# Partitioning Around Medoids Partitioning (clustering) of the data into `k` clusters around medoids , a more robust version of K-means. ```r 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) ``` ## Arguments - `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 '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. ## Returns an object of class `"pam"` representing the clustering. See `?pam.object` for details. ## Details 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`. ## Note For large datasets, `pam` may need too much memory or too much computation time since both are $O(n^2)$. Then, `clara()` is preferable, see its documentation. There is hard limit currently, $n <= 65536$, at $2^{16}$ because for larger $n$, $n(n-1)/2$ is larger than the maximal integer (`.Machine$integer.max` = $2^{31} - 1$). ## Author(s) 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. ## References 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](https://arxiv.org/abs/2008.05171)). ## See Also `agnes` for background and references; `pam.object`, `clara`, `daisy`, `partition.object`, `plot.partition`, `dist`. ## Examples ```r ## generate 25 objects, divided into 2 clusters. set.seed(17) # to get reproducible data: 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: '9' and '15' ... summary(pamx) plot(pamx) stopifnot(pamx$id.med == c(9, 15)) stopifnot(identical(pamx$clustering, rep(1:2, c(10, 15)))) ## use obs. 1 & 16 as starting medoids -- same result (for seed above, *and* 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 keep_nms <- setdiff(names(pamx), c("call", "objective"))# .$objective["build"] differ stopifnot(p2.s$id.med == c(1,16), # of course identical(pamx[keep_nms], p2m[keep_nms])) p3m <- pam(x, 3, trace.lev = 2) ## rather stupid initial medoids: (p3m. <- pam(x, 3, medoids = 3:1, trace.lev = 1)) pam(daisy(x, metric = "manhattan"), 2, diss = TRUE) data(ruspini) ## Plot similar to Figure 4 in Stryuf et al (1996) ## Not run: plot(pam(ruspini, 4), ask = TRUE) ```

pam function