Construct asymmetric eigenvector maps (AEM)

This function constructs eigenvectors of a site-by-link matrix. Weights can be applied to the links.

aem(
  aem.build.binary,
  binary.mat,
  weight,
  rm.link0 = FALSE,
  print.binary.mat = FALSE
)

Arguments

aem.build.binary: Object created by function aem.build.binary.
binary.mat: Site (n rows) by link (k columns) matrix. The 1s in the matrix represents the presence of a link influencing a site, directly or indirectly, otherwise the values are 0s.
weight: Vector of weights of length k, to be applied to the links.
rm.link0: Logical (TRUE, FALSE) determining if the links directly connecting a site to the origin (site 0) should be removed. Default value: FALSE. This parameter is only used when an object of class aem.build.binary is provided to the function.
print.binary.mat: Logical (TRUE, FALSE) determining if the site-by-link matrix used in the analysis should be printed.

Value

value: A vector of singular values associated with the AEM.
vectors: A matrix of eigenvector. Each column is an AEM eigenfunction (or variable).
mod.binary.mat: A site-by-link matrix modified through the function.

Details

If only an object of class aem.build.binary is given to this function, The argument binary.mat is not considered. binary.mat is only considered when the argument aem.build.binary is missing.

If weights are applied to the links, the length of vector weight has to take into account wether the links connecting real sites to the origin (the fictitious site 0) have been kept or removed.

Note

It sometimes happens that AEM eigenfunctions have equal singular values. In that case, different sets of AEM eigenfunctions may be produced on different plateforms.

Eigenvectors associated to an eigenvalue that is smaller than \(10^{-12}\) are considered negligeable. They have been removed from the created AEM eigenfunctions.

References

Blanchet F.G., P. Legendre and Borcard D. (2008) Modelling directional spatial processes in ecological data. Ecological Modelling, 215, 325-336.

Author

F. Guillaume Blanchet

Examples

# Construction of object of class nb (spdep)
if(require("spdep", quietly = TRUE)){
nb <- cell2nb(5,5,"queen")

# Create fictitious geographical coordinates 
xy <- cbind(1:25,expand.grid(1:5,1:5))

# Build binary site-by-link matrix
bin.mat <- aem.build.binary(nb,xy)

# Construct AEM eigenfunctions from an object of class aem.build.binary
res <- aem(aem.build.binary=bin.mat,rm.link0=FALSE)
res$values

# Illustrate 4 AEM eigenfunctions using bubble plots
opal <- palette()
palette(c("black","white"))
oldpar <- par(mfrow=c(2,2))
symbols(x=xy[,2:3], circles=abs(res$vectors[,1]), inches=FALSE, asp=1,
 fg=ifelse(sign(-res$vectors[,1])+1>0,1,0), 
 bg=ifelse(sign(res$vectors[,1])+1>0,1,0), xlab="x", ylab="y")
title("AEM 1")
symbols(x=xy[,2:3], circles=abs(res$vectors[,2]), inches=FALSE, 
asp=1, fg=ifelse(sign(-res$vectors[,2])+1>0,1,0),
 bg=ifelse(sign(res$vectors[,2])+1>0,1,0), xlab="x", ylab="y")
title("AEM 2")
symbols(x=xy[,2:3], circles=abs(res$vectors[,3]), inches=FALSE, 
asp=1, fg=ifelse(sign(-res$vectors[,3])+1>0,1,0), 
bg=ifelse(sign(res$vectors[,3])+1>0,1,0), xlab="x", ylab="y")
title("AEM 3")
symbols(x=xy[,2:3], circles=abs(res$vectors[,4]), inches=FALSE, asp=1,
 fg=ifelse(sign(-res$vectors[,4])+1>0,1,0), 
 bg=ifelse(sign(res$vectors[,4])+1>0,1,0), xlab="x", ylab="y")
title("AEM 4")

# Construct AEM eigenfunctions using only a site-by-link matrix
res2 <- aem(binary.mat=bin.mat[[1]])
res2$values

# Illustrate 4 AEM eigenfunctions using bubble plots
par(mfrow=c(2,2))
symbols(x=xy[,2:3], circles=abs(res2$vectors[,1]), inches=FALSE, 
asp=1, fg=ifelse(sign(-res2$vectors[,1])+1>0,1,0), 
bg=ifelse(sign(res2$vectors[,1])+1>0,1,0), xlab="x", ylab="y")
title("AEM 1")
symbols(x=xy[,2:3], circles=abs(res2$vectors[,2]), inches=FALSE,
asp=1, fg=ifelse(sign(-res2$vectors[,2])+1>0,1,0), 
bg=ifelse(sign(res2$vectors[,2])+1>0,1,0), xlab="x", ylab="y")
title("AEM 2")
symbols(x=xy[,2:3], circles=abs(res2$vectors[,3]), inches=FALSE,
asp=1, fg=ifelse(sign(-res2$vectors[,3])+1>0,1,0), 
bg=ifelse(sign(res2$vectors[,3])+1>0,1,0), xlab="x", ylab="y")
title("AEM 3")
symbols(x=xy[,2:3], circles=abs(res2$vectors[,4]), inches=FALSE,asp=1,
 fg=ifelse(sign(-res2$vectors[,4])+1>0,1,0), 
 bg=ifelse(sign(res2$vectors[,4])+1>0,1,0), xlab="x", ylab="y")
title("AEM 4")

palette(opal)
par(oldpar)

# Construct AEM eigenfunctions with a function of the distance
# as weights to put on the links

# Construction of object of class nb (spdep)
nb<-cell2nb(5,5,"queen")

# Create fictitious geographical coordinates
xy <- cbind(1:25,expand.grid(1:5,1:5))

# Build binary site-by-link matrix
bin.mat <- aem.build.binary(nb,xy)

# Construct a matrix of distances
long.lien.mat<-as.matrix(dist(xy))

# Extract the edges, remove the ones directly linked to site 0
lien.b<-bin.mat$edges[-1:-5,]

# Construct a vector giving the length of each edge
long.lien<-vector(length=nrow(lien.b))

for(i in 1:nrow(lien.b)){
  long.lien[i]<-long.lien.mat[lien.b[i,1],lien.b[i,2]]
}

# Construct a vector of weights based on distance
weight.vec<-1-(long.lien/max(long.lien))^2

# Construct AEM eigenfunctions from an object of class aem.build.binary
res <- aem(aem.build.binary=bin.mat,weight=weight.vec,rm.link0=TRUE)
res

# Computing Moran's I for AEMs

# Building AEMs
xy <- cbind(1:25,expand.grid(1:5,1:5))
Wdist <- 1/as.matrix(dist(xy[,2:3]))

nb <- cell2nb(5,5,"queen")
bin.mat <- aem.build.binary(nb,xy)
linkBase <- bin.mat[[2]]
link <- linkBase[-which(linkBase[,1] == 0),]
weight <- numeric()

for(i in 1:nrow(link)){
   weight[i] <- Wdist[link[i,1],link[i,2]]
}

AEM <- aem(bin.mat, weight = weight, rm.link0 = TRUE)

# Constructing asymmetric matrix
matasym <- matrix(0,ncol=25, nrow=25)

for(i in 1:nrow(link)){
    matasym[link[i,1],link[i,2]]<- weight[i]
}

# Build a listw object from the asymmetric matrix
listwAsym <-  mat2listw(matasym, style = "B", zero.policy = TRUE)

# Calculate Moran's I for AEM
MoranIAEM <- moran.randtest(AEM$vectors, listwAsym)

}
#> To access larger datasets in this package, install the spDataLarge
#> package with: `install.packages('spDataLarge',
#> repos='https://nowosad.github.io/drat/', type='source')`
#> Linking to GEOS 3.10.2, GDAL 3.4.1, PROJ 8.2.1; sf_use_s2() is TRUE
#> 
#> Attaching package: ‘spdep’
#> The following object is masked from ‘package:ade4’:
#> 
#>     mstree