Minimum Spanning Tree
Description
Function spantree
finds a minimum spanning tree
connecting all points, but disregarding dissimilarities that are at or
above the threshold or NA
.
Usage
1 2 3 4 5 6 7 8 9 10 11  spantree(d, toolong = 0)
## S3 method for class 'spantree'
as.hclust(x, ...)
## S3 method for class 'spantree'
cophenetic(x)
spandepth(x)
## S3 method for class 'spantree'
plot(x, ord, cex = 0.7, type = "p", labels, dlim,
FUN = sammon, ...)
## S3 method for class 'spantree'
lines(x, ord, display="sites", ...)

Arguments
d 
Dissimilarity data inheriting from class 
toolong 
Shortest dissimilarity regarded as 
x 
A 
ord 
An ordination configuration, or an ordination result known
by 
cex 
Character expansion factor. 
type 
Observations are plotted as points with

labels 
Text used with 
dlim 
A ceiling value used to highest 
FUN 
Ordination function to find the configuration from cophenetic dissimilarities. 
display 
Type of 
... 
Other parameters passed to functions. 
Details
Function spantree
finds a minimum spanning tree for
dissimilarities (there may be several minimum spanning trees, but the
function finds only one). Dissimilarities at or above the threshold
toolong
and NA
s are disregarded, and the spanning tree
is found through other dissimilarities. If the data are disconnected,
the function will return a disconnected tree (or a forest), and the
corresponding link is NA
. Connected subtrees can be identified
using distconnected
.
Minimum spanning tree is closesly related to single linkage
clustering, a.k.a. nearest neighbour clustering, and in genetics as
neighbour joining tree available in hclust
and
agnes
functions. The most important practical
difference is that minimum spanning tree has no concept of cluster
membership, but always joins individual points to each other. Function
as.hclust
can change the spantree
result into a
corresponding hclust
object.
Function cophenetic
finds distances between all points along
the tree segments. Function spandepth
returns the depth of
each node. The nodes of a tree are either leaves (with one link) or
internal nodes (more than one link). The leaves are recursively
removed from the tree, and the depth is the layer at with the leaf
was removed. In disconnected spantree
object (in a forest)
each tree is analysed separately and disconnected nodes not in any
tree have depth zero.
Function plot
displays the tree over a
supplied ordination configuration, and lines
adds a spanning
tree to an ordination graph. If configuration is not supplied for plot
,
the function ordinates the cophenetic dissimilarities of the
spanning tree and overlays the tree on this result. The default
ordination function is sammon
(package MASS),
because Sammon scaling emphasizes structure in the neighbourhood of
nodes and may be able to beautifully represent the tree (you may need
to set dlim
, and sometimes the results will remain
twisted). These ordination methods do not work with disconnected
trees, but you must supply the ordination configuration. Function
lines
will overlay the tree in an existing plot.
Function spantree
uses Prim's method
implemented as priorityfirst search for dense graphs (Sedgewick
1990). Function cophenetic
uses function
stepacross
with option path = "extended"
. The
spantree
is very fast, but cophenetic
is slow in very
large data sets.
Value
Function spantree
returns an object of class spantree
which is a
list with two vectors, each of length n1. The
number of links in a tree is one less the number of observations, and
the first item is omitted. The items are
kid 
The child node of the parent, starting from parent number
two. If there is no link from the parent, value will be 
dist 
Corresponding distance. If 
labels 
Names of nodes as found from the input dissimilarities. 
call 
The function call. 
Note
In principle, minimum spanning tree is equivalent to single linkage
clustering that can be performed using hclust
or
agnes
. However, these functions combine
clusters to each other and the information of the actually connected points
(the “single link”) cannot be recovered from the result. The
graphical output of a single linkage clustering plotted with
ordicluster
will look very different from an equivalent
spanning tree plotted with lines.spantree
.
Author(s)
Jari Oksanen
References
Sedgewick, R. (1990). Algorithms in C. Addison Wesley.
See Also
vegdist
or dist
for getting
dissimilarities, and hclust
or
agnes
for single linkage clustering.
Examples
1 2 3 4 5 6 7 8 9 10 11 12  data(dune)
dis < vegdist(dune)
tr < spantree(dis)
## Add tree to a metric scaling
plot(tr, cmdscale(dis), type = "t")
## Find a configuration to display the tree neatly
plot(tr, type = "t")
## Depths of nodes
depths < spandepth(tr)
plot(tr, type = "t", label = depths)
## Plot as a dendrogram
plot(as.hclust(tr))
