# Detrended Correspondence Analysis and Basic Reciprocal Averaging

### Description

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

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 | ```
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, xlim, ylim, ...)
## S3 method for class 'decorana'
text(x, display = c("sites", "species"), labels,
choices = 1:2, origin = TRUE, select, ...)
## S3 method for class 'decorana'
points(x, display = c("sites", "species"),
choices=1:2, origin = TRUE, select, ...)
## S3 method for class 'decorana'
summary(object, digits=3, origin=TRUE,
display=c("both", "species","sites","none"), ...)
## S3 method for class 'summary.decorana'
print(x, head = NA, tail = head, ...)
downweight(veg, fraction = 5)
## S3 method for class 'decorana'
scores(x, display=c("sites","species"), choices=1:4,
origin=TRUE, ...)
``` |

### Arguments

`veg` |
Community data, a matrix-like object. |

`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 |

`x, object` |
A |

`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 |

`labels` |
Optional text to be used instead of row names. |

`select` |
Items to be displayed. This can either be a logical
vector which is |

`xlim, ylim` |
the x and y limits (min,max) of the plot. |

`digits` |
Number of digits in summary output. |

`head, tail` |
Number of rows printed from the head and tail of
species and site scores. Default |

`fraction` |
Abundance fraction where downweighting begins. |

`...` |
Other arguments for |

### Details

In late 1970s, correspondence analysis became the method of choice for ordination in vegetation science, since it seemed better able to cope with non-linear species responses than principal components analysis. However, even correspondence analysis can produce an 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 successive axes are made
non-correlated, but detrending should remove all systematic dependence
between axes. Detrending is performed 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 actually works by twisting the ordination
space, so that the results look non-curved in two-dimensional projections
(‘lolly paper effect’). As a result, the points usually have an
easily recognized triangular or diamond shaped pattern, obviously an
artefact of detrending. 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.

The `summary`

method prints the ordination scores,
possible prior weights used in downweighting, and the marginal totals
after applying these weights. The `plot`

method 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 ‘decorana values’ 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 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 compatibility with
legacy software, and they should not be used.

### Value

`decorana`

returns an object of class `"decorana"`

, which has
`print`

, `summary`

and `plot`

methods.

### Note

`decorana`

uses the central numerical engine of the
original Fortran code (which is in the public domain), or about 1/3 of
the original program. I have tried to implement the original
behaviour, although a great part of preparatory steps were written in
**R** language, and may differ somewhat from the original code. However,
well-known bugs are corrected and strict criteria used (Oksanen &
Minchin 1997).

Please note that there really is no need for piecewise transformation
or even downweighting within `decorana`

, since there are more
powerful and extensive alternatives in **R**, but these options are
included for compliance with the original software. If a different
fraction of abundance is needed in downweighting, function
`downweight`

must be applied before `decorana`

. Function
`downweight`

indeed can be applied prior to correspondence
analysis, and so it can be used together with `cca`

, too.

The function finds only four axes: this is not easily changed.

### Author(s)

Mark O. Hill wrote the original Fortran code, the **R** port was by
Jari Oksanen.

### 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
`monoMDS`

may be more robust (see also
`metaMDS`

). Constrained (or ‘canonical’)
correspondence analysis can be made with `cca`

.
Orthogonal correspondence analysis can be made with
`corresp`

, or with `decorana`

or
`cca`

, but the scaling of results vary (and the one in
`decorana`

corresponds to `scaling = "sites"`

and
`hill = TRUE`

in `cca`

.). See
`predict.decorana`

for adding new points to an
ordination.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 | ```
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")
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)
``` |