Calculate Fukuyama and Sugeno index of clustering quality
calcFukuyamaSugeno(data, belongmatrix, centers, m)The original dataframe used for the clustering (n*p)
A membership matrix (n*k)
The centres of the clusters
The fuzziness parameter
A float: the Fukuyama and Sugeno index
The Fukuyama and Sugeno index (Fukuyama 1989) is the difference between the compacity of clusters and the separation of clusters. A smaller value indicates a better clustering. The formula is:
$$S(c)=\sum_{k=1}^{n} \sum_{i=1}^{c}\left(U_{i k}\right)^{m}\left(\left\|x_{k}-v_{i}\right\|^{2}-\left\|v_{i}-\bar{x}\right\|^{2}\right) 2$$
with n the number of observations, k the number of clusters and \(\bar{x}\) the mean of the dataset.
Fukuyama Y (1989). “A new method of choosing the number of clusters for the fuzzy c-mean method.” In Proc. 5th Fuzzy Syst. Symp., 1989, 247--250.
data(LyonIris)
AnalysisFields <-c("Lden","NO2","PM25","VegHautPrt","Pct0_14","Pct_65","Pct_Img",
"TxChom1564","Pct_brevet","NivVieMed")
dataset <- sf::st_drop_geometry(LyonIris[AnalysisFields])
queen <- spdep::poly2nb(LyonIris,queen=TRUE)
Wqueen <- spdep::nb2listw(queen,style="W")
result <- SFCMeans(dataset, Wqueen,k = 5, m = 1.5, alpha = 1.5, standardize = TRUE)
calcFukuyamaSugeno(result$Data,result$Belongings, result$Centers, 1.5)