Density-Based Silhouette Width (DBS) for Soft Clustering

Computes silhouette widths based on Menardi (2011) density-based method using log-ratios of posterior probabilities.

Usage

dbSilhouette(
  prob_matrix,
  average = c("median", "crisp", "fuzzy"),
  a = 2,
  sort = FALSE,
  print.summary = FALSE,
  clust_fun = NULL,
  ...
)

Arguments

prob_matrix

A numeric matrix of posterior probabilities where rows represent observations and columns represent clusters. Must sum to 1 by row. If clust_fun is provided, prob_matrix must be a string giving the name of the matrix component (e.g., "u").

average

Character string specifying the method for computing the average silhouette width. Options are:

"crisp" – unweighted (simple) average.
"fuzzy" – weighted average based on membership probability differences.
"median" – median silhouette width across observations.

Defaults to "median".

a

Numeric value controlling the fuzzifier or weight scaling in fuzzy silhouette averaging. Higher values increase the emphasis on strong membership differences. Must be positive. Defaults to 2.

sort

Logical; if TRUE, sorts the output widths data frame by cluster and descending silhouette width. Defaults to FALSE.

print.summary

Logical; if TRUE, prints a summary table of average silhouette widths and sizes for each cluster. Defaults to FALSE.

clust_fun

Optional S3 or S4 function object or function as character string specifying a clustering function that produces the proximity matrix. For example, fcm or "fcm". If provided, prob_matrix must be a string giving the name of the matrix component (e.g., "u"). Defaults to NULL.

...

Additional arguments passed to clust_fun, such as x, centers for fcm.

Value

A data frame of class "Silhouette" containing cluster assignments, nearest neighbor clusters, silhouette widths for each observation, and weights (for fuzzy clustering). The object includes the following attributes:

proximity_type: The proximity type used i.e., "similarity".
method: The silhouette calculation method used i.e., "db".
average: Character — the averaging method: "crisp", "fuzzy", or "median".

Details

Let the posterior probability matrix or cluster membership matrix as $$\Gamma = [\gamma_{ik}]_{n \times K},$$

The density-based silhouette width for observation $i$ is: $$ \mathrm{DBS}_i = \frac{\log\left( \frac{\gamma_{ik}}{\max_{k' \neq k} \gamma_{ik'}} \right)} {\max_{j=1,\dots,n} \left| \log\left( \frac{\gamma_{jk}}{\max_{k' \neq k} \gamma_{jk'}} \right) \right|} $$

#' If average = "crisp", the crisp silhouette index is calculated as ($CS$) is: $$ CS = \frac{1}{n} \sum_{i=1}^{n} S(x_i) $$ summarizing overall clustering quality.

If average = "fuzzy" and prob_matrix is provided, denoted as $\Gamma = [\gamma_{ik}]_{n \times K}$, with $\gamma_{ik}$ representing the probability of observation $i$ belonging to cluster $k$, the fuzzy silhouette index ($FS$) is calculated as: $$ FS = \frac{\sum_{i=1}^{n} w_i S(x_i) }{\sum_{i=1}^{n} w_i} $$ where $w_i = \sum_{i=1}^{n} \left( \gamma_{ik} - \max_{k' \neq k} \gamma_{ik'} \right)^{\alpha}$ is weight and $\alpha$ (the a argument) controls the emphasis on confident assignments.

If average = "median" then median Silhouette is Calculated

References

Menardi, G. (2011). Density-based silhouette diagnostics for clustering methods. Statistics and Computing, 21(3), 295–308. doi:10.1007/s11222-010-9169-0

Examples

# \donttest{
# Compare two soft clustering algorithms using dbSilhouette
# Example: FCM vs. FCM2 on iris data, using average silhouette width as a criterion
data(iris)
if (requireNamespace("ppclust", quietly = TRUE)) {
  fcm_result <- ppclust::fcm(iris[, 1:4], 3)
  out_fcm <- dbSilhouette(prob_matrix = fcm_result$u, print.summary = TRUE)
  plot(out_fcm)
  sfcm <- summary(out_fcm, print.summary = FALSE)
} else {
  message("Install 'ppclust' to run this example: install.packages('ppclust')")
}
#> ---------------------------------------
#> Median similarity db silhouette: 0.3098
#> ---------------------------------------
#> 
#>   cluster size avg.sil.width
#> 1       1   40        0.2456
#> 2       2   50        0.5074
#> 3       3   60        0.2225
#> 
#> Available attributes: names, class, row.names, proximity_type, method, average
if (requireNamespace("ppclust", quietly = TRUE)) {
  fcm2_result <- ppclust::fcm2(iris[, 1:4], 3)
  out_fcm2 <- dbSilhouette(prob_matrix = fcm2_result$u, print.summary = TRUE)
  plot(out_fcm2)
  sfcm2 <- summary(out_fcm2, print.summary = FALSE)
} else {
  message("Install 'ppclust' to run this example: install.packages('ppclust')")
}
#> ---------------------------------------
#> Median similarity db silhouette: 0.1089
#> ---------------------------------------
#> 
#>   cluster size avg.sil.width
#> 1       1   72        0.0195
#> 2       2    2        0.0575
#> 3       3   76        0.2517
#> 
#> Available attributes: names, class, row.names, proximity_type, method, average
# Compare average silhouette widths of fcm and fcm2
if (requireNamespace("ppclust", quietly = TRUE)) {
  cat("FCM average silhouette width:", sfcm$avg.width, "\n",
  "FCM2 average silhouette width:", sfcm2$avg.width, "\n")
}
#> FCM average silhouette width: 0.3097745 
#>  FCM2 average silhouette width: 0.1089476 
# }