vegan (version 2.4-2)

# predict.cca: Prediction Tools for [Constrained] Ordination (CCA, RDA, DCA, CA, PCA)

## Description

Function `predict` can be used to find site and species scores or estimates of the response data with new data sets, Function `calibrate` estimates values of constraints with new data set. Functions `fitted` and `residuals` return estimates of response data.

## Usage

```"fitted"(object, model = c("CCA", "CA", "pCCA"), type =  c("response", "working"), ...)
"fitted"(object, model = c("CCA", "CA", "pCCA", "Imaginary"), type = c("response", "working"), ...)
"residuals"(object, ...)
"predict"(object, newdata, type = c("response", "wa", "sp", "lc", "working"), rank = "full", model = c("CCA", "CA"), scaling = "none", hill = FALSE, ...)
"predict"(object, newdata, type = c("response", "wa", "sp", "lc", "working"), rank = "full", model = c("CCA", "CA"), scaling = "none", correlation = FALSE, ...)
"calibrate"(object, newdata, rank = "full", ...)
"coef"(object, ...)
"predict"(object, newdata, type = c("response", "sites", "species"), rank = 4, ...)```

## Arguments

object
A result object from `cca`, `rda`, `capscale` or `decorana`.
model
Show constrained (`"CCA"`), unconstrained (`"CA"`) or conditioned “partial” (`"pCCA"`) results. For `fitted` method of `capscale` this can also be `"Imaginary"` for imaginary components with negative eigenvalues.
newdata
New data frame to be used in prediction or in calibration. Usually this a new community data frame, but with `type = "lc"` and for constrained component with ```type = "response"``` and `type = "working"` it must be a data frame of constraints. The `newdata` must have the same number of rows as the original community data for a `cca` result with `type = "response"` or `type = "working"`. If the original model had row or column names, then new data must contain rows or columns with the same names (row names for species scores, column names for `"wa"` scores and constraint names of `"lc"` scores). In other cases the rows or columns must match directly.
type
The type of prediction, fitted values or residuals: `"response"` scales results so that the same ordination gives the same results, and `"working"` gives the values used internally, that is after Chi-square standardization in `cca` and scaling and centring in `rda`. In `capscale` the `"response"` gives the dissimilarities, and `"working"` the scaled scores that produce the dissimilarities as Euclidean distances. Alternative `"wa"` gives the site scores as weighted averages of the community data, `"lc"` the site scores as linear combinations of environmental data, and `"sp"` the species scores. In `predict.decorana` the alternatives are scores for `"sites"` or `"species"`.
rank
The rank or the number of axes used in the approximation. The default is to use all axes (full rank) of the `"model"` or all available four axes in `predict.decorana`.
scaling
logical, character, or numeric; Scaling or predicted scores with the same meaning as in `cca`, `rda` and `capscale`. See `scores.cca` for further details on acceptable values.
correlation, hill
logical; correlation-like scores or Hill's scaling as appropriate for RDA/`capscale` and CCA respectively. See `scores.cca` for additional details.
...
Other parameters to the functions.

## Value

The functions return matrices, vectors or dissimilarities as is appropriate.

## Details

Function `fitted` gives the approximation of the original data matrix or dissimilarities from the ordination result either in the scale of the response or as scaled internally by the function. Function `residuals` gives the approximation of the original data from the unconstrained ordination. With argument ```type = "response"``` the `fitted.cca` and `residuals.cca` function both give the same marginal totals as the original data matrix, and fitted and residuals do not add up to the original data. Functions `fitted.capscale` and `residuals.capscale` give the dissimilarities with `type = "response"`, but these are not additive, but the `"working"` scores are additive. All variants of `fitted` and `residuals` are defined so that for model `mod <- cca(y ~ x)`, `cca(fitted(mod))` is equal to constrained ordination, and `cca(residuals(mod))` is equal to unconstrained part of the ordination.

Function `predict` can find the estimate of the original data matrix or dissimilarities (`type = "response"`) with any rank. With `rank = "full"` it is identical to `fitted`. In addition, the function can find the species scores or site scores from the community data matrix for `cca` or `rda`. The function can be used with new data, and it can be used to add new species or site scores to existing ordinations. The function returns (weighted) orthonormal scores by default, and you must specify explicit `scaling` to add those scores to ordination diagrams. With `type = "wa"` the function finds the site scores from species scores. In that case, the new data can contain new sites, but species must match in the original and new data. With `type="sp"` the function finds species scores from site constraints (linear combination scores). In that case the new data can contain new species, but sites must match in the original and new data. With `type = "lc"` the function finds the linear combination scores for sites from environmental data. In that case the new data frame must contain all constraining and conditioning environmental variables of the model formula. With `type = "response"` or `type = "working"` the new data must contain environmental variables if constrained component is desired, and community data matrix if residual or unconstrained component is desired. With these types, the function uses `newdata` to find new `"lc"` (constrained) or `"wa"` scores (unconstrained) and then finds the response or working data from these new row scores and species scores. The original site (row) and species (column) weights are used for `type = "response"` and `type = "working"` in correspondence analysis (`cca`) and therefore the number of rows must match in the original data and `newdata`.

If a completely new data frame is created, extreme care is needed defining variables similarly as in the original model, in particular with (ordered) factors. If ordination was performed with the formula interface, the `newdata` can be a data frame or matrix, but extreme care is needed that the columns match in the original and `newdata`.

Function `calibrate.cca` finds estimates of constraints from community ordination or `"wa"` scores from `cca`, `rda` and `capscale`. This is often known as calibration, bioindication or environmental reconstruction. Basically, the method is similar to projecting site scores onto biplot arrows, but it uses regression coefficients. The function can be called with `newdata` so that cross-validation is possible. The `newdata` may contain new sites, but species must match in the original and new data. The function does not work with ‘partial’ models with `Condition` term, and it cannot be used with `newdata` for `capscale` results. The results may only be interpretable for continuous variables. Function `coef` will give the regression coefficients from centred environmental variables (constraints and conditions) to linear combination scores. The coefficients are for unstandardized environmental variables. The coefficients will be `NA` for aliased effects.

Function `predict.decorana` is similar to `predict.cca`. However, `type = "species"` is not available in detrended correspondence analysis (DCA), because detrending destroys the mutual reciprocal averaging (except for the first axis when rescaling is not used). Detrended CA does not attempt to approximate the original data matrix, so `type = "response"` has no meaning in detrended analysis (except with `rank = 1`).

## References

Greenacre, M. J. (1984). Theory and applications of correspondence analysis. Academic Press, London.

`cca`, `rda`, `capscale`, `decorana`, `vif`, `goodness.cca`.

## Examples

Run this code
``````data(dune)
data(dune.env)
mod <- cca(dune ~ A1 + Management + Condition(Moisture), data=dune.env)
# Definition of the concepts 'fitted' and 'residuals'
mod
cca(fitted(mod))
cca(residuals(mod))
# Remove rare species (freq==1) from 'cca' and find their scores
# 'passively'.
freq <- specnumber(dune, MARGIN=2)
freq
mod <- cca(dune[, freq>1] ~ A1 + Management + Condition(Moisture), dune.env)
predict(mod, type="sp", newdata=dune[, freq==1], scaling="species")
# New sites
predict(mod, type="lc", new=data.frame(A1 = 3, Management="NM", Moisture="2"), scal=2)
# Calibration and residual plot
mod <- cca(dune ~ A1 + Moisture, dune.env)
pred <- calibrate(mod)
pred
with(dune.env, plot(A1, pred[,"A1"] - A1, ylab="Prediction Error"))
abline(h=0)
``````

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