))*pack
par(mfrow=c(2,1))
matplot(x, pack, type="l", main="Ideal model")
matplot(x, noisy, type="l", main="Noisy model")
par(mfrow=c(2,2))
plot(scores(prcomp(noisy)), type="b", main="PCA", asp=1)
plot(scores(decorana(noisy, ira=1)), type="b", main="CA", asp=1)
plot(scores(decorana(noisy)), type="b", main="DCA", asp=1)
plot(scores(cca(noisy ~ x), dis="sites"), asp=1, type="b", main="CCA")
par(opar)