flexclust (version 1.3-4)

randIndex: Compare Partitions

Description

Compute the (adjusted) Rand, Jaccard and Fowlkes-Mallows index for agreement of two partitions.

Usage

comPart(x, y, type=c("ARI","RI","J","FM"))
"comPart"(x, y, type)
"comPart"(x, y, type)
"comPart"(x, y, type)
"comPart"(x, y, type)
randIndex(x, y, correct=TRUE, original=!correct)
"randIndex"(x, y, correct=TRUE, original=!correct)
"randIndex"(x, y, correct=TRUE, original=!correct)

Arguments

x

Either a 2-dimensional cross-tabulation of cluster assignments (for randIndex only), an object inheriting from class "flexclust", or an integer vector of cluster memberships.

y

An object inheriting from class "flexclust", or an integer vector of cluster memberships.

type

character vector of abbreviations of indices to compute.

correct, original

Logical, correct the Rand index for agreement by chance?

Value

Rand Index

Let $A$ denote the number of all pairs of data points which are either put into the same cluster by both partitions or put into different clusters by both partitions. Conversely, let $D$ denote the number of all pairs of data points that are put into one cluster in one partition, but into different clusters by the other partition. The partitions disagree for all pairs $D$ and agree for all pairs $A$. We can measure the agreement by the Rand index $A/(A+D)$ which is invariant with respect to permutations of cluster labels. The index has to be corrected for agreement by chance if the sizes of the clusters are not uniform (which is usually the case), or if there are many clusters, see Hubert \& Arabie (1985) for details.

Jaccard Index

If the number of clusters is very large, then usually the vast majority of pairs of points will not be in the same cluster. The Jaccard index tries to account for this by using only pairs of points that are in the same cluster in the defintion of $A$.

Fowlkes-Mallows

Let $A$ again be the pairs of points that are in the same cluster in both partitions. Fowlkes-Mallows divides this number by the geometric mean of the sums of the number of pairs in each cluster of the two partitions. This gives the probability that a pair of points which are in the same cluster in one partition are also in the same cluster in the other partition.

References

Lawrence Hubert and Phipps Arabie. Comparing partitions. Journal of Classification, 2, 193--218, 1985.

Marina Meila. Comparing clusterings - an axiomatic view. In Stefan Wrobel and Luc De Raedt, editors, Proceedings of the International Machine Learning Conference (ICML). ACM Press, 2005.

Examples

## no class correlations: corrected Rand almost zero
g1 <- sample(1:5, size=1000, replace=TRUE)
g2 <- sample(1:5, size=1000, replace=TRUE)
tab <- table(g1, g2)
randIndex(tab)

## uncorrected version will be large, because there are many points
## which are assigned to different clusters in both cases
randIndex(tab, correct=FALSE)
comPart(g1, g2)

## let pairs (g1=1,g2=1) and (g1=3,g2=3) agree better
k <- sample(1:1000, size=200)
g1[k] <- 1
g2[k] <- 1
k <- sample(1:1000, size=200)
g1[k] <- 3
g2[k] <- 3
tab <- table(g1, g2)

## the index should be larger than before
randIndex(tab, correct=TRUE, original=TRUE)
comPart(g1, g2)