The "Elbow" Method for Clustering Evaluation

Description

Determining the number of clusters in a data set by the "elbow" rule.

Usage

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## find a good k given thresholds of EV and its increment.
elbow(x,inc.thres,ev.thres,precision=3,print.warning=TRUE)

## a wrapper of `elbow' testing multiple thresholds.
elbow.batch(x,inc.thres=c(0.01,0.05,0.1),
ev.thres=c(0.95,0.9,0.8,0.75,0.67,0.5,0.33),precision=3)

## S3 method for class 'elbow'
plot(x,elbow.obj=NULL,main,xlab="k",
ylab="Explained_Variance",type="b",pch=20,col.abline="red",
lty.abline=3,if.plot.new=TRUE,print.info=TRUE,
mar=c(4,5,3,3),omi=c(0.75,0,0,0),...)

Arguments

x

a ‘css.multi’ object, generated by css.hclust

inc.thres

numeric with value(s) from 0 to 1, the threshold of the increment of EV. A single value is used in elbow while a vector of values in elbow.batch.

ev.thres

numeric with value(s) from 0 to 1, the threshold of EV. A single value is used in elbow while a vector of values in elbow.batch.

precision

integer, the number of digits to round for numerical comparison.

print.warning

logical, whether to print warning messages.

elbow.obj

a ‘elbow’ object, generated by elbow or elbow.batch

main

an overall title for the plot.

ylab

a title for the y axis.

xlab

a title for the x axis.

type

what type of plot should be drawn.
See help("plot", package="graphics").

pch

Either an integer specifying a symbol or a single character to be used as the default in plotting points (see par).

col.abline

color for straight lines through the current plot (see option col in par).

lty.abline

line type for straight lines through the current plot (see option lty in par).

if.plot.new

logical, whether to start a new plot device or not.

print.info

logical, whether to print the information of ‘elbow.obj’.

mar

A numerical vector of the form 'c(bottom, left, top, right)' which gives the number of lines of margin to be specified on the four sides of the plot (see option mar in par). The default is 'c(4, 5, 3, 3) + 0.1'.

omi

A vector of the form 'c(bottom, left, top, right)' giving the size of the outer margins in inches (see option omi in par).

...

arguments to be passed to method plot.elbow, such as graphical parameters (see par).

Details

Determining the number of clusters in a data set by the "elbow" rule and thresholds in the explained variance (EV) and its increment.

Value

Both elbow and elbow.btach return a ‘elbow’ object (if a "good" k exists), which is a list containing the following components

k number of clusters
ev explained variance given k
inc.thres the threshold of the increment in EV
ev.thres the threshold of the EV

, and with an attribute ‘meta’ that contains

description A description about the "good" k

See Also

css and css.hclust for computing Clustering Sum-of-Squares.

Examples

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## load library
require("GMD")

## simulate data around 12 points in Euclidean space
pointv <- data.frame(x=c(1,2,2,4,4,5,5,6,7,8,9,9),
y=c(1,2,8,2,4,4,5,9,9,8,1,9))
set.seed(2012)
mydata <- c()
for (i in 1:nrow(pointv)){
  mydata <- rbind(mydata,cbind(rnorm(10,pointv[i,1],0.1),
rnorm(10,pointv[i,2],0.1)))
}
mydata <- data.frame(mydata); colnames(mydata) <- c("x","y")
plot(mydata,type="p",pch=21, main="Simulated data")

## determine a "good" k using elbow
dist.obj <- dist(mydata[,1:2])
hclust.obj <- hclust(dist.obj)
css.obj <- css.hclust(dist.obj,hclust.obj)
elbow.obj <- elbow.batch(css.obj)
print(elbow.obj)

## make partition given the "good" k
k <- elbow.obj$k; cutree.obj <- cutree(hclust.obj,k=k)
mydata$cluster <- cutree.obj

## draw a elbow plot and label the data
dev.new(width=12, height=6)
par(mfcol=c(1,2),mar=c(4,5,3,3),omi=c(0.75,0,0,0))
plot(mydata$x,mydata$y,pch=as.character(mydata$cluster),
col=mydata$cluster,cex=0.75,main="Clusters of simulated data")
plot(css.obj,elbow.obj,if.plot.new=FALSE)

## clustering with more relaxed thresholds (, resulting a smaller "good" k)
elbow.obj2 <- elbow.batch(css.obj,ev.thres=0.90,inc.thres=0.05)
mydata$cluster2 <- cutree(hclust.obj,k=elbow.obj2$k)

dev.new(width=12, height=6)
par(mfcol=c(1,2), mar=c(4,5,3,3),omi=c(0.75,0,0,0))
plot(mydata$x,mydata$y,pch=as.character(mydata$cluster2),
col=mydata$cluster2,cex=0.75,main="Clusters of simulated data")
plot(css.obj,elbow.obj2,if.plot.new=FALSE)