# Clustering Large Applications Computes a `"clara"` object, a `list` representing a clustering of the data into `k` clusters. ```r clara(x, k, metric = c("euclidean", "manhattan", "jaccard"), stand = FALSE, cluster.only = FALSE, samples = 5, sampsize = min(n, 40 + 2 * k), trace = 0, medoids.x = TRUE, keep.data = medoids.x, rngR = FALSE, pamLike = FALSE, correct.d = TRUE) ``` ## Arguments - `x`: data 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 (NAs) are allowed. - `k`: integer, the number of clusters. It is required that $0 < k < n$ where $n$ is the number of observations (i.e., n = `nrow(x)`). - `metric`: character string specifying the metric to be used for calculating dissimilarities between observations. The currently available options are "euclidean", "manhattan", "jaccard". Euclidean distances are root sum-of-squares of differences, and manhattan distances are the sum of absolute differences. - `stand`: logical, indicating if 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. - `cluster.only`: logical; if true, only the clustering will be computed and returned, see details. - `samples`: integer, say $N$, the number of samples to be drawn from the dataset. The default, `N = 5`, is rather small for historical (and now back compatibility) reasons and we **recommend to set `samples` an order of magnitude larger**. - `sampsize`: integer, say $j$, the number of observations in each sample. `sampsize` should be higher than the number of clusters (`k`) and at most the number of observations ($n =$ `nrow(x)`). While computational effort is proportional to $j^2$, see note below, it may still be advisable to set $j = $`sampsize` to a **larger** value than the (historical) default. - `trace`: integer indicating a **trace level** for diagnostic output during the algorithm. - `medoids.x`: logical indicating if the medoids should be returned, identically to some rows of the input data `x`. If `FALSE`, `keep.data` must be false as well, and the medoid indices, i.e., row numbers of the medoids will still be returned (`i.med` component), and the algorithm saves space by needing one copy less of `x`. - `keep.data`: logical indicating if the (**scaled** if `stand` is true) data should be kept in the result. Setting this to `FALSE` saves memory (and hence time), but disables `clusplot()`ing of the result. Use `medoids.x = FALSE` to save even more memory. - `rngR`: logical indicating if 's random number generator should be used instead of the primitive clara()-builtin one. If true, this also means that each call to `clara()` returns a different result -- though only slightly different in good situations. - `pamLike`: logical indicating if the swap phase (see `pam`, in C code) should use the same algorithm as `pam()`. Note that from Kaufman and Rousseeuw's description this **should** have been true always, but as the original Fortran code and the subsequent port to C has always contained a small one-letter change (a typo according to Martin Maechler) with respect to PAM, the default, `pamLike = FALSE` has been chosen to remain back compatible rather than PAM compatible . - `correct.d`: logical or integer indicating that---only in the case of `NA`s present in `x`---the correct distance computation should be used instead of the wrong formula which has been present in the original Fortran code and been in use up to early 2016. Because the new correct formula is not back compatible, for the time being, a warning is signalled in this case, unless the user explicitly specifies `correct.d`. ## Returns If `cluster.only` is false (as by default), an object of class `"clara"` representing the clustering. See `clara.object` for details. If `cluster.only` is true, the result is the "clustering", an integer vector of length $n$ with entries from `1:k`. ## Details `clara` (for "euclidean" and "manhattan") is fully described in chapter 3 of Kaufman and Rousseeuw (1990). Compared to other partitioning methods such as `pam`, it can deal with much larger datasets. Internally, this is achieved by considering sub-datasets of fixed size (`sampsize`) such that the time and storage requirements become linear in $n$ rather than quadratic. Each sub-dataset is partitioned into `k` clusters using the same algorithm as in `pam`. Once `k` representative objects have been selected from the sub-dataset, each observation of the entire dataset is assigned to the nearest medoid. The mean (equivalent to the sum) of the dissimilarities of the observations to their closest medoid is used as a measure of the quality of the clustering. The sub-dataset for which the mean (or sum) is minimal, is retained. A further analysis is carried out on the final partition. Each sub-dataset is forced to contain the medoids obtained from the best sub-dataset until then. Randomly drawn observations are added to this set until `sampsize` has been reached. When `cluster.only` is true, the result is simply a (possibly named) integer vector specifying the clustering, i.e., `clara(x,k, cluster.only=TRUE)` is the same as `clara(x,k)$clustering` but computed more efficiently. ## Note By default, the random sampling is implemented with a **very** simple scheme (with period $2^{16} = 65536$) inside the Fortran code, independently of 's random number generation, and as a matter of fact, deterministically. Alternatively, we recommend setting `rngR = TRUE` which uses 's random number generators. Then, `clara()` results are made reproducible typically by using `set.seed()` before calling `clara`. The storage requirement of `clara` computation (for small `k`) is about $O(n * p) + O(j^2)$ where $j = \code{sampsize}$, and $(n,p) = \code{dim(x)}$. The CPU computing time (again assuming small `k`) is about $O(n * p * j^2 * N)$, where $N = \code{samples}$. For small datasets, the function `pam` can be used directly. What can be considered **small**, is really a function of available computing power, both memory (RAM) and speed. Originally (1990), small meant less than 100 observations; in 1997, the authors said **small (say with fewer than 200 observations)**; as of 2006, you can use `pam` with several thousand observations. ## Author(s) Kaufman and Rousseeuw (see `agnes`), originally. Metric `"jaccard"`: Kamil Kozlowski (`@ownedoutcomes.com`) and Kamil Jadeszko. All arguments from `trace` on, and most documentation and all tests by Martin Maechler. ## See Also `agnes` for background and references; `clara.object`, `pam`, `partition.object`, `plot.partition`. ## Examples ```r ## generate 500 objects, divided into 2 clusters. x <- rbind(cbind(rnorm(200,0,8), rnorm(200,0,8)), cbind(rnorm(300,50,8), rnorm(300,50,8))) clarax <- clara(x, 2, samples=50) clarax clarax$clusinfo ## using pamLike=TRUE gives the same (apart from the 'call'): all.equal(clarax[-8], clara(x, 2, samples=50, pamLike = TRUE)[-8]) plot(clarax) ## cluster.only = TRUE -- save some memory/time : clclus <- clara(x, 2, samples=50, cluster.only = TRUE) stopifnot(identical(clclus, clarax$clustering)) ## 'xclara' is an artificial data set with 3 clusters of 1000 bivariate ## objects each. data(xclara) (clx3 <- clara(xclara, 3)) ## "better" number of samples cl.3 <- clara(xclara, 3, samples=100) ## but that did not change the result here: stopifnot(cl.3$clustering == clx3$clustering) ## Plot similar to Figure 5 in Struyf et al (1996) ## Not run: plot(clx3, ask = TRUE) ## Try 100 times *different* random samples -- for reliability: nSim <- 100 nCl <- 3 # = no.classes set.seed(421)# (reproducibility) cl <- matrix(NA,nrow(xclara), nSim) for(i in 1:nSim) cl[,i] <- clara(xclara, nCl, medoids.x = FALSE, rngR = TRUE)$cluster tcl <- apply(cl,1, tabulate, nbins = nCl) ## those that are not always in same cluster (5 out of 3000 for this seed): (iDoubt <- which(apply(tcl,2, function(n) all(n < nSim)))) if(length(iDoubt)) { # (not for all seeds) tabD <- tcl[,iDoubt, drop=FALSE] dimnames(tabD) <- list(cluster = paste(1:nCl), obs = format(iDoubt)) t(tabD) # how many times in which clusters } ```

clara function