Version:	2.1.8.1
VersionNote:	Last CRAN: 2.1.8 on 2024-12-10; 2.1.7 on 2024-12-06; 2.1.6 on 2023-11-30; 2.1.5 on 2023-11-27
Date:	2025-03-11
Priority:	recommended
Title:	"Finding Groups in Data": Cluster Analysis Extended Rousseeuw et al.
Description:	Methods for Cluster analysis. Much extended the original from Peter Rousseeuw, Anja Struyf and Mia Hubert, based on Kaufman and Rousseeuw (1990) "Finding Groups in Data".
Maintainer:	Martin Maechler <maechler@stat.math.ethz.ch>
Depends:	R (≥ 3.5.0)
Imports:	graphics, grDevices, stats, utils
Suggests:	MASS, Matrix
SuggestsNote:	MASS: two examples using cov.rob() and mvrnorm(); Matrix tools for testing
Enhances:	mvoutlier, fpc, ellipse, sfsmisc
EnhancesNote:	xref-ed in man/*.Rd
LazyLoad:	yes
LazyData:	yes
ByteCompile:	yes
BuildResaveData:	no
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
URL:	https://svn.r-project.org/R-packages/trunk/cluster/
NeedsCompilation:	yes
Packaged:	2025-03-11 22:34:24 UTC; maechler
Author:	Martin Maechler [aut, cre], Peter Rousseeuw [aut] (Fortran original), Anja Struyf [aut] (S original), Mia Hubert [aut] (S original), Kurt Hornik [trl, ctb] (port to R; maintenance(1999-2000)), Matthias Studer [ctb], Pierre Roudier [ctb], Juan Gonzalez [ctb], Kamil Kozlowski [ctb], Erich Schubert [ctb] (fastpam options for pam()), Keefe Murphy [ctb] (volume.ellipsoid({d >= 3}))
Repository:	CRAN
Date/Publication:	2025-03-12 17:20:02 UTC

Agglomerative Nesting (Hierarchical Clustering)

Description

Computes agglomerative hierarchical clustering of the dataset.

Usage

agnes(x, diss = inherits(x, "dist"), metric = "euclidean",
      stand = FALSE, method = "average", par.method,
      keep.diss = n < 100, keep.data = !diss, trace.lev = 0)

Arguments

Details

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".

Value

an object of class "agnes" (which extends "twins") representing the clustering. See agnes.object for details, and methods applicable.

BACKGROUND

Cluster analysis divides a dataset into groups (clusters) of observations that are similar to each other.

Hierarchical methods

like agnes, diana, and mona construct a hierarchy of clusterings, with the number of clusters ranging from one to the number of observations.

Partitioning methods

like pam, clara, and fanny require that the number of clusters be given by the user.

Author(s)

Method "gaverage" has been contributed by Pierre Roudier, Landcare Research, New Zealand.

References

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.

See Also

agnes.object, daisy, diana, dist, hclust, plot.agnes, twins.object.

Examples

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.method = 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.plots = 2)
plot(aa.ga, which.plots = 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]))

Agglomerative Nesting (AGNES) Object

Description

The objects of class "agnes" represent an agglomerative hierarchical clustering of a dataset.

Value

A legitimate agnes object is a list with the following components:

GENERATION

This class of objects is returned from agnes.

METHODS

The "agnes" class has methods for the following generic functions: print, summary, plot, and as.dendrogram.

In addition, cutree(x, *) can be used to “cut” the dendrogram in order to produce cluster assignments.

INHERITANCE

The class "agnes" inherits from "twins". Therefore, the generic functions pltree and as.hclust are available for agnes objects. After applying as.hclust(), all its methods are available, of course.

See Also

agnes, diana, as.hclust, hclust, plot.agnes, twins.object.

cutree.

Examples

data(agriculture)
ag.ag <- agnes(agriculture)
class(ag.ag)
pltree(ag.ag) # the dendrogram

## cut the dendrogram -> get cluster assignments:
(ck3 <- cutree(ag.ag, k = 3))
(ch6 <- cutree(as.hclust(ag.ag), h = 6))
stopifnot(identical(unname(ch6), ck3))

European Union Agricultural Workforces

Description

Gross National Product (GNP) per capita and percentage of the population working in agriculture for each country belonging to the European Union in 1993.

Usage

data(agriculture)

Format

A data frame with 12 observations on 2 variables:

The row names of the data frame indicate the countries.

Details

The data seem to show two clusters, the “more agricultural” one consisting of Greece, Portugal, Spain, and Ireland.

Source

Eurostat (European Statistical Agency, 1994): Cijfers en feiten: Een statistisch portret van de Europese Unie.

References

see those in agnes.

See Also

agnes, daisy, diana.

Examples

data(agriculture)

## Compute the dissimilarities using Euclidean metric and without
## standardization
daisy(agriculture, metric = "euclidean", stand = FALSE)

## 2nd plot is similar to Figure 3 in Struyf et al (1996)
plot(pam(agriculture, 2))

## Plot similar to Figure 7 in Struyf et al (1996)
## Not run: plot(agnes(agriculture), ask = TRUE)


## Plot similar to Figure 8 in Struyf et al (1996)
## Not run: plot(diana(agriculture), ask = TRUE)

Attributes of Animals

Description

This data set considers 6 binary attributes for 20 animals.

Usage

data(animals)

Format

A data frame with 20 observations on 6 variables:

All variables are encoded as 1 = 'no', 2 = 'yes'.

Details

This dataset is useful for illustrating monothetic (only a single variable is used for each split) hierarchical clustering.

Source

Leonard Kaufman and Peter J. Rousseeuw (1990): Finding Groups in Data (pp 297ff). New York: Wiley.

References

see Struyf, Hubert & Rousseeuw (1996), in agnes.

Examples

data(animals)
apply(animals,2, table) # simple overview

ma <- mona(animals)
ma
## Plot similar to Figure 10 in Struyf et al (1996)
plot(ma)

Plot Banner (of Hierarchical Clustering)

Description

Draws a “banner”, i.e. basically a horizontal barplot visualizing the (agglomerative or divisive) hierarchical clustering or an other binary dendrogram structure.

Usage

bannerplot(x, w = rev(x$height), fromLeft = TRUE,
           main=NULL, sub=NULL, xlab = "Height",  adj = 0,
           col = c(2, 0), border = 0, axes = TRUE, frame.plot = axes,
           rev.xax = !fromLeft, xax.pretty = TRUE,
           labels = NULL, nmax.lab = 35, max.strlen = 5,
           yax.do = axes && length(x$order) <= nmax.lab,
           yaxRight = fromLeft, y.mar = 2.4 + max.strlen/2.5, ...)

Arguments

Note

This is mainly a utility called from plot.agnes, plot.diana and plot.mona.

Author(s)

Martin Maechler (from original code of Kaufman and Rousseeuw).

Examples

data(agriculture)
bannerplot(agnes(agriculture), main = "Bannerplot")

Subset of C-horizon of Kola Data

Description

This is a small rounded subset of the C-horizon data chorizon from package mvoutlier.

Usage

data(chorSub)

Format

A data frame with 61 observations on 10 variables. The variables contain scaled concentrations of chemical elements.

Details

This data set was produced from chorizon via these statements:

    data(chorizon, package = "mvoutlier")
    chorSub <- round(100*scale(chorizon[,101:110]))[190:250,]
    storage.mode(chorSub) <- "integer"
    colnames(chorSub) <- gsub("_.*", '', colnames(chorSub))
  

Source

Kola Project (1993-1998)

See Also

chorizon in package mvoutlier and other Kola data in the same package.

Examples

data(chorSub)
summary(chorSub)
pairs(chorSub, gap= .1)# some outliers

Clustering Large Applications

Description

Computes a "clara" object, a list representing a clustering of the data into k clusters.

Usage

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

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.

Value

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.

Note

By default, the random sampling is implemented with a very simple scheme (with period 2^{16} = 65536) inside the Fortran code, independently of R's random number generation, and as a matter of fact, deterministically. Alternatively, we recommend setting rngR = TRUE which uses R'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 \times 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 \times p \times j^2 \times 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 R documentation and all tests by Martin Maechler.

See Also

agnes for background and references; clara.object, pam, partition.object, plot.partition.

Examples

## 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)$clustering
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
}

Clustering Large Applications (CLARA) Object

Description

The objects of class "clara" represent a partitioning of a large dataset into clusters and are typically returned from clara.

Value

A legitimate clara object is a list with the following components:

Methods, Inheritance

The "clara" class has methods for the following generic functions: print, summary.

The class "clara" inherits from "partition". Therefore, the generic functions plot and clusplot can be used on a clara object.

See Also

clara, dissimilarity.object, partition.object, plot.partition.

Gap Statistic for Estimating the Number of Clusters

Description

clusGap() calculates a goodness of clustering measure, the “gap” statistic. For each number of clusters k, it compares \log(W(k)) with E^*[\log(W(k))] where the latter is defined via bootstrapping, i.e., simulating from a reference (H_0) distribution, a uniform distribution on the hypercube determined by the ranges of x, after first centering, and then svd (aka ‘PCA’)-rotating them when (as by default) spaceH0 = "scaledPCA".

maxSE(f, SE.f) determines the location of the maximum of f, taking a “1-SE rule” into account for the *SE* methods. The default method "firstSEmax" looks for the smallest k such that its value f(k) is not more than 1 standard error away from the first local maximum. This is similar but not the same as "Tibs2001SEmax", Tibshirani et al's recommendation of determining the number of clusters from the gap statistics and their standard deviations.

Usage

clusGap(x, FUNcluster, K.max, B = 100, d.power = 1,
        spaceH0 = c("scaledPCA", "original"),
        verbose = interactive(), ...)

maxSE(f, SE.f,
      method = c("firstSEmax", "Tibs2001SEmax", "globalSEmax",
                 "firstmax", "globalmax"),
      SE.factor = 1)

## S3 method for class 'clusGap'
print(x, method = "firstSEmax", SE.factor = 1, ...)

## S3 method for class 'clusGap'
plot(x, type = "b", xlab = "k", ylab = expression(Gap[k]),
     main = NULL, do.arrows = TRUE,
     arrowArgs = list(col="red3", length=1/16, angle=90, code=3), ...)

Arguments

Details

The main result <res>$Tab[,"gap"] of course is from bootstrapping aka Monte Carlo simulation and hence random, or equivalently, depending on the initial random seed (see set.seed()). On the other hand, in our experience, using B = 500 gives quite precise results such that the gap plot is basically unchanged after an another run.

Value

clusGap(..) returns an object of S3 class "clusGap", basically a list with components

Author(s)

This function is originally based on the functions gap of former (Bioconductor) package SAGx by Per Broberg, gapStat() from former package SLmisc by Matthias Kohl and ideas from gap() and its methods of package lga by Justin Harrington.

The current implementation is by Martin Maechler.

The implementation of spaceH0 = "original" is based on code proposed by Juan Gonzalez.

References

Tibshirani, R., Walther, G. and Hastie, T. (2001). Estimating the number of data clusters via the Gap statistic. Journal of the Royal Statistical Society B, 63, 411–423.

Tibshirani, R., Walther, G. and Hastie, T. (2000). Estimating the number of clusters in a dataset via the Gap statistic. Technical Report. Stanford.

Dudoit, S. and Fridlyand, J. (2002) A prediction-based resampling method for estimating the number of clusters in a dataset. Genome Biology 3(7). doi:10.1186/gb-2002-3-7-research0036

Per Broberg (2006). SAGx: Statistical Analysis of the GeneChip. R package version 1.9.7. https://bioconductor.org/packages/3.12/bioc/html/SAGx.html Deprecated and removed from Bioc ca. 2022

See Also

silhouette for a much simpler less sophisticated goodness of clustering measure.

cluster.stats() in package fpc for alternative measures.

Examples

### --- maxSE() methods -------------------------------------------
(mets <- eval(formals(maxSE)$method))
fk <- c(2,3,5,4,7,8,5,4)
sk <- c(1,1,2,1,1,3,1,1)/2
## use plot.clusGap():
plot(structure(class="clusGap", list(Tab = cbind(gap=fk, SE.sim=sk))))
## Note that 'firstmax' and 'globalmax' are always at 3 and 6 :
sapply(c(1/4, 1,2,4), function(SEf)
        sapply(mets, function(M) maxSE(fk, sk, method = M, SE.factor = SEf)))

### --- clusGap() -------------------------------------------------
## ridiculously nicely separated clusters in 3 D :
x <- rbind(matrix(rnorm(150,           sd = 0.1), ncol = 3),
           matrix(rnorm(150, mean = 1, sd = 0.1), ncol = 3),
           matrix(rnorm(150, mean = 2, sd = 0.1), ncol = 3),
           matrix(rnorm(150, mean = 3, sd = 0.1), ncol = 3))

## Slightly faster way to use pam (see below)
pam1 <- function(x,k) list(cluster = pam(x,k, cluster.only=TRUE))

## We do not recommend using hier.clustering here, but if you want,
## there is  factoextra::hcut () or a cheap version of it
hclusCut <- function(x, k, d.meth = "euclidean", ...)
   list(cluster = cutree(hclust(dist(x, method=d.meth), ...), k=k))

## You can manually set it before running this :    doExtras <- TRUE  # or  FALSE
if(!(exists("doExtras") && is.logical(doExtras)))
  doExtras <- cluster:::doExtras()

if(doExtras) {
  ## Note we use  B = 60 in the following examples to keep them "speedy".
  ## ---- rather keep the default B = 500 for your analysis!

  ## note we can  pass 'nstart = 20' to kmeans() :
  gskmn <- clusGap(x, FUNcluster = kmeans, nstart = 20, K.max = 8, B = 60)
  gskmn #-> its print() method
  plot(gskmn, main = "clusGap(., FUNcluster = kmeans, n.start=20, B= 60)")
  set.seed(12); system.time(
    gsPam0 <- clusGap(x, FUNcluster = pam, K.max = 8, B = 60)
  )
  set.seed(12); system.time(
    gsPam1 <- clusGap(x, FUNcluster = pam1, K.max = 8, B = 60)
  )
  ## and show that it gives the "same":
  not.eq <- c("call", "FUNcluster"); n <- names(gsPam0)
  eq <- n[!(n %in% not.eq)]
  stopifnot(identical(gsPam1[eq], gsPam0[eq]))
  print(gsPam1, method="globalSEmax")
  print(gsPam1, method="globalmax")

  print(gsHc <- clusGap(x, FUNcluster = hclusCut, K.max = 8, B = 60))

}# end {doExtras}

gs.pam.RU <- clusGap(ruspini, FUNcluster = pam1, K.max = 8, B = 60)
gs.pam.RU
plot(gs.pam.RU, main = "Gap statistic for the 'ruspini' data")
mtext("k = 4 is best .. and  k = 5  pretty close")

## This takes a minute..
## No clustering ==> k = 1 ("one cluster") should be optimal:
Z <- matrix(rnorm(256*3), 256,3)
gsP.Z <- clusGap(Z, FUNcluster = pam1, K.max = 8, B = 200)
plot(gsP.Z, main = "clusGap(<iid_rnorm_p=3>)  ==> k = 1  cluster is optimal")
gsP.Z

Bivariate Cluster Plot (of a Partitioning Object)

Description

Draws a 2-dimensional “clusplot” (clustering plot) on the current graphics device. The generic function has a default and a partition method.

Usage

clusplot(x, ...)

## S3 method for class 'partition'
clusplot(x, main = NULL, dist = NULL, ...)

Arguments

Details

The clusplot.partition() method relies on clusplot.default.

If the clustering algorithms pam, fanny and clara are applied to a data matrix of observations-by-variables then a clusplot of the resulting clustering can always be drawn. When the data matrix contains missing values and the clustering is performed with pam or fanny, the dissimilarity matrix will be given as input to clusplot. When the clustering algorithm clara was applied to a data matrix with NAs then clusplot() will replace the missing values as described in clusplot.default, because a dissimilarity matrix is not available.

Value

For the partition (and default) method: An invisible list with components Distances and Shading, as for clusplot.default, see there.

Side Effects

a 2-dimensional clusplot is created on the current graphics device.

See Also

clusplot.default for references; partition.object, pam, pam.object, clara, clara.object, fanny, fanny.object, par.

Examples

 ## For more, see ?clusplot.default

## 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)))
clusplot(pam(x, 2))
## add noise, and try again :
x4 <- cbind(x, rnorm(25), rnorm(25))
clusplot(pam(x4, 2))

Bivariate Cluster Plot (clusplot) Default Method

Description

Creates a bivariate plot visualizing a partition (clustering) of the data. All observation are represented by points in the plot, using principal components or multidimensional scaling. Around each cluster an ellipse is drawn.

Usage

## Default S3 method:
clusplot(x, clus, diss = FALSE,
          s.x.2d = mkCheckX(x, diss), stand = FALSE,
          lines = 2, shade = FALSE, color = FALSE,
          labels= 0, plotchar = TRUE,
          col.p = "dark green", col.txt = col.p,
          col.clus = if(color) c(2, 4, 6, 3) else 5, cex = 1, cex.txt = cex,
          span = TRUE,
          add = FALSE,
          xlim = NULL, ylim = NULL,
          main = paste("CLUSPLOT(", deparse1(substitute(x)),")"),
          sub = paste("These two components explain",
             round(100 * var.dec, digits = 2), "% of the point variability."),
          xlab = "Component 1", ylab = "Component 2",
          verbose = getOption("verbose"),
          ...)

Arguments

Details

clusplot uses function calls princomp(*, cor = (ncol(x) > 2)) or cmdscale(*, add=TRUE), respectively, depending on diss being false or true. These functions are data reduction techniques to represent the data in a bivariate plot.

Ellipses are then drawn to indicate the clusters. The further layout of the plot is determined by the optional arguments.

Value

An invisible list with components:

Side Effects

a visual display of the clustering is plotted on the current graphics device.

Note

When we have 4 or fewer clusters, then the color=TRUE gives every cluster a different color. When there are more than 4 clusters, clusplot uses the function pam to cluster the densities into 4 groups such that ellipses with nearly the same density get the same color. col.clus specifies the colors used.

The col.p and col.txt arguments, added for R, are recycled to have length the number of observations. If col.p has more than one value, using color = TRUE can be confusing because of a mix of point and ellipse colors.

References

Pison, G., Struyf, A. and Rousseeuw, P.J. (1999) Displaying a Clustering with CLUSPLOT, Computational Statistics and Data Analysis, 30, 381–392.

Kaufman, L. and Rousseeuw, P.J. (1990). Finding Groups in Data: An Introduction to Cluster Analysis. Wiley, New York.

Struyf, A., Hubert, M. and Rousseeuw, P.J. (1997). Integrating Robust Clustering Techniques in S-PLUS, Computational Statistics and Data Analysis, 26, 17-37.

See Also

princomp, cmdscale, pam, clara, daisy, par, identify, cov.mve, clusplot.partition.

Examples

## plotting votes.diss(dissimilarity) in a bivariate plot and
## partitioning into 2 clusters
data(votes.repub)
votes.diss <- daisy(votes.repub)
pamv <- pam(votes.diss, 2, diss = TRUE)
clusplot(pamv, shade = TRUE)
## is the same as
votes.clus <- pamv$clustering
clusplot(votes.diss, votes.clus, diss = TRUE, shade = TRUE)
## Now look at components 3 and 2 instead of 1 and 2:
str(cMDS <- cmdscale(votes.diss, k=3, add=TRUE))
clusplot(pamv, s.x.2d = list(x=cMDS$points[, c(3,2)],
                             labs=rownames(votes.repub), var.dec=NA),
         shade = TRUE, col.p = votes.clus,
         sub="", xlab = "Component 3", ylab = "Component 2")

clusplot(pamv, col.p = votes.clus, labels = 4)# color points and label ellipses
# "simple" cheap ellipses: larger than minimum volume:
# here they are *added* to the previous plot:
clusplot(pamv, span = FALSE, add = TRUE, col.clus = "midnightblue")

## Setting a small *label* size:
clusplot(votes.diss, votes.clus, diss = TRUE, labels = 3, cex.txt = 0.6)

if(dev.interactive()) { #  uses identify() *interactively* :
  clusplot(votes.diss, votes.clus, diss = TRUE, shade = TRUE, labels = 1)
  clusplot(votes.diss, votes.clus, diss = TRUE, labels = 5)# ident. only points
}

## plotting iris (data frame) in a 2-dimensional plot and partitioning
## into 3 clusters.
data(iris)
iris.x <- iris[, 1:4]
cl3 <- pam(iris.x, 3)$clustering
op <- par(mfrow= c(2,2))
clusplot(iris.x, cl3, color = TRUE)
U <- par("usr")
## zoom in :
rect(0,-1, 2,1, border = "orange", lwd=2)
clusplot(iris.x, cl3, color = TRUE, xlim = c(0,2), ylim = c(-1,1))
box(col="orange",lwd=2); mtext("sub region", font = 4, cex = 2)
##  or zoom out :
clusplot(iris.x, cl3, color = TRUE, xlim = c(-4,4), ylim = c(-4,4))
mtext("'super' region", font = 4, cex = 2)
rect(U[1],U[3], U[2],U[4], lwd=2, lty = 3)

# reset graphics
par(op)

Internal cluster functions

Description

Internal cluster functions.

Usage

meanabsdev(y)

Arguments

Details

These are not to be called by the user.

A more R-like version of meanabsdev() would default to na.rm=FALSE, as e.g., mean() or median():

  meanabsdev <- function(y, na.rm=FALSE)
       mean(abs(y - mean(y, na.rm=na.rm)), na.rm=na.rm)

Agglomerative / Divisive Coefficient for 'hclust' Objects

Description

Computes the “agglomerative coefficient” (aka “divisive coefficient” for diana), measuring the clustering structure of the dataset.

For each observation i, denote by m(i) its dissimilarity to the first cluster it is merged with, divided by the dissimilarity of the merger in the final step of the algorithm. The agglomerative coefficient is the average of all 1 - m(i). It can also be seen as the average width (or the percentage filled) of the banner plot.

coefHier() directly interfaces to the underlying C code, and “proves” that only object$heights is needed to compute the coefficient.

Because it grows with the number of observations, this measure should not be used to compare datasets of very different sizes.

Usage

coefHier(object)
coef.hclust(object, ...)
## S3 method for class 'hclust'
coef(object, ...)
## S3 method for class 'twins'
coef(object, ...)

Arguments

Value

a number specifying the agglomerative (or divisive for diana objects) coefficient as defined by Kaufman and Rousseeuw, see agnes.object $ ac or diana.object $ dc.

Examples

data(agriculture)
aa <- agnes(agriculture)
coef(aa) # really just extracts aa$ac
coef(as.hclust(aa))# recomputes
coefHier(aa)       # ditto

Dissimilarity Matrix Calculation

Description

Compute all the pairwise dissimilarities (distances) between observations in the data set. The original variables may be of mixed types. In that case, or whenever metric = "gower" is set, a generalization of Gower's formula is used, see ‘Details’ below.

Usage

daisy(x, metric = c("euclidean", "manhattan", "gower"),
      stand = FALSE, type = list(), weights = rep.int(1, p),
      warnBin = warnType, warnAsym = warnType, warnConst = warnType,
      warnType = TRUE)

Arguments

Details

The original version of daisy is fully described in chapter 1 of Kaufman and Rousseeuw (1990). Compared to dist whose input must be numeric variables, the main feature of daisy is its ability to handle other variable types as well (e.g. nominal, ordinal, (a)symmetric binary) even when different types occur in the same data set.

The handling of nominal, ordinal, and (a)symmetric binary data is achieved by using the general dissimilarity coefficient of Gower (1971). If x contains any columns of these data-types, both arguments metric and stand will be ignored and Gower's coefficient will be used as the metric. This can also be activated for purely numeric data by metric = "gower". With that, each variable (column) is first standardized by dividing each entry by the range of the corresponding variable, after subtracting the minimum value; consequently the rescaled variable has range [0,1], exactly.

Note that setting the type to symm (symmetric binary) gives the same dissimilarities as using nominal (which is chosen for non-ordered factors) only when no missing values are present, and more efficiently.

Note that daisy signals a warning when 2-valued numerical variables do not have an explicit type specified, because the reference authors recommend to consider using "asymm"; the warning may be silenced by warnBin = FALSE.

In the daisy algorithm, missing values in a row of x are not included in the dissimilarities involving that row. There are two main cases,

1. If all variables are interval scaled (and metric is not "gower"), the metric is "euclidean", and n_g is the number of columns in which neither row i and j have NAs, then the dissimilarity d(i,j) returned is \sqrt{p/n_g} (p=ncol(x)) times the Euclidean distance between the two vectors of length n_g shortened to exclude NAs. The rule is similar for the "manhattan" metric, except that the coefficient is p/n_g. If n_g = 0, the dissimilarity is NA.

2. When some variables have a type other than interval scaled, or if metric = "gower" is specified, the dissimilarity between two rows is the weighted mean of the contributions of each variable. Specifically,

   d_{ij} = d(i,j) = \frac{\sum_{k=1}^p w_k \delta_{ij}^{(k)} d_{ij}^{(k)}}{ \sum_{k=1}^p w_k \delta_{ij}^{(k)}}.

   In other words, d_{ij} is a weighted mean of d_{ij}^{(k)} with weights w_k \delta_{ij}^{(k)}, where w_k= weigths[k], \delta_{ij}^{(k)} is 0 or 1, and d_{ij}^{(k)}, the k-th variable contribution to the total distance, is a distance between x[i,k] and x[j,k], see below.

   The 0-1 weight \delta_{ij}^{(k)} becomes zero when the variable x[,k] is missing in either or both rows (i and j), or when the variable is asymmetric binary and both values are zero. In all other situations it is 1.

   The contribution d_{ij}^{(k)} of a nominal or binary variable to the total dissimilarity is 0 if both values are equal, 1 otherwise. The contribution of other variables is the absolute difference of both values, divided by the total range of that variable. Note that “standard scoring” is applied to ordinal variables, i.e., they are replaced by their integer codes 1:K. Note that this is not the same as using their ranks (since there typically are ties).

   As the individual contributions d_{ij}^{(k)} are in [0,1], the dissimilarity d_{ij} will remain in this range. If all weights w_k \delta_{ij}^{(k)} are zero, the dissimilarity is set to NA.

Value

an object of class "dissimilarity" containing the dissimilarities among the rows of x. This is typically the input for the functions pam, fanny, agnes or diana. For more details, see dissimilarity.object.

Background

Dissimilarities are used as inputs to cluster analysis and multidimensional scaling. The choice of metric may have a large impact.

Author(s)

Anja Struyf, Mia Hubert, and Peter and Rousseeuw, for the original version.
Martin Maechler improved the NA handling and type specification checking, and extended functionality to metric = "gower" and the optional weights argument.

References

Gower, J. C. (1971) A general coefficient of similarity and some of its properties, Biometrics 27, 857–874.

Kaufman, L. and Rousseeuw, P.J. (1990) Finding Groups in Data: An Introduction to Cluster Analysis. Wiley, New York.

Struyf, A., Hubert, M. and Rousseeuw, P.J. (1997) Integrating Robust Clustering Techniques in S-PLUS, Computational Statistics and Data Analysis 26, 17–37.

See Also

dissimilarity.object, dist, pam, fanny, clara, agnes, diana.

Examples

data(agriculture)
## Example 1 in ref:
##  Dissimilarities using Euclidean metric and without standardization
d.agr <- daisy(agriculture, metric = "euclidean", stand = FALSE)
d.agr
as.matrix(d.agr)[,"DK"] # via as.matrix.dist(.)
## compare with
as.matrix(daisy(agriculture, metric = "gower"))

## Example 2 in reference, extended  ---  different ways of "mixed" / "gower":

example(flower) # -> data(flower) *and* provide 'flowerN'

summary(d0    <- daisy(flower))  # -> the first 3 {0,1} treated as *N*ominal
summary(dS123 <- daisy(flower,  type = list(symm = 1:3))) # first 3 treated as *S*ymmetric
stopifnot(dS123 == d0) # i.e.,  *S*ymmetric <==> *N*ominal {for 2-level factor}
summary(dNS123<- daisy(flowerN, type = list(symm = 1:3)))
stopifnot(dS123 == d0)
## by default, however ...
summary(dA123 <- daisy(flowerN)) # .. all 3 logicals treated *A*symmetric binary (w/ warning)
summary(dA3  <- daisy(flower, type = list(asymm = 3)))
summary(dA13 <- daisy(flower, type = list(asymm = c(1, 3), ordratio = 7)))
## Mixing variable *names* and column numbers (failed in the past):
summary(dfl3 <- daisy(flower, type = list(asymm = c("V1", "V3"), symm= 2,
                                          ordratio= 7, logratio= 8)))

## If we'd treat the first 3 as simple {0,1}
Nflow <- flower
Nflow[,1:3] <- lapply(flower[,1:3], function(f) as.integer(as.character(f)))
summary(dN <- daisy(Nflow)) # w/ warning: treated binary .. 1:3 as interval
## Still, using Euclidean/Manhattan distance for {0-1} *is* identical to treating them as "N" :
stopifnot(dN == d0)
stopifnot(dN == daisy(Nflow, type = list(symm = 1:3))) # or as "S"

DIvisive ANAlysis Clustering

Description

Computes a divisive hierarchical clustering of the dataset returning an object of class diana.

Usage

diana(x, diss = inherits(x, "dist"), metric = "euclidean", stand = FALSE,
      stop.at.k = FALSE,
      keep.diss = n < 100, keep.data = !diss, trace.lev = 0)

Arguments

Details

diana is fully described in chapter 6 of Kaufman and Rousseeuw (1990). It is probably unique in computing a divisive hierarchy, whereas most other software for hierarchical clustering is agglomerative. Moreover, diana provides (a) the divisive coefficient (see diana.object) which measures the amount of clustering structure found; and (b) the banner, a novel graphical display (see plot.diana).

The diana-algorithm constructs a hierarchy of clusterings, starting with one large cluster containing all n observations. Clusters are divided until each cluster contains only a single observation.
At each stage, the cluster with the largest diameter is selected. (The diameter of a cluster is the largest dissimilarity between any two of its observations.)
To divide the selected cluster, the algorithm first looks for its most disparate observation (i.e., which has the largest average dissimilarity to the other observations of the selected cluster). This observation initiates the "splinter group". In subsequent steps, the algorithm reassigns observations that are closer to the "splinter group" than to the "old party". The result is a division of the selected cluster into two new clusters.

Value

an object of class "diana" representing the clustering; this class has methods for the following generic functions: print, summary, plot.

Further, the class "diana" inherits from "twins". Therefore, the generic function pltree can be used on a diana object, and as.hclust and as.dendrogram methods are available.

A legitimate diana object is a list with the following components:

See Also

agnes also for background and references; cutree (and as.hclust) for grouping extraction; daisy, dist, plot.diana, twins.object.

Examples

data(votes.repub)
dv <- diana(votes.repub, metric = "manhattan", stand = TRUE)
print(dv)
plot(dv) #-> plot.diana()  {w/ its own help + examples}

## Cut into 2 groups:
dv2 <- cutree(as.hclust(dv), k = 2)
table(dv2) # 8 and 42 group members
rownames(votes.repub)[dv2 == 1]

## For two groups, does the metric matter ?
dv0 <- diana(votes.repub, stand = TRUE) # default: Euclidean
dv.2 <- cutree(as.hclust(dv0), k = 2)
table(dv2 == dv.2)## identical group assignments

str(as.dendrogram(dv0)) # {via as.dendrogram.twins() method}

data(agriculture)
## Plot similar to Figure 8 in ref
## Not run: plot(diana(agriculture), ask = TRUE)

Dissimilarity Matrix Object

Description

Objects of class "dissimilarity" representing the dissimilarity matrix of a dataset.

Value

The dissimilarity matrix is symmetric, and hence its lower triangle (column wise) is represented as a vector to save storage space. If the object, is called do, and n the number of observations, i.e., n <- attr(do, "Size"), then for i < j <= n, the dissimilarity between (row) i and j is do[n*(i-1) - i*(i-1)/2 + j-i]. The length of the vector is n*(n-1)/2, i.e., of order n^2.

"dissimilarity" objects also inherit from class dist and can use dist methods, in particular, as.matrix, such that d_{ij} from above is just as.matrix(do)[i,j].

The object has the following attributes:

GENERATION

daisy returns this class of objects. Also the functions pam, clara, fanny, agnes, and diana return a dissimilarity object, as one component of their return objects.

METHODS

The "dissimilarity" class has methods for the following generic functions: print, summary.

See Also

daisy, dist, pam, clara, fanny, agnes, diana.

Compute the Ellipsoid Hull or Spanning Ellipsoid of a Point Set

Description

Compute the “ellipsoid hull” or “spanning ellipsoid”, i.e. the ellipsoid of minimal volume (‘area’ in 2D) such that all given points lie just inside or on the boundary of the ellipsoid.

Usage

ellipsoidhull(x, tol=0.01, maxit=5000,
              ret.wt = FALSE, ret.sqdist = FALSE, ret.pr = FALSE)
## S3 method for class 'ellipsoid'
print(x, digits = max(1, getOption("digits") - 2), ...)

Arguments

Details

The “spanning ellipsoid” algorithm is said to stem from Titterington(1976), in Pison et al (1999) who use it for clusplot.default.
The problem can be seen as a special case of the “Min.Vol.” ellipsoid of which a more more flexible and general implementation is cov.mve in the MASS package.

Value

an object of class "ellipsoid", basically a list with several components, comprising at least

Author(s)

Martin Maechler did the present class implementation; Rousseeuw et al did the underlying original code.

References

Pison, G., Struyf, A. and Rousseeuw, P.J. (1999) Displaying a Clustering with CLUSPLOT, Computational Statistics and Data Analysis, 30, 381–392.

D.M. Titterington (1976) Algorithms for computing D-optimal design on finite design spaces. In Proc.\ of the 1976 Conf.\ on Information Science and Systems, 213–216; John Hopkins University.

See Also

predict.ellipsoid which is also the predict method for ellipsoid objects. volume.ellipsoid for an example of ‘manual’ ellipsoid object construction;
further ellipse from package ellipse and ellipsePoints from package sfsmisc.

chull for the convex hull, clusplot which makes use of this; cov.mve.

Examples

x <- rnorm(100)
xy <- unname(cbind(x, rnorm(100) + 2*x + 10))
exy. <- ellipsoidhull(xy)
exy. # >> calling print.ellipsoid()

plot(xy, main = "ellipsoidhull(<Gauss data>) -- 'spanning points'")
lines(predict(exy.), col="blue")
points(rbind(exy.$loc), col = "red", cex = 3, pch = 13)

exy <- ellipsoidhull(xy, tol = 1e-7, ret.wt = TRUE, ret.sqdist = TRUE)
str(exy) # had small 'tol', hence many iterations
(ii <- which(zapsmall(exy $ wt) > 1e-6))
## --> only about 4 to 6  "spanning ellipsoid" points
round(exy$wt[ii],3); sum(exy$wt[ii]) # weights summing to 1
points(xy[ii,], pch = 21, cex = 2,
       col="blue", bg = adjustcolor("blue",0.25))

Fuzzy Analysis Clustering

Description

Computes a fuzzy clustering of the data into k clusters.

Usage

fanny(x, k, diss = inherits(x, "dist"), memb.exp = 2,
      metric = c("euclidean", "manhattan", "SqEuclidean"),
      stand = FALSE, iniMem.p = NULL, cluster.only = FALSE,
      keep.diss = !diss && !cluster.only && n < 100,
      keep.data = !diss && !cluster.only,
      maxit = 500, tol = 1e-15, trace.lev = 0)

Arguments

Details

In a fuzzy clustering, each observation is “spread out” over the various clusters. Denote by u_{iv} the membership of observation i to cluster v.

The memberships are nonnegative, and for a fixed observation i they sum to 1. The particular method fanny stems from chapter 4 of Kaufman and Rousseeuw (1990) (see the references in daisy) and has been extended by Martin Maechler to allow user specified memb.exp, iniMem.p, maxit, tol, etc.

Fanny aims to minimize the objective function

\sum_{v=1}^k \frac{\sum_{i=1}^n\sum_{j=1}^n u_{iv}^r u_{jv}^r d(i,j)}{ 2 \sum_{j=1}^n u_{jv}^r}

where n is the number of observations, k is the number of clusters, r is the membership exponent memb.exp and d(i,j) is the dissimilarity between observations i and j.
Note that r \to 1 gives increasingly crisper clusterings whereas r \to \infty leads to complete fuzzyness. K&R(1990), p.191 note that values too close to 1 can lead to slow convergence. Further note that even the default, r = 2 can lead to complete fuzzyness, i.e., memberships u_{iv} \equiv 1/k. In that case a warning is signalled and the user is advised to chose a smaller memb.exp (=r).

Compared to other fuzzy clustering methods, fanny has the following features: (a) it also accepts a dissimilarity matrix; (b) it is more robust to the spherical cluster assumption; (c) it provides a novel graphical display, the silhouette plot (see plot.partition).

Value

an object of class "fanny" representing the clustering. See fanny.object for details.

See Also

agnes for background and references; fanny.object, partition.object, plot.partition, daisy, dist.

Examples

## generate 10+15 objects in two clusters, plus 3 objects lying
## between those 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)),
           cbind(rnorm( 3,3.2,0.5), rnorm( 3,3.2,0.5)))
fannyx <- fanny(x, 2)
## Note that observations 26:28 are "fuzzy" (closer to # 2):
fannyx
summary(fannyx)
plot(fannyx)

(fan.x.15 <- fanny(x, 2, memb.exp = 1.5)) # 'crispier' for obs. 26:28
(fanny(x, 2, memb.exp = 3))               # more fuzzy in general

data(ruspini)
f4 <- fanny(ruspini, 4)
stopifnot(rle(f4$clustering)$lengths == c(20,23,17,15))
plot(f4, which = 1)
## Plot similar to Figure 6 in Stryuf et al (1996)
plot(fanny(ruspini, 5))

Fuzzy Analysis (FANNY) Object

Description

The objects of class "fanny" represent a fuzzy clustering of a dataset.

Value

A legitimate fanny object is a list with the following components:

GENERATION

These objects are returned from fanny.

METHODS

The "fanny" class has methods for the following generic functions: print, summary.

INHERITANCE

The class "fanny" inherits from "partition". Therefore, the generic functions plot and clusplot can be used on a fanny object.

See Also

fanny, print.fanny, dissimilarity.object, partition.object, plot.partition.

Flower Characteristics

Description

8 characteristics for 18 popular flowers.

Usage

data(flower)

Format

A data frame with 18 observations on 8 variables:

V1

winters, is binary and indicates whether the plant may be left in the garden when it freezes.

V2

shadow, is binary and shows whether the plant needs to stand in the shadow.

V3

tubers, is asymmetric binary and distinguishes between plants with tubers and plants that grow in any other way.

V4

color, is nominal and specifies the flower's color (1 = white, 2 = yellow, 3 = pink, 4 = red, 5 = blue).

V5

soil, is ordinal and indicates whether the plant grows in dry (1), normal (2), or wet (3) soil.

V6

preference, is ordinal and gives someone's preference ranking going from 1 to 18.

V7

height, is interval scaled, the plant's height in centimeters.

V8

distance, is interval scaled, the distance in centimeters that should be left between the plants.

References

Struyf, Hubert and Rousseeuw (1996), see agnes.

Examples

data(flower)
str(flower) # factors, ordered, numeric

## "Nicer" version (less numeric more self explainable) of 'flower':
flowerN <- flower
colnames(flowerN) <- c("winters", "shadow", "tubers", "color",
                       "soil", "preference", "height", "distance")
for(j in 1:3) flowerN[,j] <- (flowerN[,j] == "1")
levels(flowerN$color) <- c("1" = "white", "2" = "yellow", "3" = "pink",
                           "4" = "red", "5" = "blue")[levels(flowerN$color)]
levels(flowerN$soil)  <- c("1" = "dry", "2" = "normal", "3" = "wet")[levels(flowerN$soil)]
flowerN

## ==> example(daisy)  on how it is used

Permute Indices for Triangular Matrices

Description

Compute index vectors for extracting or reordering of lower or upper triangular matrices that are stored as contiguous vectors.

Usage

lower.to.upper.tri.inds(n)
upper.to.lower.tri.inds(n)

Arguments

Value

integer vector containing a permutation of 1:N where N = n(n-1)/2.

See Also

upper.tri, lower.tri with a related purpose.

Examples

m5 <- matrix(NA,5,5)
m <- m5; m[lower.tri(m)] <- upper.to.lower.tri.inds(5); m
m <- m5; m[upper.tri(m)] <- lower.to.upper.tri.inds(5); m

stopifnot(lower.to.upper.tri.inds(2) == 1,
          lower.to.upper.tri.inds(3) == 1:3,
          upper.to.lower.tri.inds(3) == 1:3,
     sort(upper.to.lower.tri.inds(5)) == 1:10,
     sort(lower.to.upper.tri.inds(6)) == 1:15)

Compute pam-consistent Medoids from Clustering

Description

Given a data matrix or dissimilarity x for say n observational units and a clustering, compute the pam()-consistent medoids.

Usage

medoids(x, clustering, diss = inherits(x, "dist"), USE.NAMES = FALSE, ...)

Arguments

Value

a numeric vector of length

Author(s)

Martin Maechler, after being asked how pam() could be used instead of kmeans(), starting from a previous clustering.

See Also

pam, kmeans. Further, cutree() and agnes (or hclust).

Examples

## From example(agnes):
data(votes.repub)
agn1 <- agnes(votes.repub, metric = "manhattan", stand = TRUE)
agn2 <- agnes(daisy(votes.repub), diss = TRUE, method = "complete")
agnS <- agnes(votes.repub, method = "flexible", par.method = 0.625)

for(k in 2:11) {
  print(table(cl.k <- cutree(agnS, k=k)))
  stopifnot(length(cl.k) == nrow(votes.repub), 1 <= cl.k, cl.k <= k, table(cl.k) >= 2)
  m.k <- medoids(votes.repub, cl.k)
  cat("k =", k,"; sort(medoids) = "); dput(sort(m.k), control={})
}

MONothetic Analysis Clustering of Binary Variables

Description

Returns a list representing a divisive hierarchical clustering of a dataset with binary variables only.

Usage

mona(x, trace.lev = 0)

Arguments

Details

mona is fully described in chapter 7 of Kaufman and Rousseeuw (1990). It is “monothetic” in the sense that each division is based on a single (well-chosen) variable, whereas most other hierarchical methods (including agnes and diana) are “polythetic”, i.e. they use all variables together.

The mona-algorithm constructs a hierarchy of clusterings, starting with one large cluster. Clusters are divided until all observations in the same cluster have identical values for all variables.
At each stage, all clusters are divided according to the values of one variable. A cluster is divided into one cluster with all observations having value 1 for that variable, and another cluster with all observations having value 0 for that variable.

The variable used for splitting a cluster is the variable with the maximal total association to the other variables, according to the observations in the cluster to be splitted. The association between variables f and g is given by a(f,g)*d(f,g) - b(f,g)*c(f,g), where a(f,g), b(f,g), c(f,g), and d(f,g) are the numbers in the contingency table of f and g. [That is, a(f,g) (resp. d(f,g)) is the number of observations for which f and g both have value 0 (resp. value 1); b(f,g) (resp. c(f,g)) is the number of observations for which f has value 0 (resp. 1) and g has value 1 (resp. 0).] The total association of a variable f is the sum of its associations to all variables.

Value

an object of class "mona" representing the clustering. See mona.object for details.

Missing Values (NAs)

The mona-algorithm requires “pure” 0-1 values. However, mona(x) allows x to contain (not too many) NAs. In a preliminary step, these are “imputed”, i.e., all missing values are filled in. To do this, the same measure of association between variables is used as in the algorithm. When variable f has missing values, the variable g with the largest absolute association to f is looked up. When the association between f and g is positive, any missing value of f is replaced by the value of g for the same observation. If the association between f and g is negative, then any missing value of f is replaced by the value of 1-g for the same observation.

Note

In cluster versions before 2.0.6, the algorithm entered an infinite loop in the boundary case of one variable, i.e., ncol(x) == 1, which currently signals an error (because the algorithm now in C, haes not correctly taken account of this special case).

See Also

agnes for background and references; mona.object, plot.mona.

Examples

data(animals)
ma <- mona(animals)
ma
## Plot similar to Figure 10 in Struyf et al (1996)
plot(ma)

## One place to see if/how error messages are *translated* (to 'de' / 'pl'):
ani.NA   <- animals; ani.NA[4,] <- NA
aniNA    <- within(animals, { end[2:9] <- NA })
aniN2    <- animals; aniN2[cbind(1:6, c(3, 1, 4:6, 2))] <- NA
ani.non2 <- within(animals, end[7] <- 3 )
ani.idNA <- within(animals, end[!is.na(end)] <- 1 )
try( mona(ani.NA)   ) ## error: .. object with all values missing
try( mona(aniNA)    ) ## error: .. more than half missing values
try( mona(aniN2)    ) ## error: all have at least one missing
try( mona(ani.non2) ) ## error: all must be binary
try( mona(ani.idNA) ) ## error:  ditto

Monothetic Analysis (MONA) Object

Description

The objects of class "mona" represent the divisive hierarchical clustering of a dataset with only binary variables (measurements). This class of objects is returned from mona.

Value

A legitimate mona object is a list with the following components:

METHODS

The "mona" class has methods for the following generic functions: print, summary, plot.

See Also

mona for examples etc, plot.mona.

Partitioning Around Medoids

Description

Partitioning (clustering) of the data into k clusters “around medoids”, a more robust version of K-means.

Usage

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

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.

Value

an object of class "pam" representing the clustering. See ?pam.object for details.

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 \le 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. 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. 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).

See Also

agnes for background and references; pam.object, clara, daisy, partition.object, plot.partition, dist.

Examples

## 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)

Partitioning Around Medoids (PAM) Object

Description

The objects of class "pam" represent a partitioning of a dataset into clusters.

Value

A legitimate pam object is a list with the following components:

GENERATION

These objects are returned from pam.

METHODS

The "pam" class has methods for the following generic functions: print, summary.

INHERITANCE

The class "pam" inherits from "partition". Therefore, the generic functions plot and clusplot can be used on a pam object.

See Also

pam, dissimilarity.object, partition.object, plot.partition.

Examples

## Use the silhouette widths for assessing the best number of clusters,
## following a one-dimensional example from Christian Hennig :
##
x <- c(rnorm(50), rnorm(50,mean=5), rnorm(30,mean=15))
asw <- numeric(20)
## Note that "k=1" won't work!
for (k in 2:20)
  asw[k] <- pam(x, k) $ silinfo $ avg.width
k.best <- which.max(asw)
cat("silhouette-optimal number of clusters:", k.best, "\n")

plot(1:20, asw, type= "h", main = "pam() clustering assessment",
     xlab= "k  (# clusters)", ylab = "average silhouette width")
axis(1, k.best, paste("best",k.best,sep="\n"), col = "red", col.axis = "red")

Partitioning Object

Description

The objects of class "partition" represent a partitioning of a dataset into clusters.

Value

a "partition" object is a list with the following (and typically more) components:

GENERATION

These objects are returned from pam, clara or fanny.

METHODS

The "partition" class has a method for the following generic functions: plot, clusplot.

INHERITANCE

The following classes inherit from class "partition" : "pam", "clara" and "fanny".

See pam.object, clara.object and fanny.object for details.

See Also

pam, clara, fanny.

Plant Species Traits Data

Description

This dataset constitutes a description of 136 plant species according to biological attributes (morphological or reproductive)

Usage

data(plantTraits)

Format

A data frame with 136 observations on the following 31 variables.

pdias

Diaspore mass (mg)

longindex

Seed bank longevity

durflow

Flowering duration

height

Plant height, an ordered factor with levels 1 < 2 < ... < 8.

begflow

Time of first flowering, an ordered factor with levels 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8 < 9

mycor

Mycorrhizas, an ordered factor with levels 0never < 1 sometimes< 2always

vegaer

aerial vegetative propagation, an ordered factor with levels 0never < 1 present but limited< 2important.

vegsout

underground vegetative propagation, an ordered factor with 3 levels identical to vegaer above.

autopoll

selfing pollination, an ordered factor with levels 0never < 1rare < 2 often< the rule3

insects

insect pollination, an ordered factor with 5 levels 0 < ... < 4.

wind

wind pollination, an ordered factor with 5 levels 0 < ... < 4.

lign

a binary factor with levels 0:1, indicating if plant is woody.

piq

a binary factor indicating if plant is thorny.

ros

a binary factor indicating if plant is rosette.

semiros

semi-rosette plant, a binary factor (0: no; 1: yes).

leafy

leafy plant, a binary factor.

suman

summer annual, a binary factor.

winan

winter annual, a binary factor.

monocarp

monocarpic perennial, a binary factor.

polycarp

polycarpic perennial, a binary factor.

seasaes

seasonal aestival leaves, a binary factor.

seashiv

seasonal hibernal leaves, a binary factor.

seasver

seasonal vernal leaves, a binary factor.

everalw

leaves always evergreen, a binary factor.

everparti

leaves partially evergreen, a binary factor.

elaio

fruits with an elaiosome (dispersed by ants), a binary factor.

endozoo

endozoochorous fruits, a binary factor.

epizoo

epizoochorous fruits, a binary factor.

aquat

aquatic dispersal fruits, a binary factor.

windgl

wind dispersed fruits, a binary factor.

unsp

unspecialized mechanism of seed dispersal, a binary factor.

Details

Most of factor attributes are not disjunctive. For example, a plant can be usually pollinated by insects but sometimes self-pollination can occured.

Source

Vallet, Jeanne (2005) Structuration de communautés végétales et analyse comparative de traits biologiques le long d'un gradient d'urbanisation. Mémoire de Master 2 'Ecologie-Biodiversité-Evolution'; Université Paris Sud XI, 30p.+ annexes (in french)

Examples

data(plantTraits)

## Calculation of a dissimilarity matrix
library(cluster)
dai.b <- daisy(plantTraits,
               type = list(ordratio = 4:11, symm = 12:13, asymm = 14:31))

## Hierarchical classification
agn.trts <- agnes(dai.b, method="ward")
plot(agn.trts, which.plots = 2, cex= 0.6)
plot(agn.trts, which.plots = 1)
cutree6 <- cutree(agn.trts, k=6)
cutree6

## Principal Coordinate Analysis
cmdsdai.b <- cmdscale(dai.b, k=6)
plot(cmdsdai.b[, 1:2], asp = 1, col = cutree6)

Plots of an Agglomerative Hierarchical Clustering

Description

Creates plots for visualizing an agnes object.

Usage

## S3 method for class 'agnes'
plot(x, ask = FALSE, which.plots = NULL, main = NULL,
           sub = paste("Agglomerative Coefficient = ",round(x$ac, digits = 2)),
           adj = 0, nmax.lab = 35, max.strlen = 5, xax.pretty = TRUE, ...)

Arguments

Details

When ask = TRUE, rather than producing each plot sequentially, plot.agnes displays a menu listing all the plots that can be produced. If the menu is not desired but a pause between plots is still wanted one must set par(ask= TRUE) before invoking the plot command.

The banner displays the hierarchy of clusters, and is equivalent to a tree. See Rousseeuw (1986) or chapter 5 of Kaufman and Rousseeuw (1990). The banner plots distances at which observations and clusters are merged. The observations are listed in the order found by the agnes algorithm, and the numbers in the height vector are represented as bars between the observations.

The leaves of the clustering tree are the original observations. Two branches come together at the distance between the two clusters being merged.

For more customization of the plots, rather call bannerplot and pltree(), i.e., its method pltree.twins, respectively.

directly with corresponding arguments, e.g., xlab or ylab.

Side Effects

Appropriate plots are produced on the current graphics device. This can be one or both of the following choices:
Banner
Clustering tree

Note

In the banner plot, observation labels are only printed when the number of observations is limited less than nmax.lab (35, by default), for readability. Moreover, observation labels are truncated to maximally max.strlen (5) characters.

For the dendrogram, more flexibility than via pltree() is provided by dg <- as.dendrogram(x) and plotting dg via plot.dendrogram.

References

Kaufman, L. and Rousseeuw, P.J. (1990) Finding Groups in Data: An Introduction to Cluster Analysis. Wiley, New York.

Rousseeuw, P.J. (1986). A visual display for hierarchical classification, in Data Analysis and Informatics 4; edited by E. Diday, Y. Escoufier, L. Lebart, J. Pages, Y. Schektman, and R. Tomassone. North-Holland, Amsterdam, 743–748.

Struyf, A., Hubert, M. and Rousseeuw, P.J. (1997) Integrating Robust Clustering Techniques in S-PLUS, Computational Statistics and Data Analysis, 26, 17–37.

See Also

agnes and agnes.object; bannerplot, pltree.twins, and par.

Examples

## Can also pass 'labels' to pltree() and bannerplot():
data(iris)
cS <- as.character(Sp <- iris$Species)
cS[Sp == "setosa"] <- "S"
cS[Sp == "versicolor"] <- "V"
cS[Sp == "virginica"] <- "g"
ai <- agnes(iris[, 1:4])
plot(ai, labels = cS, nmax.lab = 150)# bannerplot labels are mess

Plots of a Divisive Hierarchical Clustering

Description

Creates plots for visualizing a diana object.

Usage

## S3 method for class 'diana'
plot(x, ask = FALSE, which.plots = NULL, main = NULL,
           sub = paste("Divisive Coefficient = ", round(x$dc, digits = 2)),
           adj = 0, nmax.lab = 35, max.strlen = 5, xax.pretty = TRUE, ...)

Arguments

Details

When ask = TRUE, rather than producing each plot sequentially, plot.diana displays a menu listing all the plots that can be produced. If the menu is not desired but a pause between plots is still wanted one must set par(ask= TRUE) before invoking the plot command.

The banner displays the hierarchy of clusters, and is equivalent to a tree. See Rousseeuw (1986) or chapter 6 of Kaufman and Rousseeuw (1990). The banner plots the diameter of each cluster being splitted. The observations are listed in the order found by the diana algorithm, and the numbers in the height vector are represented as bars between the observations.

The leaves of the clustering tree are the original observations. A branch splits up at the diameter of the cluster being splitted.

Side Effects

An appropriate plot is produced on the current graphics device. This can be one or both of the following choices:
Banner
Clustering tree

Note

In the banner plot, observation labels are only printed when the number of observations is limited less than nmax.lab (35, by default), for readability. Moreover, observation labels are truncated to maximally max.strlen (5) characters.

References

see those in plot.agnes.

See Also

diana, diana.object, twins.object, par.

Examples

example(diana)# -> dv <- diana(....)

plot(dv, which.plots = 1, nmax.lab = 100)

## wider labels :
op <- par(mar = par("mar") + c(0, 2, 0,0))
plot(dv, which.plots = 1, nmax.lab = 100, max.strlen = 12)
par(op)

Banner of Monothetic Divisive Hierarchical Clusterings

Description

Creates the banner of a mona object.

Usage

## S3 method for class 'mona'
plot(x, main = paste("Banner of ", deparse1(x$call)),
          sub = NULL, xlab = "Separation step",
	  col = c(2,0), axes = TRUE, adj = 0,
          nmax.lab = 35, max.strlen = 5, ...)

Arguments

Details

Plots the separation step at which clusters are splitted. The observations are given in the order found by the mona algorithm, the numbers in the step vector are represented as bars between the observations.

When a long bar is drawn between two observations, those observations have the same value for each variable. See chapter 7 of Kaufman and Rousseeuw (1990).

Side Effects

A banner is plotted on the current graphics device.

Note

In the banner plot, observation labels are only printed when the number of observations is limited less than nmax.lab (35, by default), for readability. Moreover, observation labels are truncated to maximally max.strlen (5) characters.

References

see those in plot.agnes.

See Also

mona, mona.object, par.

Plot of a Partition of the Data Set

Description

Creates plots for visualizing a partition object.

Usage

## S3 method for class 'partition'
plot(x, ask = FALSE, which.plots = NULL,
     nmax.lab = 40, max.strlen = 5, data = x$data, dist = NULL,
     stand = FALSE, lines = 2,
     shade = FALSE, color = FALSE, labels = 0, plotchar = TRUE,
     span = TRUE, xlim = NULL, ylim = NULL, main = NULL, ...)

Arguments

Details

When ask= TRUE, rather than producing each plot sequentially, plot.partition displays a menu listing all the plots that can be produced. If the menu is not desired but a pause between plots is still wanted, call par(ask= TRUE) before invoking the plot command.

The clusplot of a cluster partition consists of a two-dimensional representation of the observations, in which the clusters are indicated by ellipses (see clusplot.partition for more details).

The silhouette plot of a nonhierarchical clustering is fully described in Rousseeuw (1987) and in chapter 2 of Kaufman and Rousseeuw (1990). For each observation i, a bar is drawn, representing its silhouette width s(i), see silhouette for details. Observations are grouped per cluster, starting with cluster 1 at the top. Observations with a large s(i) (almost 1) are very well clustered, a small s(i) (around 0) means that the observation lies between two clusters, and observations with a negative s(i) are probably placed in the wrong cluster.

A clustering can be performed for several values of k (the number of clusters). Finally, choose the value of k with the largest overall average silhouette width.

Side Effects

An appropriate plot is produced on the current graphics device. This can be one or both of the following choices:
Clusplot
Silhouette plot

Note

In the silhouette plot, observation labels are only printed when the number of observations is less than nmax.lab (40, by default), for readability. Moreover, observation labels are truncated to maximally max.strlen (5) characters.
For more flexibility, use plot(silhouette(x), ...), see plot.silhouette.

References

Rousseeuw, P.J. (1987) Silhouettes: A graphical aid to the interpretation and validation of cluster analysis. J. Comput. Appl. Math., 20, 53–65.

Further, the references in plot.agnes.

See Also

partition.object, clusplot.partition, clusplot.default, pam, pam.object, clara, clara.object, fanny, fanny.object, par.

Examples

## 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)))
plot(pam(x, 2))

## Save space not keeping data in clus.object, and still clusplot() it:
data(xclara)
cx <- clara(xclara, 3, keep.data = FALSE)
cx$data # is NULL
plot(cx, data = xclara)

Plot Clustering Tree of a Hierarchical Clustering

Description

pltree() Draws a clustering tree (“dendrogram”) on the current graphics device. We provide the twins method draws the tree of a twins object, i.e., hierarchical clustering, typically resulting from agnes() or diana().

Usage

pltree(x, ...)
## S3 method for class 'twins'
pltree(x, main = paste("Dendrogram of ", deparse1(x$call)),
             labels = NULL, ylab = "Height", ...)

Arguments

Details

Creates a plot of a clustering tree given a twins object. The leaves of the tree are the original observations. In case of an agglomerative clustering, two branches come together at the distance between the two clusters being merged. For a divisive clustering, a branch splits up at the diameter of the cluster being splitted.

Note that currently the method function simply calls plot(as.hclust(x), ...), which dispatches to plot.hclust(..). If more flexible plots are needed, consider xx <- as.dendrogram(as.hclust(x)) and plotting xx, see plot.dendrogram.

Value

a NULL value is returned.

See Also

agnes, agnes.object, diana, diana.object, hclust, par, plot.agnes, plot.diana.

Examples

data(votes.repub)
agn <- agnes(votes.repub)
pltree(agn)

dagn  <- as.dendrogram(as.hclust(agn))
dagn2 <- as.dendrogram(as.hclust(agn), hang = 0.2)
op <- par(mar = par("mar") + c(0,0,0, 2)) # more space to the right
plot(dagn2, horiz = TRUE)
plot(dagn, horiz = TRUE, center = TRUE,
     nodePar = list(lab.cex = 0.6, lab.col = "forest green", pch = NA),
     main = deparse(agn$call))
par(op)

Isotopic Composition Plutonium Batches

Description

The pluton data frame has 45 rows and 4 columns, containing percentages of isotopic composition of 45 Plutonium batches.

Usage

data(pluton)

Format

This data frame contains the following columns:

Pu238

the percentages of \ ^{238}Pu, always less than 2 percent.

Pu239

the percentages of \ ^{239}Pu, typically between 60 and 80 percent (from neutron capture of Uranium, \ ^{238}U).

Pu240

percentage of the plutonium 240 isotope.

Pu241

percentage of the plutonium 241 isotope.

Details

Note that the percentage of plutonium~242 can be computed from the other four percentages, see the examples.

In the reference below it is explained why it is very desirable to combine these plutonium patches in three groups of similar size.

Source

Available as ‘pluton.dat’ from the archive of the University of Antwerpen, ‘..../datasets/clusplot-examples.tar.gz’, no longer available.

References

Rousseeuw, P.J. and Kaufman, L and Trauwaert, E. (1996) Fuzzy clustering using scatter matrices, Computational Statistics and Data Analysis 23(1), 135–151.

Examples

data(pluton)

hist(apply(pluton,1,sum), col = "gray") # between 94% and 100%
pu5 <- pluton
pu5$Pu242 <- 100 - apply(pluton,1,sum) # the remaining isotope.
pairs(pu5)

Predict Method for Ellipsoid Objects

Description

Compute points on the ellipsoid boundary, mostly for drawing.

Usage

predict.ellipsoid(object, n.out=201, ...)
## S3 method for class 'ellipsoid'
predict(object, n.out=201, ...)
ellipsoidPoints(A, d2, loc, n.half = 201)

Arguments

Details

Note ellipsoidPoints is the workhorse function of predict.ellipsoid a standalone function and method for ellipsoid objects, see ellipsoidhull. The class of object is not checked; it must solely have valid components loc (length p), the p \times p matrix cov (corresponding to A) and d2 for the center, the shape (“covariance”) matrix and the squared average radius (or distance) or qchisq(*, p) quantile.

Unfortunately, this is only implemented for p = 2, currently; contributions for p \ge 3 are very welcome.

Value

a numeric matrix of dimension 2*n.out times p.

See Also

ellipsoidhull, volume.ellipsoid.

Examples

 ## see also  example(ellipsoidhull)

## Robust vs. L.S. covariance matrix
set.seed(143)
x <- rt(200, df=3)
y <- 3*x + rt(200, df=2)
plot(x,y, main="non-normal data (N=200)")
mtext("with classical and robust cov.matrix ellipsoids")
X <- cbind(x,y)
C.ls <- cov(X) ; m.ls <- colMeans(X)
d2.99 <- qchisq(0.99, df = 2)
lines(ellipsoidPoints(C.ls, d2.99, loc=m.ls), col="green")
if(require(MASS)) {
  Cxy <- cov.rob(cbind(x,y))
  lines(ellipsoidPoints(Cxy$cov, d2 = d2.99, loc=Cxy$center), col="red")
}# MASS

Print Method for AGNES Objects

Description

Prints the call, agglomerative coefficient, ordering of objects and distances between merging clusters ('Height') of an agnes object.

This is a method for the generic print() function for objects inheriting from class agnes, see agnes.object.

Usage

## S3 method for class 'agnes'
print(x, ...)

Arguments

See Also

summary.agnes producing more output; agnes, agnes.object, print, print.default.

Print Method for CLARA Objects

Description

Prints the best sample, medoids, clustering vector and objective function of clara object.

This is a method for the function print() for objects inheriting from class clara.

Usage

## S3 method for class 'clara'
print(x, ...)

Arguments

See Also

summary.clara producing more output; clara, clara.object, print, print.default.

Print Method for DIANA Objects

Description

Prints the ordering of objects, diameters of splitted clusters, and divisive coefficient of a diana object.

This is a method for the function print() for objects inheriting from class diana.

Usage

## S3 method for class 'diana'
print(x, ...)

Arguments

See Also

diana, diana.object, print, print.default.

Print and Summary Methods for Dissimilarity Objects

Description

Print or summarize the distances and the attributes of a dissimilarity object.

These are methods for the functions print() and summary() for dissimilarity objects. See print, print.default, or summary for the general behavior of these.

Usage

## S3 method for class 'dissimilarity'
print(x, diag = NULL, upper = NULL,
      digits = getOption("digits"), justify = "none", right = TRUE, ...)
## S3 method for class 'dissimilarity'
summary(object,
        digits = max(3, getOption("digits") - 2), ...)
## S3 method for class 'summary.dissimilarity'
print(x, ...)

Arguments

See Also

daisy, dissimilarity.object, print, print.default, print.dist.

Examples

 ## See  example(daisy)

 sd <- summary(daisy(matrix(rnorm(100), 20,5)))
 sd # -> print.summary.dissimilarity(.)
 str(sd)

Print and Summary Methods for FANNY Objects

Description

Prints the objective function, membership coefficients and clustering vector of fanny object.

This is a method for the function print() for objects inheriting from class fanny.

Usage

## S3 method for class 'fanny'
print(x, digits = getOption("digits"), ...)
## S3 method for class 'fanny'
summary(object, ...)
## S3 method for class 'summary.fanny'
print(x, digits = getOption("digits"), ...)

Arguments

See Also

fanny, fanny.object, print, print.default.

Print Method for MONA Objects

Description

Prints the ordering of objects, separation steps, and used variables of a mona object.

This is a method for the function print() for objects inheriting from class mona.

Usage

## S3 method for class 'mona'
print(x, ...)

Arguments

See Also

mona, mona.object, print, print.default.

Print Method for PAM Objects

Description

Prints the medoids, clustering vector and objective function of pam object.

This is a method for the function print() for objects inheriting from class pam.

Usage

## S3 method for class 'pam'
print(x, ...)

Arguments

See Also

pam, pam.object, print, print.default.

Ruspini Data

Description

The Ruspini data set, consisting of 75 points in four groups that is popular for illustrating clustering techniques.

Usage

data(ruspini)

Format

A data frame with 75 observations on 2 variables giving the x and y coordinates of the points, respectively.

Source

E. H. Ruspini (1970) Numerical methods for fuzzy clustering. Inform. Sci. 2, 319–350.

References

see those in agnes.

Examples

data(ruspini)

## Plot similar to Figure 4 in Stryuf et al (1996)
## Not run: plot(pam(ruspini, 4), ask = TRUE)


## Plot similar to Figure 6 in Stryuf et al (1996)
plot(fanny(ruspini, 5))

Compute or Extract Silhouette Information from Clustering

Description

Compute silhouette information according to a given clustering in k clusters.

Usage

silhouette(x, ...)
## Default S3 method:
  silhouette(x, dist, dmatrix, ...)
## S3 method for class 'partition'
silhouette(x, ...)
## S3 method for class 'clara'
silhouette(x, full = FALSE, subset = NULL, ...)

sortSilhouette(object, ...)
## S3 method for class 'silhouette'
summary(object, FUN = mean, ...)
## S3 method for class 'silhouette'
plot(x, nmax.lab = 40, max.strlen = 5,
     main = NULL, sub = NULL, xlab = expression("Silhouette width "* s[i]),
     col = "gray",  do.col.sort = length(col) > 1, border = 0,
     cex.names = par("cex.axis"), do.n.k = TRUE, do.clus.stat = TRUE, ...)

Arguments

Details

For each observation i, the silhouette width s(i) is defined as follows:
Put a(i) = average dissimilarity between i and all other points of the cluster to which i belongs (if i is the only observation in its cluster, s(i) := 0 without further calculations). For all other clusters C, put d(i,C) = average dissimilarity of i to all observations of C. The smallest of these d(i,C) is b(i) := \min_C d(i,C), and can be seen as the dissimilarity between i and its “neighbor” cluster, i.e., the nearest one to which it does not belong. Finally,

s(i) := \frac{b(i) - a(i) }{max(a(i), b(i))}.

silhouette.default() is now based on C code donated by Romain Francois (the R version being still available as cluster:::silhouetteR).

Observations with a large s(i) (almost 1) are very well clustered, a small s(i) (around 0) means that the observation lies between two clusters, and observations with a negative s(i) are probably placed in the wrong cluster.

Value

silhouette() returns an object, sil, of class silhouette which is an n \times 3 matrix with attributes. For each observation i, sil[i,] contains the cluster to which i belongs as well as the neighbor cluster of i (the cluster, not containing i, for which the average dissimilarity between its observations and i is minimal), and the silhouette width s(i) of the observation. The colnames correspondingly are c("cluster", "neighbor", "sil_width").

summary(sil) returns an object of class summary.silhouette, a list with components

si.summary:

numerical summary of the individual silhouette widths s(i).

clus.avg.widths:

numeric (rank 1) array of clusterwise means of silhouette widths where mean = FUN is used.

avg.width:

the total mean FUN(s) where s are the individual silhouette widths.

clus.sizes:

table of the k cluster sizes.

call:

if available, the call creating sil.

Ordered:

logical identical to attr(sil, "Ordered"), see below.

sortSilhouette(sil) orders the rows of sil as in the silhouette plot, by cluster (increasingly) and decreasing silhouette width s(i).
attr(sil, "Ordered") is a logical indicating if sil is ordered as by sortSilhouette(). In that case, rownames(sil) will contain case labels or numbers, and
attr(sil, "iOrd") the ordering index vector.

Note

While silhouette() is intrinsic to the partition clusterings, and hence has a (trivial) method for these, it is straightforward to get silhouettes from hierarchical clusterings from silhouette.default() with cutree() and distance as input.

By default, for clara() partitions, the silhouette is just for the best random subset used. Use full = TRUE to compute (and later possibly plot) the full silhouette.

References

Rousseeuw, P.J. (1987) Silhouettes: A graphical aid to the interpretation and validation of cluster analysis. J. Comput. Appl. Math., 20, 53–65.

chapter 2 of Kaufman and Rousseeuw (1990), see the references in plot.agnes.

See Also

partition.object, plot.partition.

Examples

data(ruspini)
pr4 <- pam(ruspini, 4)
str(si <- silhouette(pr4))
(ssi <- summary(si))
plot(si) # silhouette plot
plot(si, col = c("red", "green", "blue", "purple"))# with cluster-wise coloring

si2 <- silhouette(pr4$clustering, dist(ruspini, "canberra"))
summary(si2) # has small values: "canberra"'s fault
plot(si2, nmax= 80, cex.names=0.6)

op <- par(mfrow= c(3,2), oma= c(0,0, 3, 0),
          mgp= c(1.6,.8,0), mar= .1+c(4,2,2,2))
for(k in 2:6)
   plot(silhouette(pam(ruspini, k=k)), main = paste("k = ",k), do.n.k=FALSE)
mtext("PAM(Ruspini) as in Kaufman & Rousseeuw, p.101",
      outer = TRUE, font = par("font.main"), cex = par("cex.main")); frame()

## the same with cluster-wise colours:
c6 <- c("tomato", "forest green", "dark blue", "purple2", "goldenrod4", "gray20")
for(k in 2:6)
   plot(silhouette(pam(ruspini, k=k)), main = paste("k = ",k), do.n.k=FALSE,
        col = c6[1:k])
par(op)

## clara(): standard silhouette is just for the best random subset
data(xclara)
set.seed(7)
str(xc1k <- xclara[ sample(nrow(xclara), size = 1000) ,]) # rownames == indices
cl3 <- clara(xc1k, 3)
plot(silhouette(cl3))# only of the "best" subset of 46
## The full silhouette: internally needs large (36 MB) dist object:
sf <- silhouette(cl3, full = TRUE) ## this is the same as
s.full <- silhouette(cl3$clustering, daisy(xc1k))
stopifnot(all.equal(sf, s.full, check.attributes = FALSE, tolerance = 0))
## color dependent on original "3 groups of each 1000": % __FIXME ??__
plot(sf, col = 2+ as.integer(names(cl3$clustering) ) %/% 1000,
     main ="plot(silhouette(clara(.), full = TRUE))")

## Silhouette for a hierarchical clustering:
ar <- agnes(ruspini)
si3 <- silhouette(cutree(ar, k = 5), # k = 4 gave the same as pam() above
    	           daisy(ruspini))
stopifnot(is.data.frame(di3 <- as.data.frame(si3)))
plot(si3, nmax = 80, cex.names = 0.5)
## 2 groups: Agnes() wasn't too good:
si4 <- silhouette(cutree(ar, k = 2), daisy(ruspini))
plot(si4, nmax = 80, cex.names = 0.5)

Sample Size of Dissimilarity Like Object

Description

Returns the number of observations (sample size) corresponding to a dissimilarity like object, or equivalently, the number of rows or columns of a matrix when only the lower or upper triangular part (without diagonal) is given.

It is nothing else but the inverse function of f(n) = n(n-1)/2.

Usage

sizeDiss(d)

Arguments

Value

a number; n if length(d) == n(n-1)/2, NA otherwise.

See Also

dissimilarity.object and also as.dist for class dissimilarity and dist objects which have a Size attribute.

Examples

sizeDiss(1:10)# 5, since 10 == 5 * (5 - 1) / 2
sizeDiss(1:9) # NA

n <- 1:100
stopifnot(n == sapply( n*(n-1)/2, function(n) sizeDiss(logical(n))))

Summary Method for 'agnes' Objects

Description

Returns (and prints) a summary list for an agnes object. Printing gives more output than the corresponding print.agnes method.

Usage

## S3 method for class 'agnes'
summary(object, ...)
## S3 method for class 'summary.agnes'
print(x, ...)

Arguments

See Also

agnes, agnes.object.

Examples

data(agriculture)
summary(agnes(agriculture))

Summary Method for 'clara' Objects

Description

Returns (and prints) a summary list for a clara object. Printing gives more output than the corresponding print.clara method.

Usage

## S3 method for class 'clara'
summary(object, ...)
## S3 method for class 'summary.clara'
print(x, ...)

Arguments

See Also

clara.object

Examples

## generate 2000 objects, divided into 5 clusters.
set.seed(47)
x <- rbind(cbind(rnorm(400, 0,4), rnorm(400, 0,4)),
           cbind(rnorm(400,10,8), rnorm(400,40,6)),
           cbind(rnorm(400,30,4), rnorm(400, 0,4)),
           cbind(rnorm(400,40,4), rnorm(400,20,2)),
           cbind(rnorm(400,50,4), rnorm(400,50,4))
)
clx5 <- clara(x, 5)
## Mis'classification' table:
table(rep(1:5, rep(400,5)), clx5$clustering) # -> 1 "error"
summary(clx5)

## Graphically:
par(mfrow = c(3,1), mgp = c(1.5, 0.6, 0), mar = par("mar") - c(0,0,2,0))
plot(x, col = rep(2:6, rep(400,5)))
plot(clx5)

Summary Method for 'diana' Objects

Description

Returns (and prints) a summary list for a diana object.

Usage

## S3 method for class 'diana'
summary(object, ...)
## S3 method for class 'summary.diana'
print(x, ...)

Arguments

See Also

diana, diana.object.

Summary Method for 'mona' Objects

Description

Returns (and prints) a summary list for a mona object.

Usage

## S3 method for class 'mona'
summary(object, ...)
## S3 method for class 'summary.mona'
print(x, ...)

Arguments

See Also

mona, mona.object.

Summary Method for PAM Objects

Description

Summarize a pam object and return an object of class summary.pam. There's a print method for the latter.

Usage

## S3 method for class 'pam'
summary(object, ...)
## S3 method for class 'summary.pam'
print(x, ...)

Arguments

See Also

pam, pam.object.

Hierarchical Clustering Object

Description

The objects of class "twins" represent an agglomerative or divisive (polythetic) hierarchical clustering of a dataset.

Value

See agnes.object and diana.object for details.

GENERATION

This class of objects is returned from agnes or diana.

METHODS

The "twins" class has a method for the following generic function: pltree.

INHERITANCE

The following classes inherit from class "twins" : "agnes" and "diana".

See Also

agnes,diana.

Compute the Volume (of an Ellipsoid)

Description

Compute the volume of geometric R object. This is a generic function and has a method for ellipsoid objects (typically resulting from ellipsoidhull().

Usage

volume(object, ...)
## S3 method for class 'ellipsoid'
volume(object, log = FALSE, ...)

Arguments

Value

a number, the volume V (or \log(V) if log = TRUE) of the given object.

Author(s)

Martin Maechler (2002, extracting from former clusplot code); Keefe Murphy (2019) provided code for dimensions d > 2.

See Also

ellipsoidhull for spanning ellipsoid computation.

Examples

## example(ellipsoidhull) # which defines 'ellipsoid' object <namefoo>

myEl <- structure(list(cov = rbind(c(3,1),1:2), loc = c(0,0), d2 = 10),
                   class = "ellipsoid")
volume(myEl)# i.e. "area" here (d = 2)
myEl # also mentions the "volume"

set.seed(1)
d5 <- matrix(rt(500, df=3), 100,5)
e5 <- ellipsoidhull(d5)

Votes for Republican Candidate in Presidential Elections

Description

A data frame with the percents of votes given to the republican candidate in presidential elections from 1856 to 1976. Rows represent the 50 states, and columns the 31 elections.

Usage

data(votes.repub)

Source

S. Peterson (1973): A Statistical History of the American Presidential Elections. New York: Frederick Ungar Publishing Co.

Data from 1964 to 1976 is from R. M. Scammon, American Votes 12, Congressional Quarterly.

Bivariate Data Set with 3 Clusters

Description

An artificial data set consisting of 3000 points in 3 quite well-separated clusters.

Usage

data(xclara)

Format

A data frame with 3000 observations on 2 numeric variables (named V1 and V2) giving the x and y coordinates of the points, respectively.

Note

Our version of the xclara is slightly more rounded than the one from read.table("xclara.dat") and the relative difference measured by all.equal is 1.15e-7 for V1 and 1.17e-7 for V2 which suggests that our version has been the result of a options(digits = 7) formatting.

Previously (before May 2017), it was claimed the three cluster were each of size 1000, which is clearly wrong. pam(*, 3) gives cluster sizes of 899, 1149, and 952, which apart from seven “outliers” (or “mislabellings”) correspond to observation indices \{1:900\}, \{901:2050\}, and \{2051:3000\}, see the example.

Source

Sample data set accompanying the reference below (file ‘xclara.dat’ in side ‘clus_examples.tar.gz’).

References

Anja Struyf, Mia Hubert & Peter J. Rousseeuw (1996) Clustering in an Object-Oriented Environment. Journal of Statistical Software 1. doi:10.18637/jss.v001.i04

Examples

## Visualization: Assuming groups are defined as {1:1000}, {1001:2000}, {2001:3000}
plot(xclara, cex = 3/4, col = rep(1:3, each=1000))
p.ID <- c(78, 1411, 2535) ## PAM's medoid indices  == pam(xclara, 3)$id.med
text(xclara[p.ID,], labels = 1:3, cex=2, col=1:3)

 px <- pam(xclara, 3) ## takes ~2 seconds
 cxcl <- px$clustering ; iCl <- split(seq_along(cxcl), cxcl)
 boxplot(iCl, range = 0.7, horizontal=TRUE,
         main = "Indices of the 3 clusters of  pam(xclara, 3)")

 ## Look more closely now:
 bxCl <- boxplot(iCl, range = 0.7, plot=FALSE)
 ## We see 3 + 2 + 2 = 7  clear "outlier"s  or "wrong group" observations:
 with(bxCl, rbind(out, group))
 ## out   1038 1451 1610   30  327  562  770
 ## group    1    1    1    2    2    3    3
 ## Apart from these, what are the robust ranges of indices? -- Robust range:
 t(iR <- bxCl$stats[c(1,5),])
 ##    1  900
 ##  901 2050
 ## 2051 3000
 gc <- adjustcolor("gray20",1/2)
 abline(v = iR, col = gc, lty=3)
 axis(3, at = c(0, iR[2,]), padj = 1.2, col=gc, col.axis=gc)