Agglomerative Nesting (Hierarchical Clustering)
Computes agglomerative hierarchical clustering of the dataset.
agnes(x, diss = inherits(x, "dist"), metric = "euclidean",
stand = FALSE, method = "average", par.method,
keep.diss = n < 100, keep.data = !diss, trace.lev = 0)x |
data matrix or data frame, or dissimilarity matrix, depending on the
value of the In case of a matrix or data frame, each row corresponds to an observation, and each column corresponds to a variable. All variables must be numeric. Missing values (NAs) are allowed. In case of a dissimilarity matrix, |
diss |
logical flag: if TRUE (default for |
metric |
character string specifying the metric to be used for calculating
dissimilarities between observations.
The currently available options are |
stand |
logical flag: if TRUE, then the measurements in |
method |
character string defining the clustering method. The six methods
implemented are
The default is |
par.method |
If |
keep.diss, keep.data |
logicals indicating if the dissimilarities
and/or input data |
trace.lev |
integer specifying a trace level for printing
diagnostics during the algorithm. Default |
agnes is fully described in chapter 5 of Kaufman and Rousseeuw (1990).
Compared to other agglomerative clustering methods such as hclust,
agnes has the following features: (a) it yields the
agglomerative coefficient (see agnes.object)
which measures the amount of clustering structure found; and (b)
apart from the usual tree it also provides the banner, a novel
graphical display (see plot.agnes).
The agnes-algorithm constructs a hierarchy of clusterings.
At first, each observation is a small cluster by itself. Clusters are
merged until only one large cluster remains which contains all the
observations. At each stage the two nearest clusters are combined
to form one larger cluster.
For method="average", the distance between two clusters is the
average of the dissimilarities between the points in one cluster and the
points in the other cluster.
In method="single", we use the smallest dissimilarity between a
point in the first cluster and a point in the second cluster (nearest
neighbor method).
When method="complete", we use the largest dissimilarity
between a point in the first cluster and a point in the second cluster
(furthest neighbor method).
The method = "flexible" allows (and requires) more details:
The Lance-Williams formula specifies how dissimilarities are
computed when clusters are agglomerated (equation (32) in K&R(1990),
p.237). If clusters C_1 and C_2 are agglomerated into a
new cluster, the dissimilarity between their union and another
cluster Q is given by
D(C_1 \cup C_2, Q) = α_1 * D(C_1, Q) + α_2 * D(C_2, Q) + β * D(C_1,C_2) + γ * |D(C_1, Q) - D(C_2, Q)|,
where the four coefficients (α_1, α_2, β, γ)
are specified by the vector par.method, either directly as vector of
length 4, or (more conveniently) if par.method is of length 1,
say = α, par.method is extended to
give the “Flexible Strategy” (K&R(1990), p.236 f) with
Lance-Williams coefficients (α_1 = α_2 = α, β =
1 - 2α, γ=0).
Also, if length(par.method) == 3, γ = 0 is set.
Care and expertise is probably needed when using method = "flexible"
particularly for the case when par.method is specified of
longer length than one. Since cluster version 2.0, choices
leading to invalid merge structures now signal an error (from
the C code already).
The weighted average (method="weighted") is the same as
method="flexible", par.method = 0.5. Further,
method= "single" is equivalent to method="flexible", par.method = c(.5,.5,0,-.5), and
method="complete" is equivalent to method="flexible", par.method = c(.5,.5,0,+.5).
The method = "gaverage" is a generalization of "average", aka
“flexible UPGMA” method, and is (a generalization of the approach)
detailed in Belbin et al. (1992). As "flexible", it uses the
Lance-Williams formula above for dissimilarity updating, but with
α_1 and α_2 not constant, but proportional to
the sizes n_1 and n_2 of the clusters C_1 and
C_2 respectively, i.e,
α_j = α'_j * n_1/(n_1 + n_2),
where α'_1, α'_2 are determined from par.method,
either directly as (α_1, α_2, β, γ) or
(α_1, α_2, β) with γ = 0, or (less flexibly,
but more conveniently) as follows:
Belbin et al proposed “flexible beta”, i.e. the user would only
specify β (as par.method), sensibly in
-1 ≤ β < 1,
and β determines α'_1 and α'_2 as
α'_j = 1 - β,
and γ = 0.
This β may be specified by par.method (as length 1 vector),
and if par.method is not specified, a default value of -0.1 is used,
as Belbin et al recommend taking a β value around -0.1 as a general
agglomerative hierarchical clustering strategy.
Note that method = "gaverage", par.method = 0 (or par.method =
c(1,1,0,0)) is equivalent to the agnes() default method "average".
an object of class "agnes" (which extends "twins")
representing the clustering. See agnes.object for
details, and methods applicable.
Cluster analysis divides a dataset into groups (clusters) of observations that are similar to each other.
Method "gaverage" has been contributed by Pierre Roudier, Landcare
Research, New Zealand.
Kaufman, L. and Rousseeuw, P.J. (1990). (=: “K&R(1990)”) Finding Groups in Data: An Introduction to Cluster Analysis. Wiley, New York.
Anja Struyf, Mia Hubert and Peter J. Rousseeuw (1996) Clustering in an Object-Oriented Environment. Journal of Statistical Software 1. doi: 10.18637/jss.v001.i04
Struyf, A., Hubert, M. and Rousseeuw, P.J. (1997). Integrating Robust Clustering Techniques in S-PLUS, Computational Statistics and Data Analysis, 26, 17–37.
Lance, G.N., and W.T. Williams (1966). A General Theory of Classifactory Sorting Strategies, I. Hierarchical Systems. Computer J. 9, 373–380.
Belbin, L., Faith, D.P. and Milligan, G.W. (1992). A Comparison of Two Approaches to Beta-Flexible Clustering. Multivariate Behavioral Research, 27, 417–433.
data(votes.repub)
agn1 <- agnes(votes.repub, metric = "manhattan", stand = TRUE)
agn1
plot(agn1)
op <- par(mfrow=c(2,2))
agn2 <- agnes(daisy(votes.repub), diss = TRUE, method = "complete")
plot(agn2)
## alpha = 0.625 ==> beta = -1/4 is "recommended" by some
agnS <- agnes(votes.repub, method = "flexible", par.meth = 0.625)
plot(agnS)
par(op)
## "show" equivalence of three "flexible" special cases
d.vr <- daisy(votes.repub)
a.wgt <- agnes(d.vr, method = "weighted")
a.sing <- agnes(d.vr, method = "single")
a.comp <- agnes(d.vr, method = "complete")
iC <- -(6:7) # not using 'call' and 'method' for comparisons
stopifnot(
all.equal(a.wgt [iC], agnes(d.vr, method="flexible", par.method = 0.5)[iC]) ,
all.equal(a.sing[iC], agnes(d.vr, method="flex", par.method= c(.5,.5,0, -.5))[iC]),
all.equal(a.comp[iC], agnes(d.vr, method="flex", par.method= c(.5,.5,0, +.5))[iC]))
## Exploring the dendrogram structure
(d2 <- as.dendrogram(agn2)) # two main branches
d2[[1]] # the first branch
d2[[2]] # the 2nd one { 8 + 42 = 50 }
d2[[1]][[1]]# first sub-branch of branch 1 .. and shorter form
identical(d2[[c(1,1)]],
d2[[1]][[1]])
## a "textual picture" of the dendrogram :
str(d2)
data(agriculture)
## Plot similar to Figure 7 in ref
## Not run: plot(agnes(agriculture), ask = TRUE)
data(animals)
aa.a <- agnes(animals) # default method = "average"
aa.ga <- agnes(animals, method = "gaverage")
op <- par(mfcol=1:2, mgp=c(1.5, 0.6, 0), mar=c(.1+ c(4,3,2,1)),
cex.main=0.8)
plot(aa.a, which.plot = 2)
plot(aa.ga, which.plot = 2)
par(op)
## Show how "gaverage" is a "generalized average":
aa.ga.0 <- agnes(animals, method = "gaverage", par.method = 0)
stopifnot(all.equal(aa.ga.0[iC], aa.a[iC]))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.