## Multivariate spatial analysis

### Description

This function provides a multivariate extension of the univariate method of spatial autocorrelation analysis. It provides a spatial ordination by maximizing the product of variance by spatial autocorrelation.

### Usage

```multispati(dudi, listw, scannf = TRUE, nfposi = 2, nfnega = 0)

## S3 method for class 'multispati'
summary(object, ...)

## S3 method for class 'multispati'
print(x, ...)

## S3 method for class 'multispati'
plot(x, xax = 1, yax = 2, pos = -1, storeData = TRUE, plot = TRUE, ...)
```

### Arguments

 `dudi` an object of class `dudi` obtained by the simple analysis of a data table `listw` an object of class `listw` created for example by `nb2listw` `scannf` a logical value indicating whether the eigenvalues barplot should be displayed `nfposi` an integer indicating the number of axes with positive autocorrelation `nfnega` an integer indicating the number of axes with negative autocorrelation `...` further arguments passed to or from other methods `x, object` an object of class `multispati` `xax, yax` the numbers of the x-axis and the y-axis `pos` an integer indicating the position of the environment where the data are stored, relative to the environment where the function is called. Useful only if `storeData` is `FALSE` `storeData` a logical indicating if the data should be stored in the returned object. If `FALSE`, only the names of the data arguments are stored `plot` a logical indicating if the graphics is displayed

### 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)

Stéphane Dray stephane.dray@univ-lyon1.fr with contributions by Daniel Chessel, Sebastien Ollier and Thibaut Jombart

### 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.

`dudi`,`mat2listw`

### Examples

```

if (require(spdep, quiet = TRUE) & require(ade4, quiet = TRUE)) {
data(mafragh)
maf.xy <- mafragh\$xy
maf.flo <- mafragh\$flo
maf.listw <- nb2listw(mafragh\$nb)
g1 <- s.label(maf.xy, nb = mafragh\$nb, plab.cex = 0.75)
} else {
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
Ibounds <- moran.bounds(eval(as.list(obj\$call)\$listw))
Imin <- Ibounds[1]
Imax <- Ibounds[2]
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

g2 <- s1d.barchart(maf.coa\$eig, p1d.hori = FALSE, plot = FALSE)
g3 <- s1d.barchart(maf.coa.ms\$eig, p1d.hori = FALSE, plot = FALSE)
g4 <- s.corcircle(maf.coa.ms\$as, plot = FALSE)
G1 <- ADEgS(list(g2, g3, g4), layout = c(1, 3))
} else {
par(mfrow = c(1, 3))
barplot(maf.coa\$eig)
barplot(maf.coa.ms\$eig)
s.corcircle(maf.coa.ms\$as)
par(mfrow = c(1, 1))
}

g5 <- s.value(maf.xy, -maf.coa\$li[, 1], plot = FALSE)
g6 <- s.value(maf.xy, -maf.coa\$li[, 2], plot = FALSE)
g7 <- s.value(maf.xy, maf.coa.ms\$li[, 1], plot = FALSE)
g8 <- s.value(maf.xy, maf.coa.ms\$li[, 2], plot = FALSE)
G2 <- ADEgS(list(g5, g6, g7, g8), layout = c(2, 2))
} else {
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))
}

w1 <- -maf.coa\$li[, 1:2]
w1m <- apply(w1, 2, lag.listw, x = maf.listw)
w1.ms <- maf.coa.ms\$li[, 1:2]
w1.msm <- apply(w1.ms, 2, lag.listw, x = maf.listw)
g9 <- s.match(w1, w1m, plab.cex = 0.75, plot = FALSE)
g10 <- s.match(w1.ms, w1.msm, plab.cex = 0.75, plot = FALSE)
G3 <- cbindADEg(g9, g10, plot = TRUE)
} else {
par(mfrow = c(1,2))
s.match(w1, w1m, clab = 0.75)
s.match(w1.ms, w1.msm, clab = 0.75)
par(mfrow = c(1, 1))
}

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

```