## Multivariate spatial analysis

### Description

This function ensures a multivariate extension of the univariate method of spatial autocorrelation analysis. By accounting for the spatial dependence of data observations and their multivariate covariance simultaneously, complex interactions among many variables are analysed. Using a methodological scheme borrowed from duality diagram analysis, a strategy for the exploratory analysis of spatial pattern in the multivariate is developped.

### Usage

```multispati(dudi, listw, scannf = TRUE, nfposi = 2, nfnega = 0)
## S3 method for class 'multispati':
plot(x, xax = 1, yax = 2, ...)
## S3 method for class 'multispati':
summary(object, ...)
## S3 method for class 'multispati':
print(x, ...)
```

### Arguments

 `dudi` an object of class `dudi` for the duality diagram analysis `listw` an object of class `listw` for the spatial dependence of data observations `scannf` a logical value indicating whether the eigenvalues bar plot should be displayed `nfposi` an integer indicating the number of kept positive axes `nfnega` an integer indicating the number of kept negative axes `x, object` an object of class `multispati` `xax, yax` the numbers of the x-axis and the y-axis `...` further arguments passed to or from other methods

### Details

This analysis generalizes the Wartenberg's multivariate spatial correlation analysis to various duality diagrams created by the functions (`dudi.pca`, `dudi.coa`, `dudi.acm`, `dudi.mix`...) If dudi is a duality diagram created by the function `dudi.pca` and listw gives spatial weights created by a row normalized coding scheme, the analysis is equivalent to Wartenberg's analysis.

We note X the data frame with the variables, Q the column weights matrix and D the row weights matrix associated to the duality diagram dudi. We note L the neighbouring weights matrix associated to listw. Then, the `'multispati'` analysis gives principal axes v that maximize the product of spatial autocorrelation and inertia of row scores :

I(XQv)*\|\|XQv\|\|^2 = t(v)t(Q)t(X)DLXQv

### Value

Returns an object of class `multispati`, which contains the following elements :
 `eig` a numeric vector containing the eigenvalues `nfposi` integer, number of kept axes associated to positive eigenvalues `nfnega` integer, number of kept axes associated to negative eigenvalues `c1` principle axes (v), data frame with p rows and (nfposi + nfnega) columns `li` principal components (XQv), data frame with n rows and (nfposi + nfnega) columns `ls` lag vector onto the principal axes (LXQv), data frame with n rows and (nfposi + nfnega) columns `as` principal axes of the dudi analysis (u) onto principal axes of multispati (t(u)Qv), data frame with dudi\\$nf rows and (nfposi + nfnega) columns

### Author(s)

Daniel Chessel
Sebastien Ollier ollier@biomserv.univ-lyon1.fr
Thibaut Jombart jombart@biomserv.univ-lyon1.fr

### References

Dray, S., Said, S. and Debias, F. (2008) Spatial ordination of vegetation data using a generalization of Wartenberg's multivariate spatial correlation. Journal of vegetation science, 19, 45–56.

Grunsky, E. C. and Agterberg, F. P. (1988) Spatial and multivariate analysis of geochemical data from metavolcanic rocks in the Ben Nevis area, Ontario. Mathematical Geology, 20, 825–861.

Switzer, P. and Green, A.A. (1984) Min/max autocorrelation factors for multivariate spatial imagery. Tech. rep. 6, Stanford University.

Thioulouse, J., Chessel, D. and Champely, S. (1995) Multivariate analysis of spatial patterns: a unified approach to local and global structures. Environmental and Ecological Statistics, 2, 1–14.

Wartenberg, D. E. (1985) Multivariate spatial correlation: a method for exploratory geographical analysis. Geographical Analysis, 17, 263–283.

Jombart, T., Devillard, S., Dufour, A.-B. and Pontier, D. A spatially explicit multivariate method to disentangle global and local patterns of genetic variability. Submitted to Genetics.

`dudi`,`mat2listw`

### Examples

```## Not run:
if (require(maptools, quiet = TRUE) & require(spdep, quiet = TRUE)) {
data(mafragh)
maf.xy <- mafragh\$xy
maf.flo <- mafragh\$flo
maf.listw <- nb2listw(neig2nb(mafragh\$neig))
s.label(maf.xy, neig = mafragh\$neig, clab = 0.75)
maf.coa <- dudi.coa(maf.flo,scannf = FALSE)
maf.coa.ms <- multispati(maf.coa, maf.listw, scannf = FALSE, nfposi = 2,
nfnega = 2)
maf.coa.ms

### detail eigenvalues components
fgraph <- function(obj){
# use multispati summary
sum.obj <- summary(obj)
# compute Imin and Imax
L <- listw2mat(eval(as.list(obj\$call)\$listw))
Imin <- min(eigen(0.5*(L+t(L)))\$values)
Imax <- max(eigen(0.5*(L+t(L)))\$values)
I0 <- -1/(nrow(obj\$li)-1)
# create labels
labels <- lapply(1:length(obj\$eig),function(i) bquote(lambda[.(i)]))
# draw the plot
xmax <- eval(as.list(obj\$call)\$dudi)\$eig[1]*1.1
par(las=1)
var <- sum.obj[,2]
moran <- sum.obj[,3]
plot(x=var,y=moran,type='n',xlab='Inertia',ylab="Spatial autocorrelation (I)",
xlim=c(0,xmax),ylim=c(Imin*1.1,Imax*1.1),yaxt='n')
text(x=var,y=moran,do.call(expression,labels))
ytick <- c(I0,round(seq(Imin,Imax,le=5),1))
ytlab <- as.character(round(seq(Imin,Imax,le=5),1))
ytlab <- c(as.character(round(I0,1)),as.character(round(Imin,1)),
ytlab[2:4],as.character(round(Imax,1)))
axis(side=2,at=ytick,labels=ytlab)
rect(0,Imin,xmax,Imax,lty=2)
segments(0,I0,xmax,I0,lty=2)
abline(v=0)
title("Spatial and inertia components of the eigenvalues")
}
fgraph(maf.coa.ms)
## end eigenvalues details

par(mfrow = c(1,3))
barplot(maf.coa\$eig)
barplot(maf.coa.ms\$eig)
s.corcircle(maf.coa.ms\$as)

par(mfrow = c(2,2))
s.value(maf.xy, -maf.coa\$li[,1])
s.value(maf.xy, -maf.coa\$li[,2])
s.value(maf.xy, maf.coa.ms\$li[,1])
s.value(maf.xy, maf.coa.ms\$li[,2])
par(mfrow = c(1,1))

par(mfrow = c(1,2))
w1 <- -maf.coa\$li[,1:2]
w1m <- apply(w1, 2, lag.listw, x = maf.listw)
s.match(w1, w1m, clab = 0.75)
w1.ms <- maf.coa.ms\$li[,1:2]
w1.msm <- apply(w1.ms, 2, lag.listw, x = maf.listw)
s.match(w1.ms, w1.msm, clab = 0.75)
par(mfrow = c(1,1))

maf.pca <- dudi.pca(mafragh\$mil, scannf = FALSE)
multispati.randtest(maf.pca, maf.listw)
maf.pca.ms <- multispati(maf.pca, maf.listw, scannf=FALSE)
plot(maf.pca.ms)
}

## End(Not run)```

### Worked out examples

```
> ### Name: multispati
> ### Title: Multivariate spatial analysis
> ### Aliases: multispati plot.multispati summary.multispati print.multispati
> ### Keywords: multivariate spatial
>
> ### ** Examples
>
> if (require(maptools, quiet = TRUE) & require(spdep, quiet = TRUE)) {
+     data(mafragh)
+     maf.xy <- mafragh\$xy
+     maf.flo <- mafragh\$flo
+     maf.listw <- nb2listw(neig2nb(mafragh\$neig))
+     s.label(maf.xy, neig = mafragh\$neig, clab = 0.75)
+     maf.coa <- dudi.coa(maf.flo,scannf = FALSE)
+     maf.coa.ms <- multispati(maf.coa, maf.listw, scannf = FALSE, nfposi = 2,
+         nfnega = 2)
+     maf.coa.ms
+
+     ### detail eigenvalues components
+     fgraph <- function(obj){
+       # use multispati summary
+       sum.obj <- summary(obj)
+       # compute Imin and Imax
+       L <- listw2mat(eval(as.list(obj\$call)\$listw))
+       Imin <- min(eigen(0.5*(L+t(L)))\$values)
+       Imax <- max(eigen(0.5*(L+t(L)))\$values)
+       I0 <- -1/(nrow(obj\$li)-1)
+       # create labels
+       labels <- lapply(1:length(obj\$eig),function(i) bquote(lambda[.(i)]))
+       # draw the plot
+       xmax <- eval(as.list(obj\$call)\$dudi)\$eig[1]*1.1
+       par(las=1)
+       var <- sum.obj[,2]
+       moran <- sum.obj[,3]
+       plot(x=var,y=moran,type='n',xlab='Inertia',ylab="Spatial autocorrelation (I)",
+            xlim=c(0,xmax),ylim=c(Imin*1.1,Imax*1.1),yaxt='n')
+       text(x=var,y=moran,do.call(expression,labels))
+       ytick <- c(I0,round(seq(Imin,Imax,le=5),1))
+       ytlab <- as.character(round(seq(Imin,Imax,le=5),1))
+       ytlab <- c(as.character(round(I0,1)),as.character(round(Imin,1)),
+            ytlab[2:4],as.character(round(Imax,1)))
+       axis(side=2,at=ytick,labels=ytlab)
+       rect(0,Imin,xmax,Imax,lty=2)
+       segments(0,I0,xmax,I0,lty=2)
+       abline(v=0)
+       title("Spatial and inertia components of the eigenvalues")
+     }
+     fgraph(maf.coa.ms)
+     ## end eigenvalues details
+
+
+     par(mfrow = c(1,3))
+     barplot(maf.coa\$eig)
+     barplot(maf.coa.ms\$eig)
+     s.corcircle(maf.coa.ms\$as)
+
+     par(mfrow = c(2,2))
+     s.value(maf.xy, -maf.coa\$li[,1])
+     s.value(maf.xy, -maf.coa\$li[,2])
+     s.value(maf.xy, maf.coa.ms\$li[,1])
+     s.value(maf.xy, maf.coa.ms\$li[,2])
+     par(mfrow = c(1,1))
```
```+
+     par(mfrow = c(1,2))
+     w1 <- -maf.coa\$li[,1:2]
+     w1m <- apply(w1, 2, lag.listw, x = maf.listw)
+     s.match(w1, w1m, clab = 0.75)
+     w1.ms <- maf.coa.ms\$li[,1:2]
+     w1.msm <- apply(w1.ms, 2, lag.listw, x = maf.listw)
+     s.match(w1.ms, w1.msm, clab = 0.75)
+     par(mfrow = c(1,1))
```
```+
+     maf.pca <- dudi.pca(mafragh\$mil, scannf = FALSE)
+     multispati.randtest(maf.pca, maf.listw)
+     maf.pca.ms <- multispati(maf.pca, maf.listw, scannf=FALSE)
+     plot(maf.pca.ms)
+ }
deldir 0.0-12

Please note: The process for determining duplicated points
has changed from that used in version 0.0-9 (and previously).

Multivariate Spatial Analysis
Call: multispati(dudi = maf.coa, listw = maf.listw, scannf = FALSE,
nfposi = 2, nfnega = 2)

Scores from the initial duality diagramm:
var       cum     ratio     moran
RS1 0.8691476 0.8691476 0.1043473 0.7250457
RS2 0.6491089 1.5182565 0.1822775 0.4834366

Multispati eigenvalues decomposition:
eig       var      moran
CS1   0.68545912 0.8332937  0.8225901
CS2   0.37853390 0.5865926  0.6453097
CS54 -0.08077788 0.2482426 -0.3253990
CS55 -0.08900757 0.2749796 -0.3236879
```
```>
>
>
>
```