# adespatial: Moran's Eigenvector Maps and related methods for the spatial multiscale analysis of ecological communities In adespatial: Multivariate Multiscale Spatial Analysis

The package adespatial contains functions for the multiscale analysis of spatial multivariate data. It implements some new functions and reimplements existing functions that were available in packages of the sedaR project hosted on R-Forge (spacemakeR, packfor, AEM, etc.). It can be seen as a bridge between packages dealing with mutltivariate data (e.g., ade4, @Dray2007) and packages that deals with spatial data (spdep). In adespatial, the spatial information is considered as a spatial weighting matrix, object of class listw provided by the spdep package (Figure 1). It allows to build Moran's Eigenvector Maps (MEM, @Dray2006) that are orthogonal vectors maximizing the spatial autocorrelation (measured by Moran's index of autocorrelation). These spatial predictors can be used in multivariate statistical methods to provide spatially-explicit multiscale tools [@Dray2012]. This document provides a description of the main functionalities of the package.

Figure 1: Schematic representation of the functioning of the adespatial package. Classes are represented in pink frames and functions in blue frames. Classes and functions provided by adespatial are in bold.

To run the different analysis described, several packages are required and are loaded:

library(adespatial)
library(spdep)
library(maptools)


# Spatial Neighborhood

Spatial neighborhoods are managed in spdep as objects of class nb. It corresponds to the notion of connectivity matrices discussed in @Dray2006 and can be represented by an unweighted graph. Various functions are devoted to create nb objects from geographic coordinates of sites. We present different alternatives according to the design of the sampling scheme.

## Surface data

The function poly2nb allows to define neighborhood when the sampling sites are polygons and not points (two regions are neighbors if they share a common boundary).

data(mafragh)
class(mafragh$Spatial) par(mar = c(0, 0, 3, 0)) xx <- poly2nb(mafragh$Spatial)
plot(mafragh$Spatial, border = "grey") plot(xx, coordinates(mafragh$Spatial), add = TRUE, pch = 20, col = "red")
title(main="Neighborhood for polygons")


## Regular grids

If the sampling scheme is based on grid of 10 rows and 8 columns, spatial coordinates can be easily generated:

par(mar = rep(0,4))
xygrid <- expand.grid(x = 1:10, y = 1:8)
plot(xygrid, pch = 20, asp = 1)


For a regular grid, spatial neighborhood can be created with the function cell2nb. Two types of neighborhood can be defined. The queen specification considered horizontal, vertical and diagonal edges:

par(mar = c(0, 0, 3, 0))

nb1 <- cell2nb(10, 8, type = "queen")

plot(nb1, xygrid, col = "red", pch = 20)
title(main = "Queen neighborhood")

nb1


The rook specification considered only horizontal and vertical edges:

par(mar = c(0, 0, 3, 0))
nb2 <- cell2nb(10, 8, type = "rook")

plot(nb2, xygrid, col = "red", pch = 20)
title(main = "Rook neighborhood")

nb2


## Transects

The easiest way to deal with transects is to consider them as grids with only one row:

par(mar = c(0, 0, 3, 0))
xytransect <- expand.grid(1:20, 1)
nb3 <- cell2nb(20, 1)

plot(nb3, xytransect, col = "red", pch = 20)
title(main = "Transect of 20 sites")

summary(nb3)


All sites have two neighbors except the first and the last one.

## Irregular samplings

There are many ways to define neighborhood in the case of irregular samplings. We consider a random sampling with 10 sites:

par(mar = c(0, 0, 3, 0))
set.seed(3)
xyir <- matrix(runif(20), 10, 2)
plot(xyir, pch = 20, main = "Irregular sampling with 10 sites")


The most intuitive way is to consider that sites are neighbors (or not) according to the distances between them. This definition is provided by the dnearneigh function:

par(mar = c(0, 0, 3, 0), mfrow = c(2, 2))
nbnear1 <- dnearneigh(xyir, 0, 0.2)
nbnear2 <- dnearneigh(xyir, 0, 0.3)
nbnear3 <- dnearneigh(xyir, 0, 0.5)
nbnear4 <- dnearneigh(xyir, 0, 1.5)

plot(nbnear1, xyir, col = "red", pch = 20)
title(main = "neighbors if 0<d<0.2")
plot(nbnear2, xyir, col = "red", pch = 20)
title(main = "neighbors if 0<d<0.3")
plot(nbnear3, xyir, col = "red", pch = 20)
title(main = "neighbors if 0<d<0.5")
plot(nbnear4, xyir, col = "red", pch = 20)
title(main = "neighbors if 0<d<1.5")


Using a distance-based criteria could lead to unbalanced graphs. For instance, if the maximum distance is too low, some points have no neighbors:

nbnear1


On the other hand, if the maximum distance is to high, all sites could connected to the 9 others:

nbnear4


It is also possible to possible to define neighborhood by a criteria based on nearest neighbors. However, this option can lead to non-symmetric neighborhood: if site A is the nearest neighbor of site B, it does not mean that site B is the nearest neighbor of site A.

The function knearneigh creates an object of class knn. It can be transformed into a nb object with the function knn2nb. This function has an argument sym which can be set to TRUE to force the output neighborhood to symmetry.

par(mar = c(0, 0, 3, 0), mfrow = c(1, 2))
knn1 <- knearneigh(xyir, k = 1)
nbknn1 <- knn2nb(knn1, sym = TRUE)
knn2 <- knearneigh(xyir, k = 2)
nbknn2 <- knn2nb(knn2, sym = TRUE)

plot(nbknn1, xyir, col = "red", pch = 20)
title(main = "Nearest neighbors (k=1)")
plot(nbknn2, xyir, col = "red", pch = 20)
title(main="Nearest neighbors (k=2)")


This definition of neighborhood can lead to unconnected subgraphs. The function n.comp.nb finds the number of disjoint connected subgraphs:

n.comp.nb(nbknn1)
n.comp.nb(nbknn2)


More elaborate procedures are available to define neighborhood. For instance, Delaunay triangulation is obtained with the function tri2nb. It requires the package deldir. Other graph-based procedures are also available:

par(mar = c(0, 0, 3, 0), mfrow = c(2, 2))
nbtri <- tri2nb(xyir)
nbgab <- graph2nb(gabrielneigh(xyir), sym = TRUE)
nbrel <- graph2nb(relativeneigh(xyir), sym = TRUE)
nbsoi <- graph2nb(soi.graph(nbtri, xyir), sym = TRUE)

plot(nbtri, xyir, col = "red", pch = 20)
title(main="Delaunay triangulation")
plot(nbgab, xyir, col = "red", pch = 20)
title(main = "Gabriel Graph")
plot(nbrel, xyir, col = "red", pch = 20)
title(main = "Relative Neighbor Graph")
plot(nbsoi, xyir, col = "red", pch = 20)
title(main = "Sphere of Influence Graph")


The function chooseCN provides a simple way to build spatial neighborhoods. It is a wrap up to many of the spdep functions presented above. The function createlistw discussed in section XX is an interactive graphical interface that allows to generate R code to build neighborhood objects.

## Manipulation of nb objects

A nb object is a list of neighbors. The neighbors of the first site are in the first element of the list:

nbgab[[1]]


Various tools are provided by spdep to deal with these objects. For instance, it is possible to identify differences between two neighborhoods:

diffnb(nbsoi,nbrel)


Usually, it can be useful to remove some connections due to edge effects. In this case, the function edit.nb provides an interactive tool to add or delete connections.

The function include.self allows to include a site itself in its own list of neighbors:

str(nbsoi)
str(include.self(nbsoi))


The spdep package provides many other tools to manipulate nb objects:

intersect.nb(nb.obj1, nb.obj2)
union.nb(nb.obj1, nb.obj2)
setdiff.nb(nb.obj1, nb.obj2)
complement.nb(nb.obj)

nblag(neighbours, maxlag)


# Spatial weighting matrices

Spatial weighting matrices are computed by a transformation of the spatial neighborhood objects. In R, they are not stored as matrices but as objects of the class listw. This format is more efficient than a matrix representation to manage large data sets. An object of class listw can be easily created from an object of class nb with the function nb2listw.

Different objects listw can be obtained from a nb object. The argument style allows to define a transformation of the matrix such as standardization by row sum, by total sum or binary coding, etc. General spatial weights can be introduced by the argument glist. This allows to introduce, for instance, a weighting relative to the distances between the points. For this task, the function nbdists is very useful as it computes Euclidean distance between neighbor sites defined by an nb object.

To obtain a simple row-standardization, the function is simply called by:

nb2listw(nbgab)


More sophisticated forms of spatial weighting matrices can be defined. For instance, it is possible to weight edges between neighbors as functions of geographic distances. In a fist step, distances between neighbors are obtained by the function \texttt{nbdists}:

distgab <- nbdists(nbgab, xyir)
str(distgab)


Then, spatial weights are defined as a function of distance (e.g. $1-d_{ij}/max(d_{ij})$):

fdist <- lapply(distgab, function(x) 1-x/max(dist(xyir)))


And the spatial weighting matrix is then created:

listwgab <- nb2listw(nbgab, glist = fdist, style = "B")
listwgab
names(listwgab)
listwgab$neighbours[[1]] listwgab$weights[[1]]


The matrix representation of a listw object can also be obtained:

print(listw2mat(listwgab),digits=3)


To facilitate the building of spatial neighborhoods (nb object) and associated spatial weighting matrices (listw object), the package adespatial provides an interactive graphical interface. The interface is launched by the call createlistw() assuming that spatial coordinates are still stored in an object of the R session (Figure 2).

Figure 2: The interactive interface provided by the function createlistw.

# Spatial predictors

The package adespatial provide different tools to build spatial predictors that can be incorporated in multivariate analysis. They are orthogonal vectors stored in a object of class orthobasisSp. Orthogonal polynomials of geographic coordinates can be computed by the function orthobasis.poly whereas traditional principal coordinates of neighbour matrices (PCNM, @Borcard2002) are obtained by the function dbmem. The more flexible Moran's eigenvectors maps (MEMs) of a spatial weighting matrix are computed by the functions scores.listw or mem of the adespatial package. These two functions are exactly identical and return an object of class orthobasisSp.

mem.gab <- mem(listwgab)
mem.gab


This object contains MEMs, stored as a data.frame and other attributes:

str(mem.gab)


The eigenvalues associated to MEMs are stored in the attribute called values:

par(mar = c(0, 2, 3, 0))
barplot(attr(mem.gab, "values"),
main = "Eigenvalues of the spatial weighting matrix", cex.main = 0.7)


A plot method is provided to represent MEMs. By default, eigenvectors are represented as a table (sites as rows, MEMs as columns):

plot(mem.gab)


The previous representation is not really informative and MEMs can be represented in the geographical space as maps if the argument SpORcoords is documented:

plot(mem.gab, SpORcoords = xyir, nb = nbgab)


Moran's I can be computed and tested for each eigenvector with the moran.randtest function:

moranI <- moran.randtest(mem.gab, listwgab, 99)
moranI


By default, the function moran.randtest tests against the alternative hypothesis of positive autocorrelation (alter = "greater") but this can be modified by setting the argument alter to "less" or "two-sided". The function is not only devoted to MEMs and can be used to compute spatial autocorrelations for all kind of variables.

As demonstrated in @Dray2006, eigenvalues and Moran's I are equal (post-multiply by a constant):

attr(mem.gab, "values") / moranI$obs  Then, it is possible to map only positive significant eigenvectors (i.e., MEMs with significant positive spatial autocorrelation): signi <- which(moranI$p < 0.05)
signi
plot(mem.gab[,signi], SpORcoords = xyir, nb = nbgab)


# Data-driven selection of a spatial weighting matrix

The choice of a spatial weighting matrix is an important step and @Dray2006 proposed a data-driven procedure of selection based on AICc. The function ortho.AIC orders variables and returns AICc for all models of one, two, ..., $p$ variables. We illustrate its use with the oribatid data-set which is available in the ade4 package. Data are Hellinger-transformed and then the linear trend is removed:

data(oribatid)
fau <- sqrt(oribatid$fau / outer(apply(oribatid$fau, 1, sum),
rep(1, ncol(oribatid$fau)), "*")) faudt <- resid(lm(as.matrix(fau) ~ as.matrix(oribatid$xy)))


For instance, we consider the binary spatial weighting matrix based on the Delaunay triangulation.

nbtri <- tri2nb(as.matrix(oribatid$xy)) sc.tri <- scores.listw(nb2listw(nbtri, style = "B")) AIC.tri <- ortho.AIC(faudt, sc.tri) head(AIC.tri)  The minimum value and the rank of the corresponding are obtained easily: min(AIC.tri, na.rm = TRUE) which.min(AIC.tri)  Note that the order of the variables can also be obtained from the function ortho.AIC by setting the ord.var argument to TRUE. In this case, the returned object is a list of two vectors: AIC.tri <- ortho.AIC(faudt, sc.tri, ord.var = TRUE) head(AIC.tri$AICc)
head(AIC.tri$ord)  The user-friendly function test.W simplifies the procedure of selection of a spatial weighting matrix. It takes at least two arguments: a response matrix and an object of the class nb. If only two arguments are considered, the function prints the results for the best model. All the results are stored in the element best of the list. It contains eigenvectors and eigenvalues of the spatial weighting matrix considered and the results of the AIC-based procedure. tri.res <- test.W(faudt,nbtri) names(tri.res) names(tri.res$best)


The function can also be used to estimate the best values of parameters if we consider a function of the distance. This can be illustrated with the function $f_2=1-(x^\alpha)/dmax^\alpha$ with the connectivity defined by Delaunay triangulation. We considered the sequence of integers between 2 and 10 for $\alpha$.

f2 <- function(x, dmax, y) {
1 - (x ^ y) / (dmax) ^ y
}

maxi <- max(unlist(nbdists(nbtri, as.matrix(oribatid$xy)))) tri.f2 <- test.W(faudt, nbtri, f = f2, y = 2:10, dmax = maxi, xy = as.matrix(oribatid$xy))


In this case, the element best contains the results for the best values of the parameter $\alpha$.

names(tri.f2$best)  Lastly, the function test.W can be used to evaluate different definitions of neighborhood. We illustrate this possibility by the definition of a sequence of neighborhood by distance criteria. Firstly, we choose the range of values to be tested with an empirical multivariate variogram using the function variogmultiv. The function has been applied to oribatid mites data: mvspec <- variogmultiv(faudt, oribatid$xy, nclass = 20)
plot(mvspec$d, mvspec$var, type = 'b', pch = 20, xlab = "Distance", ylab = "C(distance)")


We will construct ten neighborhood matrices with a distance criterion varying along the sequence of 10 evenly distributed values between 1.012 and 4 m:

dxy <- seq(give.thresh(dist(oribatid$xy)), 4, length = 10) nbdnnlist <- lapply(dxy, dnearneigh, x = as.matrix(oribatid$xy), d1 = 0)


Then, the function test.W can be applied to this list of neighborhood matrices:

dnn.bin <- lapply(nbdnnlist, test.W, Y = faudt)


The object dnn.bin is a list with the results of test.W for each neighborhood matrix:

length(dnn.bin)


For each neighborhood matrix, we can find the lowest \textit{AICc}:

minAIC <- sapply(dnn.bin, function(x) min(x$best$AIC$AICc, na.rm = TRUE))  And select the best spatial weighting matrix corresponding to a distance of r round(dxy[which.min(minAIC)],3) m: which.min(minAIC) dxy[which.min(minAIC)]  A similar approach can be used with a spatial weighting function: f2 <- function(x,dmax,y) {1-(x^y)/(dmax)^y}  It is a little bit more complicate if some parameters (here dmax) vary with the neighborhood matrix: dnn.f2 <- lapply(nbdnnlist, function(x) test.W(x, Y = faudt, f = f2, y = 2:10, dmax = max(unlist(nbdists(x, as.matrix(oribatid$xy)))), xy = as.matrix(oribatid$xy))) minAIC <- sapply(dnn.f2, function(x) min(x$best$AIC$AICc, na.rm = TRUE))
min(minAIC)
which.min(minAIC)
dnn.f2[[which.min(minAIC)]]$all  Lastly, Eigenvectors of the best spatial weighting matrix can be mapped. They are represented by the order given by the selection procedure. The third MEM explains the largest part of the oribatid community, then it is the second and the eighth: plot(dnn.f2[[7]]$best$MEM[, dnn.f2[[7]]$best$AIC$ord[1:3]], oribatid\$xy)