eigenmap: Spatial eigenvector maps

Description

Function to calculate spatial eigenvector maps of a set of locations in a space with an arbitrary number of dimension.

Usage

eigenmap(x,opt.coord=NA,weighting=Wf.sqrd,boundaries,wpar,select=.Machine$double.eps^0.5)
Wf.sqrd(D)
Wf.binary(D,boundaries)
Wf.PCNM(D,boundaries)
Wf.Drayf1(D,boundaries)
Wf.Drayf2(D,boundaries,wpar=1)
Wf.Drayf3(D,boundaries,wpar=1)
eigenmap.score(object,target)

Arguments

A set of coordinates defined in one (numeric vector) or many (a coordinate x dimension matrix) dimensions or, alternatively, a distance matrix provided by dist. Coordinates are treated as cartesian coordinates and the distances between them are assumed to be Euclidean.

opt.coord

Coordinates to be used when a distance matrix is provided as x. Used for plotting purposes.

weighting

The function to obtain the edge weighting matrix. That function must have the raw distances as a first parameter, optionally a second parameter named boundaries giving the boundaries of the within which locations are regarded as neighbour and a third parameter named wpar containing any other weighting function parameter.

Wf.binary (default value) the spatial weighting matrix is simply the connectivity matrix,

Wf.PCNM is $a_{i, j} = 1 - (d_{i, j} / (4 * b o u n d a r i e s_{2}))^{2}$

Wf.Drayf1 is $a_{i, j} = 1 - (d_{i, j} / d_{m a x})$ where $d_{m} a x$ is the distance between the two most distant locations in the set,

Wf.Drayf2 is $a_{i, j} = 1 - (d_{i, j} / d_{m a x})^{w p a r}$ ,

Wf.Drayf3 is $a_{i, j} = 1 / d_{i, j}^{w p a r}$ , and

Wf.sqrd consists in taking $w_{i, j} = - 0.5 * d_{i, j}$ and does not involve any trunction.

Functions Wf.Drayf1, Wf.Drayf2, and Wf.Drayf3 were proposed by Dray et al. (2006) and function PCNM was proposed by Legendre and Legendre (2012). The Wf.sqrd weighting approach is equivallent to submitting the elementwise square-root of the distance matrix to a principal coordinate analysis. That option is not much documented in the ecological litterature, but is actually equivallent, for evenly spaced transect or surfaces (square or rectangle), to using the basis functions of type II discrete cosine basis transforms.

boundaries

(optional) Threshold values (minimum and maximum) used to obtain the connectivity matrix. Pairs of location whose distance to one another are between these values are considered as neighbours ( $b_{i, j} = 1$ ) whereas values located below the mininum and above the maximum are considered as equivallent or distant, respectively ( $b_{i, j} = 0$ in both cases). Defaults are $0$ for the minimum value and NA for the maximum. Values NA indicates the function to take the minimum value that allow every locations to form a single cluster following single linkage clustering as a maximum value (obtained from hclust

wpar

Weighting function parameters.

select

The smallest absolute eigenvalue for eigenfunctions to be considered as a suitable predictive variables. Default value depends on one's particular computer and is set to the square-root of .Machine$double.eps

A distance matrix.

object

A eigenmap-class object.

target

A set of distances between the sampling locations (passed to eigenmap using x) and the target locations where spatially-explicit predictions are to be made.

Value

eigenmap returns a eigenmap-class object and eigenmap.score returns a the scores on for each target locations

Details

Spatial eigenvector maps are sets of eigenfunctions obtained from the locations of the observations in a structuring framework, e.g., space, time, or in a graph. It is obtained by eigenvalue decomposition of a spatial weighting matrix, computed as described in Dray et al. (2006) and Legendre & Legendre (2012, Section 14.2). That square matrix is Gower-centred before eigen-decomposition. The spatial weighting matrix is the Hadamard product of a connectivity matrix $B$ and an edge weighting matrix $A$ . The function described herein handles user-chosen truncation parameters to calculate $B$ and provides a default approach to estimate these parameters should they be missing. It also offers four different ways of computing $A$ through parameters weighting and wpar.

In is noteworthy that in the present implementation, matrix $B$ is not obtained using a minimum spanning tree as suggested by Dray et al. (2006) but using a simpler approach whereby every distances within a user-defined trunction interval are taken as neighbour.

Functions Wf.sqrd, Wf.binary, Wf.PCNM, Wf.Drayf1, Wf.Drayf2, and Wf.Drayf3 are not intested to be called as is but through eigenmap (and within eigenmap.score). Other, user-defined, function can be used by eigenmap and should be visible to if one wants to call eigenmap.score to obtain predictors.

For eigenmap.score, the distances between sampling locations and the targets locations must be of the same type as those that had been passed to eigenmap. If cartesian coordinates were passed to x, the distances to target must be Euclidean.

References

Borcard, D. and Legendre, P. 2002. All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices. Ecol. Model. 153: 51-68

Dray, S.; Legendre, P. and Peres-Neto, P. 2006. Spatial modelling: a comprehensive framework for principal coordinate analysis of neighbor matrices (PCNM). Ecol. Modelling 196: 483-493

Legendre, P. and Legendre, L. 2012. Numerical Ecology, 3rd English edition. Elsevier Science B.V., Amsterdam, The Neatherlands.

Examples

Run this code

# NOT RUN {
#
### Example 1: A linear transect.
#
data(Salmon)
#
## No boundaries provided for a function that requires them: a warning is issued
map <- eigenmap(x=Salmon[,"Position"],weighting=Wf.binary)
map # plot(map)
#
## Boundaries are provided: the function is happy
map <- eigenmap(x=Salmon[,"Position"],weighting=Wf.binary,boundaries=c(0,20))
map # plot(map)
#
map <- eigenmap(x=Salmon[,"Position"],weighting=Wf.Drayf1,boundaries=c(0,20))
map # plot(map)
#
map <- eigenmap(x=Salmon[,"Position"],weighting=Wf.Drayf2,boundaries=c(0,20))
map # plot(map)
#
map <- eigenmap(x=Salmon[,"Position"],weighting=Wf.Drayf3,boundaries=c(0,20),wpar=2)
map # plot(map)
#
map <- eigenmap(x=Salmon[,"Position"],weighting=Wf.PCNM,boundaries=c(0,20))
map # plot(map)
#
map <- eigenmap(x=Salmon[,"Position"],weighting=Wf.sqrd)
map # plot(map)
#
### Example 2: Using predictior scores
#
smpl <- c(4,7,10,14,34,56,61,64)  # A sample to discard
map <- eigenmap(x=Salmon[-smpl,"Position"],weighting=Wf.sqrd)
scr <- eigenmap.score(object=map,target=as.matrix(dist(Salmon[,"Position"]))[,-smpl])
all(round(scr[-smpl,] - map$U, 10) == 0) # Scores of sampling points are the eigenvectors
scr[smpl,]
#
wh <- 5L   # You can try with other vectors.
plot(map$U[,wh]~Salmon[-smpl,"Position"], ylab = expression(U[5]),
     xlab = "Position along transect")
points(y=scr[smpl,wh],x=Salmon[smpl,"Position"],pch=21,bg="black")
#
map <- eigenmap(x=Salmon[-smpl,"Position"],weighting=Wf.binary,boundaries=c(0,20))
scr <- eigenmap.score(object=map,target=as.matrix(dist(Salmon[,"Position"]))[smpl,-smpl])
#
wh <- 1L   # You can try with other vectors.
plot(map$U[,wh]~Salmon[-smpl,"Position"], ylab = expression(U[1]),
     xlab = "Position along transect (m)")
points(y=scr[,wh],x=Salmon[smpl,"Position"],pch=21,bg="black")
#
map <- eigenmap(x=Salmon[-smpl,"Position"],weighting=Wf.PCNM,boundaries=c(0,100))
scr <- eigenmap.score(object=map,target=as.matrix(dist(Salmon[,"Position"]))[smpl,-smpl])
#
wh <- 1L   # You can try with other vectors.
plot(map$U[,wh]~Salmon[-smpl,"Position"], ylab = expression(U[1]),
     xlab = "Position along transect (m)")
points(y=scr[,wh],x=Salmon[smpl,"Position"],pch=21,bg="black")
#
### Example 3: A unevenly sampled surface.
#
data(Mite)
map <- eigenmap(x=as.matrix(mite.geo),weighting=Wf.sqrd)
map # plot(map)
#
# }

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