Boundary detection using Monmonier algorithm

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Description

The Monmonier's algorithm detects boundaries among vertices of a valuated graph. This is achieved by finding the path exhibiting the largest distances between connected vertices.

The highest distance between two connected vertices (i.e. neighbours) is found, giving the starting point of the path. Then, the algorithm seeks the highest distance between immediate neighbours, and so on until a threshold value is attained. This threshold can be chosen from the plot of sorted distances between connected vertices: a boundary will likely result in an abrupt decrease of these values.

When several paths are looked for, the previous paths are taken into account, and cannot be either crossed or redrawn. Monmonier's algorithm can be used to assess the boundaries between patches of homogeneous observations.

Although Monmonier algorithm was initially designed for Voronoi tesselation, this implementation generalizes this algorithm to different connection networks. The optimize.monmonier function produces a monmonier object by trying several starting points, and returning the best boundary (i.e. largest sum of local distances). This is designed to avoid the algorithm to be trapped by a single strong local difference inside an homogeneous patch.

Usage

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monmonier(xy, dist, cn, threshold=NULL, bd.length=NULL, nrun=1,
skip.local.diff=rep(0,nrun),scanthres=is.null(threshold), allowLoop=TRUE)

optimize.monmonier(xy, dist, cn, ntry=10, bd.length=NULL, return.best=TRUE,
display.graph=TRUE, threshold=NULL, scanthres=is.null(threshold), allowLoop=TRUE)

## S3 method for class 'monmonier'
plot(x, variable=NULL,
displayed.runs=1:x$nrun, add.arrows=TRUE,
col='blue', lty=1, bwd=4, clegend=1, csize=0.7,
method=c('squaresize','greylevel'), sub='', csub=1, possub='topleft',
cneig=1, pixmap=NULL, contour=NULL, area=NULL, add.plot=FALSE, ...)

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

Arguments

xy

a matrix yielding the spatial coordinates of the objects, with two columns respectively giving X and Y

dist

an object of class dist, giving the distances between the objects

cn

a connection network of class nb (package spdep)

threshold

a number giving the minimal distance between two neighbours crossed by the path; by default, this is the third quartile of all the distances between neighbours

bd.length

an optional integer giving the requested length of the boundaries (in number of local differences)

nrun

is a integer giving the number of runs of the algorithm, that is, the number of paths to search, being one by default

skip.local.diff

is a vector of integers, whose length is the number of paths (nrun); each integer gives the number of starting point to skip, to avoid being stuck in a local difference between two neighbours into an homogeneous patch; none are skipped by default

scanthres

a logical stating whether the threshold sould be chosen from the barplot of sorted distances between neighbours

allowLoop

a logical specifying whether the boundary can loop (TRUE, default) or not (FALSE)

ntry

an integer giving the number of different starting points tried.

return.best

a logical stating whether the best monmonier object should be returned (TRUE, default) or not (FALSE)

display.graph

a logical whether the scores of each try should be plotted (TRUE, default) or not

x

a monmonier object

variable

a variable to be plotted using s.value (package ade4)

displayed.runs

an integer vector giving the rank of the paths to represent

add.arrows

a logical, stating whether arrows should indicate the direction of the path (TRUE) or not (FALSE, used by default)

col

a characters vector giving the colors to be used for each boundary; recycled is needed; 'blue' is used by default

lty

a characters vector giving the type of line to be used for each boundary; 1 is used by default

bwd

a number giving the boundary width factor, applying to every segments of the paths; 4 is used by default

clegend

like in s.value, the size factor of the legend if a variable is represented

csize

like in s.value, the size factor of the squares used to represent a variable

method

like in s.value, a character giving the method to be used to represent the variable, either 'squaresize' (by default) or 'greylevel'

sub

a string of characters giving the subtitle of the plot

csub

the size factor of the subtitle

possub

the position of the subtitle; available choices are 'topleft' (by default), 'topright', 'bottomleft', and 'bottomright'

cneig

the size factor of the connection network

pixmap

an object of the class pixmap displayed in the map background

contour

a data frame with 4 columns to plot the contour of the map: each row gives a segment (x1,y1,x2,y2)

area

a data frame of class 'area' to plot a set of surface units in contour

add.plot

a logical stating whether the plot should be added to the current one (TRUE), or displayed in a new window (FALSE, by default)

...

further arguments passed to other methods

Details

The function monmonier returns a list of the class monmonier, which contains the general informations about the algorithm, and about each run. When displayed, the width of the boundaries reflects their 'strength'. Let a segment MN be part of the path, M being the middle of AB, N of CD. Then the boundary width for MN is proportionnal to (d(AB)+d(CD))/2.

As there is no perfect method to display graphically a quantitative variable (see for instance the differences between the two methods of s.value), the boundaries provided by this algorithm seem sometimes more reliable than the boundaries our eyes perceive (or miss).

Value

Returns an object of class monmonier, which contains the following elements :

run1 (run2, ...)

for each run, a list containing a dataframe giving the path coordinates, and a vector of the distances between neighbours of the path

nrun

the number of runs performed, i.e. the number of boundaries in the monmonier object

threshold

the threshold value, minimal distance between neighbours accounted for by the algorithm

xy

the matrix of spatial coordinates

cn

the connection network of class nb

call

the call of the function

Author(s)

Thibaut Jombart t.jombart@imperial.ac.uk

References

Monmonier, M. (1973) Maximum-difference barriers: an alternative numerical regionalization method. Geographic Analysis, 3, 245–261.

Manni, F., Guerard, E. and Heyer, E. (2004) Geographic patterns of (genetic, morphologic, linguistic) variation: how barriers can be detected by "Monmonier's algorithm". Human Biology, 76, 173–190

See Also

spca,edit.nb

Examples

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if(require(spdep)){

### non-interactive example

# est-west separation
load(system.file("files/mondata1.rda",package="adegenet"))
cn1 <- chooseCN(mondata1$xy,type=2,ask=FALSE)
mon1 <- monmonier(mondata1$xy,dist(mondata1$x1),cn1,threshold=2)
plot(mon1,mondata1$x1)
plot(mon1,mondata1$x1,met="greylevel",add.arr=FALSE,col="red",bwd=6,lty=2)

# square in the middle
load(system.file("files/mondata2.rda",package="adegenet"))
cn2 <- chooseCN(mondata2$xy,type=1,ask=FALSE)
mon2 <- monmonier(mondata2$xy,dist(mondata2$x2),cn2,threshold=2)
plot(mon2,mondata2$x2,method="greylevel",add.arr=FALSE,bwd=6,col="red",csize=.5)

### genetic data example
## Not run: 
data(sim2pop)

if(require(hierfstat)){
## try and find the Fst
fstat(sim2pop,fst=TRUE)
# Fst = 0.038
}

## run monmonier algorithm

# build connection network
gab <- chooseCN(sim2pop@other$xy,ask=FALSE,type=2)

# filter random noise 
pca1 <- dudi.pca(sim2pop@tab,scale=FALSE, scannf=FALSE, nf=1)

# run the algorithm
mon1 <- monmonier(sim2pop@other$xy,dist(pca1$l1[,1]),gab,scanthres=FALSE)

# graphical display 
plot(mon1,var=pca1$l1[,1])
temp <- sim2pop@pop
levels(temp) <- c(17,19)
temp <- as.numeric(as.character(temp))
plot(mon1)
points(sim2pop@other$xy,pch=temp,cex=2)
legend("topright",leg=c("Pop A", "Pop B"),pch=c(17,19))


### interactive example

# north-south separation
xy <- matrix(runif(120,0,10), ncol=2)
x1 <- rnorm(60)
x1[xy[,2] > 5] <- x1[xy[,2] > 5]+3
cn1 <- chooseCN(xy,type=1,ask=FALSE)
mon1 <- optimize.monmonier(xy,dist(x1)^2,cn1,ntry=10)

# graphics
plot(mon1,x1,met="greylevel",csize=.6)

# island in the middle
x2 <- rnorm(60)
sel <- (xy[,1]>3.5 & xy[,2]>3.5 & xy[,1]<6.5 & xy[,2]<6.5)
x2[sel] <- x2[sel]+4
cn2 <- chooseCN(xy,type=1,ask=FALSE)
mon2 <- optimize.monmonier(xy,dist(x2)^2,cn2,ntry=10)

# graphics
plot(mon2,x2,method="greylevel",add.arr=FALSE,bwd=6,col="red",csize=.5)

## End(Not run)
}

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