calcGD53: Generalized Dunn’s index (53)

In geocmeans: Implementing Methods for Spatial Fuzzy Unsupervised Classification

Generalized Dunn’s index (53)

Description

Calculate the Generalized Dunn’s index (v53) of clustering quality.

Usage

calcGD53(data, belongmatrix, centers)

Arguments

data	The original dataframe used for the clustering (n*p)
belongmatrix	A membership matrix (n*k)
centers	The centres of the clusters

Details

The Generalized Dunn’s index \insertCiteda2020incrementalgeocmeans is a ratio of the worst pair-wise separation of clusters and the worst compactness of clusters. A higher value indicates a better clustering. The formula is:

GD_{r s}=\frac{\min_{i \neq j}\left[\delta_{r}\left(\omega_{i}, \omega_{j}\right)\right]}{\max_{k}\left[\Delta_{s}\left(\omega_{k}\right)\right]}

The numerator is a measure of the minimal separation between all the clusters i and j given by the formula:

\delta_{r}\left(\omega_{i}, \omega_{j}\right)=\frac{\sum_{l=1}^{n}\left\|\boldsymbol{x_{l}}-\boldsymbol{c_{i}}\right\|^{\frac{1}{2}} . u_{il}+\sum_{l=1}^{n}\left\|\boldsymbol{x_{l}}-\boldsymbol{c_{j}}\right\|^{\frac{1}{2}} . u_{jl}}{\sum{u_{i}} + \sum{u_{j}}}

where u is the membership matrix and u_{i} is the column of u describing the membership of the n observations to cluster i. c_{i} is the center of the cluster i.

The denominator is a measure of the maximal dispersion of all clusters, given by the formula:

\frac{2*\sum_{l=1}^{n}\left\|\boldsymbol{x}_{l}-\boldsymbol{c_{i}}\right\|^{\frac{1}{2}}}{\sum{u_{i}}}

Value

A float: the Generalized Dunn’s index (53)

References

\insertAllCited

Examples

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)
calcGD53(result$Data, result$Belongings, result$Centers)

geocmeans documentation built on Sept. 12, 2023, 9:06 a.m.