cluster (version 2.0.4)

clusGap: 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 distribution (a uniform distribution on the hypercube determined by the ranges of the first centered, and then svd-rotated x).

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

Arguments

x

numeric matrix or data.frame.

FUNcluster

a function which accepts as first argument a (data) matrix like x, second argument, say $k, k >= 2$, the number of clusters desired, and returns a list with a component named (or shortened to) cluster which is a vector of length n = nrow(x) of integers in 1:k determining the clustering or grouping of the n observations.

K.max

the maximum number of clusters to consider, must be at least two.

B

integer, number of Monte Carlo (“bootstrap”) samples.

d.power

a positive integer specifying the power $p$ which is applied to the euclidean distances (dist) before they are summed up to give $W(k)$. The default, d.power = 1, corresponds to the “historical” R implementation, whereas d.power = 2 corresponds to what Tibshirani et al had proposed. This was found by Juan Gonzalez, in 2016-02.

verbose

integer or logical, determining if “progress” output should be printed. The default prints one bit per bootstrap sample.

...

optionally further arguments for FUNcluster(), see kmeans example below.

f

numeric vector of ‘function values’, of length $K$, whose (“1 SE respected”) maximum we want.

SE.f

numeric vector of length $K$ of standard errors of f.

method

character string indicating how the “optimal” number of clusters, $k^$, is computed from the gap statistics (and their standard deviations), or more generally how the location $k^$ of the maximum of $f[k]$ should be determined.

"globalmax":

simply corresponds to the global maximum, i.e., is which.max(f)

"firstmax":

gives the location of the first local maximum.

"Tibs2001SEmax":

uses the criterion, Tibshirani et al (2001) proposed: “the smallest $k$ such that $f(k) \ge f(k+1) - s_{k+1}$”. Note that this chooses $k = 1$ when all standard deviations are larger than the differences $f(k+1) - f(k)$.

"firstSEmax":

location of the first $f()$ value which is not larger than the first local maximum minus SE.factor * SE.f[], i.e, within an “f S.E.” range of that maximum (see also SE.factor).

This, the default, has been proposed by Martin Maechler in 2012, when adding clusGap() to the cluster package, after having seen the "globalSEmax" proposal (in code) and read the "Tibs2001SEmax" proposal.

"globalSEmax":

(used in Dudoit and Fridlyand (2002), supposedly following Tibshirani's proposition): location of the first $f()$ value which is not larger than the global maximum minus SE.factor * SE.f[], i.e, within an “f S.E.” range of that maximum (see also SE.factor).

See the examples for a comparison in a simple case.

SE.factor

[When method contains "SE"] Determining the optimal number of clusters, Tibshirani et al. proposed the “1 S.E.”-rule. Using an SE.factor $f$, the “f S.E.”-rule is used, more generally.

Value

an object of S3 class "clusGap", basically a list with components

Tab

a matrix with K.max rows and 4 columns, named "logW", "E.logW", "gap", and "SE.sim", where gap = E.logW - logW, and SE.sim corresponds to the standard error of gap, SE.sim[k]=$s[k]$, where $s[k] := sqrt(1 + 1/B) sd^*(gap[])$, and $sd^*()$ is the standard deviation of the simulated (“bootstrapped”) gap values.

n

number of observations, i.e., nrow(x).

B

input B

FUNcluster

input function FUNcluster

Details

The main result $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.

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.

Per Broberg (2006). SAGx: Statistical Analysis of the GeneChip. R package version 1.9.7. http://home.swipnet.se/pibroberg/expression_hemsida1.html

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))

doExtras <- cluster:::doExtras()
## or set it explicitly to TRUE for the following
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, FUN = kmeans, nstart = 20, K.max = 8, B = 60)
gskmn #-> its print() method
plot(gskmn, main = "clusGap(., FUN = kmeans, n.start=20, B= 60)")
set.seed(12); system.time(
  gsPam0 <- clusGap(x, FUN = pam, K.max = 8, B = 60)
)
set.seed(12); system.time(
  gsPam1 <- clusGap(x, FUN = pam1, K.max = 8, B = 60)
)
## and show that it gives the same:
stopifnot(identical(gsPam1[-4], gsPam0[-4]))
gsPam1
print(gsPam1, method="globalSEmax")
print(gsPam1, method="globalmax")
}

gs.pam.RU <- clusGap(ruspini, FUN = 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, FUN = pam1, K.max = 8, B = 200)
plot(gsP.Z)
gsP.Z