fuzzyconcordance {ConsRankClass} R Documentation

## Normalized Degree of Concordance (NDC) and Adjusted Concordance Index (ACI)

### Description

Given two fuzzy (Ruspini) partitions, it compute the NDC and the ACI. NDC is the fuzzy version of the Rand Index, as well as ACI is the fuzzy version of the Adjusted Rand Index

### Usage

fuzzyconcordance(P, Q, nperms = 1000)


### Arguments

 P A fuzzy partition. It has to be a matrix with n rows and k columns. Each column is expression of the degree of membership of the i-th row over the k partitions (see details). Q A fuzzy partition. It has to be a matrix with n rows and h columns. Each column is expression of the degree of membership of the i-th row over the h partitions (see details). nperms number of permutations necessary to compute ACI. Default: 1000

### Details

Both P and Q, or only one of those, can be crisp (or hard) partitions. In this case, each row must contain either 0 or 1, and the sum of the i-th row must be 1. In other words, either P or Q (or both) are expressed in terms of dummy coding. If both partitions are crisp, then NDC is equal to Rand Index and ACI is equal to Adjusted Rand Index. This function can be used to externally validate the output of any fuzzy clustering method

### Value

A list containing:

 ACI the Adjusted Concordance Index NDC the Normalized Degree of Concordance

### Author(s)

Antonio D'Ambrosio antdambr@unina.it

### References

D’Ambrosio, A., Amodio, S., Iorio, C., Pandolfo, G. and Siciliano, R. (2021). Adjusted Concordance Index: an Extension of the Adjusted Rand Index to Fuzzy Partitions. Journal of Classification vol. 38(1), pp. 112–128 (2021). DOI: 10.1007/s00357-020-09367-0

Hullermeier, E., Rifqi, M., Henzgen, S., and Senge, R. (2012). Comparing fuzzy partitions: a generalization of the Rand index and related measures. IEEE Transactions on Fuzzy Systems, 20(3), 546–556. DOI: 10.1109/TFUZZ.2011.2179303

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### Examples

#two random fuzzy partitions
P = rbind(c(0.5259,  0.1656,    0.3085),
c(0.5623,    0.1036,    0.3341),
c(0.2508,    0.1849,    0.5643),
c(0.5654,    0.1934,    0.2413),
c(0.4529,    0.1679,    0.3792),
c(0.2390,    0.1758,    0.5852),
c(0.3114,    0.1743,    0.5143),
c(0.4188,    0.1392,    0.4420),
c(0.5830,    0.1655,    0.2514),
c(0.5860,    0.1171,    0.2969),
c(0.2630,    0.1706,    0.5664),
c(0.5882,    0.1032,    0.3086),
c(0.5829,    0.1277,    0.2894),
c(0.3942,    0.1046,    0.5012),
c(0.5201,    0.1097,    0.3702),
c(0.2568,    0.1823,    0.5609),
c(0.3687,    0.1695,    0.4618),
c(0.5663,    0.1317,    0.3020),
c(0.5169,    0.1950,    0.2881),
c(0.5838,    0.1034,    0.3128))

Q = rbind(c(0.4494,    0.3755,    0.1751),
c(0.5219,    0.3526,    0.1255),
c(0.3432,    0.5062,    0.1506),
c(0.3120,    0.5181,    0.1699),
c(0.5362,    0.2747,    0.1891),
c(0.4082,    0.3959,    0.1959),
c(0.4670,    0.3782,    0.1547),
c(0.4276,    0.4585,    0.1139),
c(0.4013,    0.4837,    0.1149),
c(0.3724,    0.5019,    0.1258),
c(0.5055,    0.3104,    0.1841),
c(0.4027,    0.4719,    0.1254),
c(0.3565,    0.4620,    0.1814),
c(0.6106,    0.2650,    0.1244),
c(0.5595,    0.2476,    0.1929),
c(0.4657,    0.3993,    0.1350),
c(0.2964,    0.5839,    0.1197),
c(0.5387,    0.3362,    0.1251),
c(0.4043,    0.4341,    0.1616),
c(0.5631,    0.2895,    0.1473))

ci <- fuzzyconcordance(P,Q)

#generate a random fuzzy partition with two components (clusters)
Q2 <- matrix(runif(20),ncol=1)
Q2 <- cbind(Q2,1-Q2)

ci2 <- fuzzyconcordance(P,Q2)

#generate a random crisp partition
P2 <- t(rmultinom(20,1,c(0.3,0.3,0.4)))

ci3 <- fuzzyconcordance(P2,Q)
#--------------------
## Not run:
# install.packages("Rankcluster")
library("Rankcluster") # model-based clustering algorithm for
#  ranking data by Biernacki and Jacques (2013)
#  <doi:10.1016/j.csda.2012.08.008>
data(APA)
set.seed(136) #for reproducibility
rcres <- rankclust(APA$data,K=3) # solution with 3 centers, it takes about 75 seconds ## ccares <- cca(APA$data,k=3) #solution with 3 components, it takes about 7 seconds
##
ci <- fuzzyconcordance(rcres[3]@tik,ccares$pk) ci$ACI  # 0.0226 means that the two partitions are similar (see NDC below),
# but their similarity is mainly due to chance
ci\$NDC

## End(Not run)



[Package ConsRankClass version 1.0.1 Index]