Extract Eigenvalues from an Ordination Object

Function extracts eigenvalues from an object that has them. Many multivariate methods return such objects.

Usage

eigenvals(x, ...)
# S3 method for cca
eigenvals(x, model = c("all", "unconstrained", "constrained"),
          constrained = NULL, ...)
# S3 method for decorana
eigenvals(x, kind = c("additive", "axiswise", "decorana"),
           ...)
# S3 method for eigenvals
summary(object, ...)

Arguments

x: An object from which to extract eigenvalues.
object: An eigenvals result object.
model: Which eigenvalues to return for objects that inherit from class "cca" only.
constrained: Return only constrained eigenvalues. Deprecated as of vegan 2.5-0. Use model instead.
kind: Kind of eigenvalues returned for decorana. Only "additive" eigenvalues can be used for reporting importances of components in summary. "axiswise" gives the non-additive eigenvalues, and "decorana" the decorana values (see decorana for details).
...: Other arguments to the functions (usually ignored)

Details

This is a generic function that has methods for cca, wcmdscale, pcnm, prcomp, princomp, dudi (of ade4), and pca and pco (of labdsv) result objects. The default method also extracts eigenvalues if the result looks like being from eigen or svd. Functions prcomp and princomp contain square roots of eigenvalues that all called standard deviations, but eigenvals function returns their squares. Function svd contains singular values, but function eigenvals returns their squares. For constrained ordination methods cca, rda and capscale the function returns the both constrained and unconstrained eigenvalues concatenated in one vector, but the partial component will be ignored. However, with argument constrained = TRUE only constrained eigenvalues are returned.

The summary of eigenvals result returns eigenvalues, proportion explained and cumulative proportion explained. The result object can have some negative eigenvalues (wcmdscale, dbrda, pcnm) which correspond to imaginary axes of Euclidean mapping of non-Euclidean distances (Gower 1985). In these case real axes (corresponding to positive eigenvalues) will "explain" proportion >1 of total variation, and negative eigenvalues bring the cumulative proportion to 1. capscale will only find the positive eigenvalues and only these are used in finding proportions. For decorana the importances and cumulative proportions are only reported for kind = "additive", because other alternatives do not add up to total inertia of the input data.

Value

An object of class "eigenvals", which is a vector of eigenvalues.

The summary method returns an object of class

"summary.eigenvals", which is a matrix.

Author

Jari Oksanen.

References

Gower, J. C. (1985). Properties of Euclidean and non-Euclidean distance matrices. Linear Algebra and its Applications 67, 81--97.

Examples

data(varespec)
data(varechem)
mod <- cca(varespec ~ Al + P + K, varechem)
ev <- eigenvals(mod)
ev
#>      CCA1      CCA2      CCA3       CA1       CA2       CA3       CA4       CA5 
#> 0.3615566 0.1699600 0.1126167 0.3500372 0.2200788 0.1850741 0.1551179 0.1351054 
#>       CA6       CA7       CA8       CA9      CA10      CA11      CA12      CA13 
#> 0.1002670 0.0772991 0.0536938 0.0365603 0.0350887 0.0282291 0.0170651 0.0122474 
#>      CA14      CA15      CA16      CA17      CA18      CA19      CA20 
#> 0.0101910 0.0094701 0.0055090 0.0030529 0.0025118 0.0019485 0.0005178 
summary(ev)
#> Importance of components:
#>                         CCA1    CCA2    CCA3    CA1    CA2     CA3     CA4
#> Eigenvalue            0.3616 0.16996 0.11262 0.3500 0.2201 0.18507 0.15512
#> Proportion Explained  0.1736 0.08159 0.05406 0.1680 0.1056 0.08884 0.07446
#> Cumulative Proportion 0.1736 0.25514 0.30920 0.4772 0.5829 0.67172 0.74618
#>                           CA5     CA6     CA7     CA8     CA9    CA10    CA11
#> Eigenvalue            0.13511 0.10027 0.07730 0.05369 0.03656 0.03509 0.02823
#> Proportion Explained  0.06485 0.04813 0.03711 0.02577 0.01755 0.01684 0.01355
#> Cumulative Proportion 0.81104 0.85917 0.89627 0.92205 0.93960 0.95644 0.96999
#>                           CA12     CA13     CA14     CA15     CA16     CA17
#> Eigenvalue            0.017065 0.012247 0.010191 0.009470 0.005509 0.003053
#> Proportion Explained  0.008192 0.005879 0.004892 0.004546 0.002644 0.001465
#> Cumulative Proportion 0.978183 0.984062 0.988954 0.993500 0.996145 0.997610
#>                           CA18      CA19      CA20
#> Eigenvalue            0.002512 0.0019485 0.0005178
#> Proportion Explained  0.001206 0.0009353 0.0002486
#> Cumulative Proportion 0.998816 0.9997514 1.0000000

## choose which eignevalues to return
eigenvals(mod, model = "unconstrained")
#>       CA1       CA2       CA3       CA4       CA5       CA6       CA7       CA8 
#> 0.3500372 0.2200788 0.1850741 0.1551179 0.1351054 0.1002670 0.0772991 0.0536938 
#>       CA9      CA10      CA11      CA12      CA13      CA14      CA15      CA16 
#> 0.0365603 0.0350887 0.0282291 0.0170651 0.0122474 0.0101910 0.0094701 0.0055090 
#>      CA17      CA18      CA19      CA20 
#> 0.0030529 0.0025118 0.0019485 0.0005178