`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.

```
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 clusGap
print(x, method = "firstSEmax", SE.factor = 1, …)

# S3 method for 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), …)

x

numeric matrix or `data.frame`

.

FUNcluster

a `function`

which accepts as first
argument a (data) matrix like `x`

, second argument, say
\(k, k\geq 2\), 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.

spaceH0

a `character`

string specifying the
space of the \(H_0\) distribution (of *no* cluster). Both
`"scaledPCA"`

and `"original"`

use a uniform distribution
in a hyper cube and had been mentioned in the reference;
`"original"`

been added after a proposal (including code) by
Juan Gonzalez.

verbose

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

…

(for `clusGap()`

:) 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, \(\hat k\), is computed from the gap statistics (and their standard deviations), or more generally how the location \(\hat k\) 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.

type, xlab, ylab, main

arguments with the same meaning as in
`plot.default()`

, with different default.

do.arrows

logical indicating if (1 SE -)“error bars”
should be drawn, via `arrows()`

.

arrowArgs

a list of arguments passed to `arrows()`

;
the default, notably `angle`

and `code`

, provide a style
matching usual error bars.

`clusGap(..)`

returns an object of S3 class `"clusGap"`

,
basically a list with components

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 \(s_k := \sqrt{1 + 1/B} sd^*(gap_j)\), and \(sd^*()\) is the standard deviation of the
simulated (“bootstrapped”) gap values.

the `clusGap(..)`

`call`

.

the `spaceH0`

argument (`match.arg()`

ed).

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

.

input `B`

input function `FUNcluster`

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.

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

`silhouette`

for a much simpler less sophisticated
goodness of clustering measure.

`cluster.stats()`

in package fpc for
alternative measures.

# NOT RUN { ### --- 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 could set it 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, 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": 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, FUN = hclusCut, K.max = 8, B = 60)) }# end {doExtras} 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") # } # NOT RUN { ## 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, main = "clusGap(<iid_rnorm_p=3>) ==> k = 1 cluster is optimal") gsP.Z # } # NOT RUN { <!-- %end{dont..} --> # }