geocmeans (version 0.3.4)

calcGD43: Generalized Dunn’s index (43)

Description

Calculate the Generalized Dunn’s index (v43) of clustering quality.

Usage

calcGD43(data, belongmatrix, centers)

Value

A float: the Generalized Dunn’s index (43)

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 da2020incrementalgeocmeans 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:

$$\delta_{r}\left(\omega_{i}, \omega_{j}\right)=\left\|\boldsymbol{c}_{i}-\boldsymbol{c}_{j}\right\|$$

which is basically the Euclidean distance between the centres of clusters \(c_{i}\) and \(c_{j}\)

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

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

References

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)
calcGD43(result$Data, result$Belongings, result$Centers)