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 multivariate
data (e.g., `ade4`

, Dray and Dufour
(2007)) and
packages that deals with spatial data (`sp`

,
`spdep`

). In `adespatial`

, many methods consider
the spatial information as a spatial weighting matrix (SWM), object of
class `listw`

provided by the `spdep`

package (Figure 1). The SWM is defined as the Hadamard
product (element-wise product) of a connectivity matrix by a weighting
matrix. The binary connectivity matrix (spatial neighborhood, object of
class `nb`

) defines the pairs of connected and unconnected
samples, while the weighting matrix allows weighting the connections,
for instance to define that the strength of the connection between two
samples decreases with the geographic distance.

Once SWM is defined, it can be used to build Moran’s Eigenvector Maps (MEM, Dray, Legendre, and Peres-Neto (2006)) that are orthogonal vectors maximizing the spatial autocorrelation (measured by Moran’s coefficient). These spatial predictors can be used in multivariate statistical methods to provide spatially-explicit multiscale tools (Dray et al. 2012). 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(ade4)
library(adespatial)
```

```
## Registered S3 methods overwritten by 'adegraphics':
## method from
## biplot.dudi ade4
## kplot.foucart ade4
## kplot.mcoa ade4
## kplot.mfa ade4
## kplot.pta ade4
## kplot.sepan ade4
## kplot.statis ade4
## scatter.coa ade4
## scatter.dudi ade4
## scatter.nipals ade4
## scatter.pco ade4
## score.acm ade4
## score.mix ade4
## score.pca ade4
## screeplot.dudi ade4
```

```
## Registered S3 method overwritten by 'spdep':
## method from
## plot.mst ape
```

```
## Registered S3 methods overwritten by 'adespatial':
## method from
## plot.multispati adegraphics
## print.multispati ade4
## summary.multispati ade4
```

```
##
## Attachement du package : 'adespatial'
```

```
## L'objet suivant est masqué depuis 'package:ade4':
##
## multispati
```

`library(adegraphics)`

```
##
## Attachement du package : 'adegraphics'
```

```
## Les objets suivants sont masqués depuis 'package:ade4':
##
## kplotsepan.coa, s.arrow, s.class, s.corcircle, s.distri, s.image,
## s.label, s.logo, s.match, s.traject, s.value, table.value,
## triangle.class
```

`library(spdep)`

`## Le chargement a nécessité le package : sp`

`## Le chargement a nécessité le package : spData`

```
## To access larger datasets in this package, install the spDataLarge
## package with: `install.packages('spDataLarge',
## repos='https://nowosad.github.io/drat/', type='source')`
```

`## Le chargement a nécessité le package : sf`

`## Linking to GEOS 3.10.2, GDAL 3.4.1, PROJ 8.2.1; sf_use_s2() is TRUE`

```
##
## Attachement du package : 'spdep'
```

```
## L'objet suivant est masqué depuis 'package:ade4':
##
## mstree
```

`library(sp)`

The Mafragh data set is used to illustrate several methods. It is
available in the `ade4`

package and stored as a
`list`

:

```
data("mafragh")
class(mafragh)
```

`## [1] "list"`

`names(mafragh)`

```
## [1] "xy" "flo" "neig" "env"
## [5] "partition" "area" "tre" "traits"
## [9] "nb" "Spatial" "spenames" "Spatial.contour"
```

`dim(mafragh$flo)`

`## [1] 97 56`

The `data.frame`

`mafragh$flo`

is a floristic
table that contains the abundance of 56 plant species in 97 sites in
Algeria. Names of species are listed in `mafragh$spenames`

.
The geographic coordinates of the sites are given in
`mafragh$xy`

.

`str(mafragh$env)`

```
## 'data.frame': 97 obs. of 11 variables:
## $ Clay : num 0.73 0.75 0.74 0.23 0.73 0.72 0.52 0.42 0.74 0.53 ...
## $ Silt : num 0.24 0.24 0.24 0.26 0.24 0.22 0.46 0.49 0.24 0.44 ...
## $ Sand : num 0.03 0.02 0.02 0.49 0.03 0.03 0.02 0.08 0.02 0.04 ...
## $ K2O : num 1.3 0.8 1.7 0.3 1.3 1.7 0.8 1.4 1.7 1.4 ...
## $ Mg++ : num 9.2 10.7 8.6 2 9.2 6 6 19.6 8.6 17 ...
## $ Na+/100g : num 4.2 10.4 10.8 1.2 4.2 10.7 18.4 2.5 10.8 11.7 ...
## $ K+ : num 1.2 1.4 1.9 0.3 1.2 1.3 2.2 1.3 1.9 0.5 ...
## $ Conductivity: num 7.9 11.5 10.4 0.6 7.9 14.5 15.2 2.9 10.4 16.9 ...
## $ Retention : num 41.8 42.4 41.4 22.3 41.8 42.7 37.4 35.4 41.4 38 ...
## $ Na+/l : num 48.7 66 24 2.2 48.7 ...
## $ Elevation : num 6 2 2 6 6 4 4 3 3 6 ...
```

The `data.frame`

`mafragh$env`

contains 11
quantitative environmental variables. A map of the study area is also
available (`mafragh$Spatial.contour`

) that can be used as a
background to display the sampling design:

```
<- as.matrix(mafragh$xy)
mxy rownames(mxy) <- NULL
s.label(mxy, ppoint.pch = 15, ppoint.col = "darkseagreen4", Sp = mafragh$Spatial.contour)
```

A more detailed description of the data set is available at http://pbil.univ-lyon1.fr/R/pdf/pps053.pdf (in French).

The functionalities of the `adegraphics`

package (Siberchicot et al. 2017) can be used to
design simple thematic maps to represent data. For instance, it is
possible to represent the spatial distribution of two species using the
`s.Spatial`

function applied on Voronoi polygons contained in
`mafragh$Spatial`

:

`$spenames[c(1, 11), ] mafragh`

```
## scientific code
## Sp1 Arisarum vulgare Arvu
## Sp11 Bolboschoenus maritimus Boma
```

```
<- colorRampPalette(c("white", "darkseagreen2", "darkseagreen3", "palegreen4"))
fpalette <- SpatialPolygonsDataFrame(Sr = mafragh$Spatial, data = mafragh$flo, match.ID = FALSE)
sp.flo s.Spatial(sp.flo[,c(1, 11)], col = fpalette(3), nclass = 3)
```

Spatial neighborhoods are managed in `spdep`

as objects of
class `nb`

. It corresponds to the notion of connectivity
matrices discussed in Dray, Legendre, and
Peres-Neto (2006) and can be represented by an
unweighted graph. Various functions allow to create `nb`

objects from geographic coordinates of sites. We present different
alternatives according to the design of the sampling scheme.

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). The resulting object can be
plotted on a geographical map using the `s.Spatial`

function
of the `adegraphics`

package (Siberchicot et al.
2017).

```
data(mafragh)
class(mafragh$Spatial)
```

```
## [1] "SpatialPolygons"
## attr(,"package")
## [1] "sp"
```

```
<- poly2nb(mafragh$Spatial)
nb.maf s.Spatial(mafragh$Spatial, nb = nb.maf, plabel.cex = 0, pnb.edge.col = 'red')
```

If the sampling scheme is based on regular sampling (e.g., grid of 8 rows and 10 columns), spatial coordinates can be easily generated:

```
<- expand.grid(x = 1:10, y = 1:8)
xygrid s.label(xygrid, plabel.cex = 0)
```

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 whereas the `rook`

specification considered
only horizontal and vertical edges:

```
<- cell2nb(8, 10, type = "queen")
nb2.q <- cell2nb(8, 10, type = "rook")
nb2.r s.label(xygrid, nb = nb2.q, plabel.cex = 0, main = "Queen neighborhood")
```

`s.label(xygrid, nb = nb2.r, plabel.cex = 0, main = "Rook neighborhood")`

The function `cell2nb`

is the easiest way to deal with
transects by considering a grid with only one row:

```
<- expand.grid(1:20, 1)
xytransect <- cell2nb(20, 1)
nb3
summary(nb3)
```

```
## Neighbour list object:
## Number of regions: 20
## Number of nonzero links: 38
## Percentage nonzero weights: 9.5
## Average number of links: 1.9
## Link number distribution:
##
## 1 2
## 2 18
## 2 least connected regions:
## 1:1 1:20 with 1 link
## 18 most connected regions:
## 1:2 1:3 1:4 1:5 1:6 1:7 1:8 1:9 1:10 1:11 1:12 1:13 1:14 1:15 1:16 1:17 1:18 1:19 with 2 links
```

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

There are many ways to define the neighborhood in the case of
irregular samplings. We consider a random subsample of 20 sites of the
`mafragh`

data set to better illustrate the differences
between methods:

```
set.seed(3)
<- mxy[sample(1:nrow(mafragh$xy), 20),]
xyir s.label(xyir, main = "Irregular sampling with 20 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:

```
<- dnearneigh(xyir, 0, 50)
nbnear1 <- dnearneigh(xyir, 0, 305)
nbnear2
<- s.label(xyir, nb = nbnear1, pnb.edge.col = "red", main = "neighbors if 0<d<50", plot = FALSE)
g1 <- s.label(xyir, nb = nbnear2, pnb.edge.col = "red", main = "neighbors if 0<d<305", plot = FALSE)
g2 cbindADEg(g1, g2, plot = TRUE)
```

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`

```
## Neighbour list object:
## Number of regions: 20
## Number of nonzero links: 38
## Percentage nonzero weights: 9.5
## Average number of links: 1.9
```

On the other hand, if the maximum distance is too high, all sites are connected:

` nbnear2`

```
## Neighbour list object:
## Number of regions: 20
## Number of nonzero links: 354
## Percentage nonzero weights: 88.5
## Average number of links: 17.7
```

It is also 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.

```
<- knearneigh(xyir, k = 1)
knn1 <- knn2nb(knn1, sym = TRUE)
nbknn1 <- knearneigh(xyir, k = 2)
knn2 <- knn2nb(knn2, sym = TRUE)
nbknn2
<- s.label(xyir, nb = nbknn1, pnb.edge.col = "red", main = "Nearest neighbors (k=1)", plot = FALSE)
g1 <- s.label(xyir, nb = nbknn2, pnb.edge.col = "red", main = "Nearest neighbors (k=2)", plot = FALSE)
g2 cbindADEg(g1, g2, plot = TRUE)
```

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)`

```
## $nc
## [1] 7
##
## $comp.id
## [1] 1 2 3 1 4 5 3 6 2 3 3 7 1 6 5 4 7 5 7 3
```

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:

```
<- tri2nb(xyir)
nbtri <- graph2nb(gabrielneigh(xyir), sym = TRUE)
nbgab <- graph2nb(relativeneigh(xyir), sym = TRUE)
nbrel
<- s.label(xyir, nb = nbtri, pnb.edge.col = "red", main = "Delaunay", plot = FALSE)
g1 <- s.label(xyir, nb = nbgab, pnb.edge.col = "red", main = "Gabriel", plot = FALSE)
g2 <- s.label(xyir, nb = nbrel, pnb.edge.col = "red", main = "Relative", plot = FALSE)
g3
ADEgS(list(g1, g2, g3))
```

The `adespatial`

functions `chooseCN`

and
`listw.candidates`

provides simple ways to build spatial
neighborhoods. They are wrappers of many of the `spdep`

functions presented above. The function `listw.explore`

is an
interactive graphical interface that allows to generate R code to build
neighborhood objects (see Figure 2).

`nb`

objectsA `nb`

object is not stored as a matrix. It is a list of
neighbors. The neighbors of the first site are in the first element of
the list:

`1]] nbgab[[`

`## [1] 6 13`

Various tools are provided by `spdep`

to deal with these
objects. For instance, it is possible to identify differences between
two neighborhoods:

`diffnb(nbgab, nbrel)`

```
## Neighbour list object:
## Number of regions: 20
## Number of nonzero links: 14
## Percentage nonzero weights: 3.5
## Average number of links: 0.7
## 10 regions with no links:
## 4 5 7 9 10 11 16 17 18 20
```

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 in
its own list of neighbors (self-loops). 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)
droplinks(nb, drop, sym = TRUE)
nblag(neighbours, maxlag)
```

A spatial weighting matrices (SWM) is computed by a transformation of a spatial neighborhood. We consider the Gabriel graph for the full data set:

`<- graph2nb(gabrielneigh(mxy), sym = TRUE) nbgab `

In R, SWM 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)`

```
## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 97
## Number of nonzero links: 450
## Percentage nonzero weights: 4.782655
## Average number of links: 4.639175
##
## Weights style: W
## Weights constants summary:
## n nn S0 S1 S2
## W 97 9409 97 45.3915 395.3193
```

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 :

```
<- nbdists(nbgab, mxy)
distgab 1]] nbgab[[
```

`## [1] 2 4 5 6`

`1]] distgab[[`

`## [1] 16.63971 21.34986 14.54966 16.99176`

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

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

And the spatial weighting matrix is then created:

```
<- nb2listw(nbgab, glist = fdist)
listwgab listwgab
```

```
## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 97
## Number of nonzero links: 450
## Percentage nonzero weights: 4.782655
## Average number of links: 4.639175
##
## Weights style: W
## Weights constants summary:
## n nn S0 S1 S2
## W 97 9409 97 45.41085 395.21
```

`names(listwgab)`

`## [1] "style" "neighbours" "weights"`

`$neighbours[[1]] listwgab`

`## [1] 2 4 5 6`

`$weights[[1]] listwgab`

`## [1] 0.2505174 0.2472375 0.2519728 0.2502723`

The matrix representation of a `listw`

object can also be
obtained:

`print(listw2mat(listwgab)[1:10, 1:10], digits = 3)`

```
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## 1 0.000 0.251 0.000 0.247 0.252 0.250 0.000 0.000 0.000 0
## 2 0.250 0.000 0.250 0.000 0.000 0.250 0.250 0.000 0.000 0
## 3 0.000 0.251 0.000 0.000 0.000 0.000 0.251 0.249 0.249 0
## 4 0.199 0.000 0.000 0.000 0.201 0.200 0.000 0.000 0.000 0
## 5 0.503 0.000 0.000 0.497 0.000 0.000 0.000 0.000 0.000 0
## 6 0.201 0.201 0.000 0.199 0.000 0.000 0.202 0.000 0.000 0
## 7 0.000 0.200 0.200 0.000 0.000 0.201 0.000 0.199 0.000 0
## 8 0.000 0.000 0.167 0.000 0.000 0.000 0.167 0.000 0.167 0
## 9 0.000 0.000 0.333 0.000 0.000 0.000 0.000 0.335 0.000 0
## 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0
```

To facilitate the building of spatial neighborhoods (`nb`

object) and associated spatial weighting matrices (`listw`

object), the package `adespatial`

provides several tools. An
interactive graphical interface is launched by the call
`listw.explore()`

assuming that spatial coordinates are still
stored in an object of the R session (Figure
2).

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 principal coordinates of neighbour matrices (PCNM, Borcard and Legendre (2002)) are obtained by the function
`dbmem`

.

The Moran’s Eigenvectors Maps (MEMs) provide the most flexible framework. If we consider the \(n \times n\) spatial weighting matrix \(\mathbf{W} = [w_{ij}]\), they are the \(n-1\) eigenvectors obtained by the diagonalization of the doubly-centred SWM:

\[ \mathbf{\Omega V}= \mathbf{V \Lambda} \]

where \(\mathbf{\Omega} = \mathbf{HWH}\) is the doubly-centred SWM and \(\mathbf{H} = \left ( \mathbf{I}-\mathbf{11}\hspace{-0.05cm}^{\top}\hspace{-0.05cm}/n \right )\) is the centring operator.

MEMs are orthogonal vectors with a unit norm that maximize Moran’s coefficient of spatial autocorrelation (Griffith 1996; Dray et al. 2012) and are stored in matrix \(\mathbf{V}\).

MEMs are provided by the functions `scores.listw`

or
`mem`

of the `adespatial`

package. These two
functions are exactly identical (both are kept for historical reasons
and compatibility) and return an object of class
`orthobasisSp`

.

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

```
## Orthobasis with 97 rows and 96 columns
## Only 6 rows and 4 columns are shown
## MEM1 MEM2 MEM3 MEM4
## 1 0.9251530 -2.050270 -0.6159371 -1.13648688
## 2 0.8495416 -1.859746 -0.4163876 -0.57971608
## 3 0.8092292 -1.699300 -0.1970169 0.02251458
## 4 1.0455937 -2.177654 -0.7488499 -1.45727142
## 5 0.7098875 -1.571499 -0.5144638 -1.00604362
## 6 0.9629486 -2.017900 -0.5572747 -0.92335694
```

This object contains MEMs, stored as a `data.frame`

and
other attributes:

`class(mem.gab)`

`## [1] "orthobasisSp" "orthobasis" "data.frame"`

`names(attributes(mem.gab))`

`## [1] "names" "class" "row.names" "values" "weights" "call"`

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

:

```
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). This representation is usually not informative and it is
better to map MEMs in the geographical space by documenting the argument
`SpORcoords`

:

`plot(mem.gab[,c(1, 5, 10, 20, 30, 40, 50, 60, 70)], SpORcoords = mxy)`

or using the more flexible `s.value`

function:

`s.value(mxy, mem.gab[,c(1, 5, 10, 20, 30, 40, 50, 60, 70)], symbol = "circle", ppoint.cex = 0.6)`

Moran’s I can be computed and tested for each eigenvector with the
`moran.randtest`

function:

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

As demonstrated in Dray, Legendre, and Peres-Neto (2006), eigenvalues and Moran’s I are equal (post-multiply by a constant):

`head(attr(mem.gab, "values") / moranI$obs)`

```
## MEM1.statistic MEM2.statistic MEM3.statistic MEM4.statistic MEM5.statistic
## 0.01030928 0.01030928 0.01030928 0.01030928 0.01030928
## MEM6.statistic
## 0.01030928
```

In the previous sections, we considered only the spatial information
as a geographic map, a spatial weighting matrix (SWM) or a basis of
spatial predictors (e.g., MEM). In the next sections, we will show how
the spatial information can be integrated to analyze multivariate data
and identify multiscale spatial patterns. The dataset
`mafragh`

available in `ade4`

will be used to
illustrate the different methods.

The SWM can be used to compute the level of spatial autocorrelation of a quantitative variable using the Moran’s coefficient. If we consider the \(n \times 1\) vector \(\mathbf{x} = \left ( {x_1 \cdots x_n } \right )\hspace{-0.05cm}^{\top}\hspace{-0.05cm}\) containing measurements of a quantitative variable for \(n\) sites and \(\mathbf{W} = [w_{ij}]\) the SWM. The usual formulation for Moran’s coefficient (MC) of spatial autocorrelation is: \[ MC(\mathbf{x}) = \frac{n\sum\nolimits_{\left( 2 \right)} {w_{ij} (x_i -\bar {x})(x_j -\bar {x})} }{\sum\nolimits_{\left( 2 \right)} {w_{ij} } \sum\nolimits_{i = 1}^n {(x_i -\bar {x})^2} }\mbox{ where }\sum\nolimits_{\left( 2 \right)} =\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n } \mbox{ with }i\ne j \]

MC can be rewritten using matrix notation: \[ MC(\mathbf{x}) = \frac{n}{\mathbf{1}\hspace{-0.05cm}^{\top}\hspace{-0.05cm}\mathbf{W1}}\frac{\mathbf{z}\hspace{-0.05cm}^{\top}\hspace{-0.05cm}{\mathbf{Wz}}}{\mathbf{z}\hspace{-0.05cm}^{\top}\hspace{-0.05cm}\mathbf{z}} \] where \(\mathbf{z} = \left ( \mathbf{I}-\mathbf{1}\mathbf{1}\hspace{-0.05cm}^{\top}\hspace{-0.05cm}/n \right )\mathbf{x}\) is the vector of centred values (i.e., \(z_i = x_i-\bar{x}\)).

The function `moran.mc`

of the `spdep`

package
allows to compute and test, by permutation, the significance of the
Moran’s coefficient. A wrapper is provided by the
`moran.randtest`

function to test simultaneously and
independently the spatial structure for several variables.

```
<- SpatialPolygonsDataFrame(Sr = mafragh$Spatial, data = mafragh$env, match.ID = FALSE)
sp.env <- s.Spatial(sp.env, col = fpalette(6), nclass = 6) maps.env
```

```
<- moran.randtest(mafragh$env, listwgab, nrepet = 999)
MC.env MC.env
```

```
## class: krandtest lightkrandtest
## Monte-Carlo tests
## Call: moran.randtest(x = mafragh$env, listw = listwgab, nrepet = 999)
##
## Number of tests: 11
##
## Adjustment method for multiple comparisons: none
## Permutation number: 999
## Test Obs Std.Obs Alter Pvalue
## 1 Clay 0.4464655 6.552261 greater 0.001
## 2 Silt 0.3967605 5.946167 greater 0.001
## 3 Sand 0.1218959 2.114549 greater 0.033
## 4 K2O 0.2916865 4.470113 greater 0.001
## 5 Mg++ 0.2040580 3.220200 greater 0.003
## 6 Na+/100g 0.3404142 5.138034 greater 0.001
## 7 K+ 0.6696787 10.193028 greater 0.001
## 8 Conductivity 0.3843430 5.891744 greater 0.001
## 9 Retention 0.2217547 3.609803 greater 0.003
## 10 Na+/l 0.3075238 4.791118 greater 0.001
## 11 Elevation 0.6136770 9.047577 greater 0.001
```

For a given SWM, the upper and lower bounds of MC are equal to \(\lambda_{max}
(n/\mathbf{1}\hspace{-0.05cm}^{\top}\hspace{-0.05cm}\mathbf{W1})\)
and \(\lambda_{min}
(n/\mathbf{1}\hspace{-0.05cm}^{\top}\hspace{-0.05cm}\mathbf{W1})\)
where \(\lambda_{max}\) and \(\lambda_{min}\) are the extreme eigenvalues
of \(\mathbf{\Omega}\). These extreme
values are returned by the `moran.bounds`

function:

```
<- moran.bounds(listwgab)
mc.bounds mc.bounds
```

```
## Imin Imax
## -0.9474872 1.0098330
```

Hence, it is possible to display Moran’s coefficients computed on environmental variables and the minimum and maximum values for the given SWM:

```
<- s1d.barchart(MC.env$obs, labels = MC.env$names, plot = FALSE, xlim = 1.1 * mc.bounds, paxes.draw = TRUE, pgrid.draw = FALSE)
env.maps addline(env.maps, v = mc.bounds, plot = TRUE, pline.col = 'red', pline.lty = 3)
```

The standard test based on MC is not able to detect the coexistence
of positive and negative autocorrelation structures (i.e., it leads to a
non-significant test). The `moranNP.randtest`

function allows
to decompose the standard MC statistic into two additive parts and thus
to test for positive and negative autocorrelation separately (Dray
2011). For instance, we can test the spatial distribution of
Magnesium. Only positive autocorrelation is detected:

```
<- moranNP.randtest(mafragh$env[,5], listwgab, nrepet = 999, alter = "two-sided")
NP.Mg NP.Mg
```

```
## class: krandtest lightkrandtest
## Monte-Carlo tests
## Call: moranNP.randtest(x = mafragh$env[, 5], listw = listwgab, nrepet = 999,
## alter = "two-sided")
##
## Number of tests: 2
##
## Adjustment method for multiple comparisons: none
## Permutation number: 999
## Test Obs Std.Obs Alter Pvalue
## 1 I+ 0.3611756 3.743501 greater 0.002
## 2 I- -0.1571176 1.552759 less 0.944
```

`plot(NP.Mg)`

`sum(NP.Mg$obs)`

`## [1] 0.204058`

`$obs[5] MC.env`

```
## Mg++.statistic
## 0.204058
```

When multivariate data are considered, it is possible to search for spatial structures by computing univariate statistics (e.g., Moran’s Coefficient) on each variable separately. Another alternative is to summarize data by multivariate methods and then detect spatial structures using the output of the analysis. For instance, we applied a centred principal component analysis on the abundance data:

`<- dudi.pca(mafragh$flo, scale = FALSE, scannf = FALSE, nf = 2) pca.hell `

MC can be computed for PCA scores and the associated spatial structures can be visualized on a map:

`moran.randtest(pca.hell$li, listw = listwgab)`

```
## class: krandtest lightkrandtest
## Monte-Carlo tests
## Call: moran.randtest(x = pca.hell$li, listw = listwgab)
##
## Number of tests: 2
##
## Adjustment method for multiple comparisons: none
## Permutation number: 999
## Test Obs Std.Obs Alter Pvalue
## 1 Axis1 0.4830837 7.405295 greater 0.001
## 2 Axis2 0.4613738 7.119489 greater 0.001
```

`s.value(mxy, pca.hell$li, Sp = mafragh$Spatial.contour, symbol = "circle", col = c("white", "palegreen4"), ppoint.cex = 0.6)`

PCA results are highly spatially structured. Results can be
optimized, compared to this two-step procedure, by searching directly
for multivariate spatial structures. The `multispati`

function implements a method (Dray, Saïd, and Débias
2008) that search for axes maximizing the product of variance
(multivariate aspect) by MC (spatial aspect):

`<- multispati(pca.hell, listw = listwgab, scannf = F) ms.hell `

The `summary`

method can be applied on the resulting
object to compare the results of initial analysis (axis 1 (RS1) and axis
2 (RS2)) and those of the multispati method (CS1 and CS2):

`summary(ms.hell)`

```
##
## Multivariate Spatial Analysis
## Call: multispati(dudi = pca.hell, listw = listwgab, scannf = F)
##
## Scores from the initial duality diagram:
## var cum ratio moran
## RS1 5.331174 5.331174 0.2834660 0.4830837
## RS2 1.972986 7.304159 0.3883725 0.4613738
##
## Multispati eigenvalues decomposition:
## eig var moran
## CS1 2.933824 4.833900 0.6069269
## CS2 1.210573 1.892671 0.6396110
```

The MULTISPATI analysis allows to better identify spatial structures (higher MC values).

Scores of the analysis can be mapped and highlight some spatial patterns of community composition:

`<- s.value(mafragh$xy, ms.hell$li, Sp = mafragh$Spatial.contour, symbol = "circle", col = c("white", "palegreen4"), ppoint.cex = 0.6) g.ms.maps `

The spatial distribution of several species can be inserted to facilitate the interpretation of the outputs of the analysis:

```
<- s.arrow(ms.hell$c1, plot = FALSE)
g.ms.spe <- s.value(mxy, mafragh$flo[, c(12,11,31,16)],
g.abund Sp = mafragh$Spatial.contour, symbol = "circle", col = c("black", "palegreen4"), plegend.drawKey = FALSE, ppoint.cex = 0.4, plot = FALSE)
<- list(c(0.05, 0.65), c(0.01, 0.25), c(0.74, 0.58), c(0.55, 0.05))
p1 for (i in 1:4)
<- insert(g.abund[[i]], g.ms.spe, posi = p1[[i]], ratio = 0.25, plot = FALSE)
g.ms.spe g.ms.spe
```