ade4 (version 1.5-2)

multispati: 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

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

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 : $$I(XQv)*\|XQv\|^2 = v^{t}Q^{t}X^{t}DLXQv$$

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.

See Also

dudi,mat2listw

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)
}