Partitioning (clustering) of the data into `k`

clusters “around
medoids”, a more robust version of K-means.

```
pam(x, k, diss = inherits(x, "dist"),
metric = c("euclidean", "manhattan"),
medoids = if(is.numeric(nstart)) "random",
nstart = if(variant == "faster") 1 else NA,
stand = FALSE, cluster.only = FALSE,
do.swap = TRUE,
keep.diss = !diss && !cluster.only && n < 100,
keep.data = !diss && !cluster.only,
variant = c("original", "o_1", "o_2", "f_3", "f_4", "f_5", "faster"),
pamonce = FALSE, trace.lev = 0)
```

an object of class `"pam"`

representing the clustering. See

`?pam.object`

for details.

- x
data matrix or data frame, or dissimilarity matrix or object, 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 (or logical). Missing values (

`NA`

s)*are*allowed---as long as every pair of observations has at least one case not missing.In case of a dissimilarity matrix,

`x`

is typically the output of`daisy`

or`dist`

. Also a vector of 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 (`NA`

s) are*not*allowed.- k
positive integer specifying the number of clusters, less than the number of observations.

- diss
logical flag: if TRUE (default for

`dist`

or`dissimilarity`

objects), then`x`

will be considered as a dissimilarity matrix. If FALSE, then`x`

will be considered 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.- medoids
NULL (default) or length-

`k`

vector of integer indices (in`1:n`

) specifying initial medoids instead of using the ‘*build*’ algorithm.- nstart
used only when

`medoids = "random"`

: specifies the*number*of random “starts”; this argument corresponds to the one of`kmeans()`

(from R's package stats).- stand
logical; if true, 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.- cluster.only
logical; if true, only the clustering will be computed and returned, see details.

- do.swap
logical indicating if the

**swap**phase should happen. The default,`TRUE`

, correspond to the original algorithm. On the other hand, the**swap**phase is much more computer intensive than the**build**one for large \(n\), so can be skipped by`do.swap = FALSE`

.- 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*.- pamonce
logical or integer in

`0:6`

specifying algorithmic short cuts as proposed by Reynolds et al. (2006), and Schubert and Rousseeuw (2019, 2021) see below.- variant
a

`character`

string denoting the variant of PAM algorithm to use; a more self-documenting version of`pamonce`

which should be used preferably; note that`"faster"`

not only uses`pamonce = 6`

but also`nstart = 1`

and hence`medoids = "random"`

by default.- trace.lev
integer specifying a trace level for printing diagnostics during the build and swap phase of the algorithm. Default

`0`

does not print anything; higher values print increasingly more.

Kaufman and Rousseeuw's orginal Fortran code was translated to C
and augmented in several ways, e.g. to allow `cluster.only=TRUE`

or `do.swap=FALSE`

, by Martin Maechler.

Matthias Studer, Univ.Geneva provided the `pamonce`

(`1`

and `2`

)
implementation.

Erich Schubert, TU Dortmund contributed the `pamonce`

(`3`

to `6`

)
implementation.

The basic `pam`

algorithm is fully described in chapter 2 of
Kaufman and Rousseeuw(1990). Compared to the k-means approach in `kmeans`

, the
function `pam`

has the following features: (a) it also accepts a
dissimilarity matrix; (b) it is more robust because it minimizes a sum
of dissimilarities instead of a sum of squared euclidean distances;
(c) it provides a novel graphical display, the silhouette plot (see
`plot.partition`

) (d) it allows to select the number of clusters
using `mean(silhouette(pr)[, "sil_width"])`

on the result
`pr <- pam(..)`

, or directly its component
`pr$silinfo$avg.width`

, see also `pam.object`

.

When `cluster.only`

is true, the result is simply a (possibly
named) integer vector specifying the clustering, i.e.,

`pam(x,k, cluster.only=TRUE)`

is the same as

`pam(x,k)$clustering`

but computed more efficiently.

The `pam`

-algorithm is based on the search for `k`

representative objects or medoids among the observations of the
dataset. These observations should represent the structure of the
data. After finding a set of `k`

medoids, `k`

clusters are
constructed by assigning each observation to the nearest medoid. The
goal is to find `k`

representative objects which minimize the sum
of the dissimilarities of the observations to their closest
representative object.

By default, when `medoids`

are not specified, the algorithm first
looks for a good initial set of medoids (this is called the
**build** phase). Then it finds a local minimum for the
objective function, that is, a solution such that there is no single
switch of an observation with a medoid (i.e. a ‘swap’) that will
decrease the objective (this is called the **swap** phase).

When the `medoids`

are specified (or randomly generated), their order does *not*
matter; in general, the algorithms have been designed to not depend on
the order of the observations.

The `pamonce`

option, new in cluster 1.14.2 (Jan. 2012), has been
proposed by Matthias Studer, University of Geneva, based on the
findings by Reynolds et al. (2006) and was extended by Erich Schubert,
TU Dortmund, with the FastPAM optimizations.

The default `FALSE`

(or integer `0`

) corresponds to the
original “swap” algorithm, whereas `pamonce = 1`

(or
`TRUE`

), corresponds to the first proposal ....
and `pamonce = 2`

additionally implements the second proposal as
well.

The key ideas of ‘FastPAM’ (Schubert and Rousseeuw, 2019) are implemented except for the linear approximate build as follows:

`pamonce = 3`

:reduces the runtime by a factor of O(k) by exploiting that points cannot be closest to all current medoids at the same time.

`pamonce = 4`

:additionally allows executing multiple swaps per iteration, usually reducing the number of iterations.

`pamonce = 5`

:adds minor optimizations copied from the

`pamonce = 2`

approach, and is expected to be the fastest of the ‘FastPam’ variants included.

‘FasterPAM’ (Schubert and Rousseeuw, 2021) is implemented via

`pamonce = 6`

:execute each swap which improves results immediately, and hence typically multiple swaps per iteration; this swapping algorithm runs in \(O(n^2)\) rather than \(O(n(n-k)k)\) time which is much faster for all but small \(k\).

In addition, ‘FasterPAM’ uses *random* initialization of the
medoids (instead of the ‘*build*’ phase) to avoid the
\(O(n^2 k)\) initialization cost of the build algorithm. In particular
for large k, this yields a much faster algorithm, while preserving a
similar result quality.

One may decide to use *repeated* random initialization by setting
`nstart > 1`

.

Reynolds, A., Richards, G., de la Iglesia, B. and Rayward-Smith, V. (1992)
Clustering rules: A comparison of partitioning and hierarchical
clustering algorithms;
*Journal of Mathematical Modelling and Algorithms* **5**,
475--504. tools:::Rd_expr_doi("10.1007/s10852-005-9022-1").

Erich Schubert and Peter J. Rousseeuw (2019) Faster k-Medoids Clustering: Improving the PAM, CLARA, and CLARANS Algorithms; SISAP 2020, 171--187. tools:::Rd_expr_doi("10.1007/978-3-030-32047-8_16").

Erich Schubert and Peter J. Rousseeuw (2021) Fast and Eager k-Medoids Clustering: O(k) Runtime Improvement of the PAM, CLARA, and CLARANS Algorithms; Preprint, to appear in Information Systems (https://arxiv.org/abs/2008.05171).

`agnes`

for background and references;
`pam.object`

, `clara`

, `daisy`

,
`partition.object`

, `plot.partition`

,
`dist`

.

```
## generate 25 objects, divided into 2 clusters.
x <- rbind(cbind(rnorm(10,0,0.5), rnorm(10,0,0.5)),
cbind(rnorm(15,5,0.5), rnorm(15,5,0.5)))
pamx <- pam(x, 2)
pamx # Medoids: '7' and '25' ...
summary(pamx)
plot(pamx)
## use obs. 1 & 16 as starting medoids -- same result (typically)
(p2m <- pam(x, 2, medoids = c(1,16)))
## no _build_ *and* no _swap_ phase: just cluster all obs. around (1, 16):
p2.s <- pam(x, 2, medoids = c(1,16), do.swap = FALSE)
p2.s
p3m <- pam(x, 3, trace = 2)
## rather stupid initial medoids:
(p3m. <- pam(x, 3, medoids = 3:1, trace = 1))
# \dontshow{
ii <- pmatch(c("obj","call"), names(pamx))
stopifnot(all.equal(pamx [-ii], p2m [-ii], tolerance=1e-14),
all.equal(pamx$objective[2], p2m$objective[2], tolerance=1e-14))
# }
pam(daisy(x, metric = "manhattan"), 2, diss = TRUE)
data(ruspini)
## Plot similar to Figure 4 in Stryuf et al (1996)
if (FALSE) plot(pam(ruspini, 4), ask = TRUE)
plot(pam(ruspini, 4))
```

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