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 `diss`

argument.

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, `x`

is typically the output of
`daisy`

or `dist`

.
Also a vector with length n*(n-1)/2 is allowed (where n is the number
of observations), and will be interpreted in the same way as the
output of the above-mentioned functions. Missing values (NAs) are not
allowed.

diss

logical flag: if TRUE (default for `dist`

or
`dissimilarity`

objects), then `x`

is assumed to be a
dissimilarity matrix. If FALSE, then `x`

is treated as
a matrix of observations by variables.

metric

character string specifying the metric to be used for calculating
dissimilarities between observations.
The currently available options are `"euclidean"`

and `"manhattan"`

.
Euclidean distances are root sum-of-squares of differences, and
manhattan distances are the sum of absolute differences.
If `x`

is already a dissimilarity matrix, then this argument will
be ignored.

stand

logical flag: if TRUE, then the measurements in `x`

are
standardized before calculating the dissimilarities. Measurements
are standardized for each variable (column), by subtracting the
variable's mean value and dividing by the variable's mean absolute
deviation. If `x`

is already a dissimilarity matrix, then this
argument will be ignored.

method

character string defining the clustering method. The six methods
implemented are
`"average"`

([unweighted pair-]group [arithMetic] average method, aka ‘UPGMA’),
`"single"`

(single linkage), `"complete"`

(complete linkage),
`"ward"`

(Ward's method),
`"weighted"`

(weighted average linkage, aka ‘WPGMA’), its generalization
`"flexible"`

which uses (a constant version of)
the Lance-Williams formula and the `par.method`

argument, and
`"gaverage"`

a generalized `"average"`

aka “flexible
UPGMA” method also using the Lance-Williams formula and `par.method`

.

The default is `"average"`

.

par.method

If `method`

is `"flexible"`

or `"gaverage"`

, a numeric
vector of length 1, 3, or 4, (with a default for `"gaverage"`

), see in
the details section.

keep.diss, keep.data

logicals indicating if the dissimilarities
and/or input data `x`

should be kept in the result. Setting
these to `FALSE`

can give much smaller results and hence even save
memory allocation *time*.

trace.lev

integer specifying a trace level for printing
diagnostics during the algorithm. Default `0`

does not print
anything; higher values print increasingly more.

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.

`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
$$
D(C_1 \cup C_2, Q) = \alpha_1 * D(C_1, Q) + \alpha_2 * D(C_2, Q) +
\beta * D(C_1,C_2) + \gamma * |D(C_1, Q) - D(C_2, Q)|,
$$
where the four coefficients \((\alpha_1, \alpha_2, \beta, \gamma)\)
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 \(= \alpha\), `par.method`

is extended to
give the “Flexible Strategy” (K&R(1990), p.236 f) with
Lance-Williams coefficients \((\alpha_1 = \alpha_2 = \alpha, \beta =
1 - 2\alpha, \gamma=0)\).
Also, if `length(par.method) == 3`

, \(\gamma = 0\) 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
\(\alpha_1\) and \(\alpha_2\) not constant, but *proportional* to
the *sizes* \(n_1\) and \(n_2\) of the clusters \(C_1\) and
\(C_2\) respectively, i.e,
$$\alpha_j = \alpha'_j \frac{n_1}{n_1+n_2},$$
where \(\alpha'_1\), \(\alpha'_2\) are determined from `par.method`

,
either directly as \((\alpha_1, \alpha_2, \beta, \gamma)\) or
\((\alpha_1, \alpha_2, \beta)\) with \(\gamma = 0\), or (less flexibly,
but more conveniently) as follows:

Belbin et al proposed “flexible beta”, i.e. the user would only
specify \(\beta\) (as `par.method`

), sensibly in
$$-1 \leq \beta < 1,$$
and \(\beta\) determines \(\alpha'_1\) and \(\alpha'_2\) as
$$\alpha'_j = 1 - \beta,$$ and \(\gamma = 0\).

This \(\beta\) 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 \(\beta\) 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"`

.

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**.
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.

`agnes.object`

, `daisy`

, `diana`

,
`dist`

, `hclust`

, `plot.agnes`

,
`twins.object`

.

# NOT RUN { 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) # } # NOT RUN { 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) # } # NOT RUN { ## 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])) # }