¿Where can I find the references that describes the methods used in n_clusters?

[luifrancgom]

When using n_clusters in the documentation it is mention that " ... The main n_clusters function proposes to run all of them, and find out the number of clusters that is suggested by the majority of methods ..." But there is not a description about what methods are used.

Using the example pointed out you can extract all the methods used in the following way:

library(parameters)
library(tidyverse)

data <- iris |> 
  as_tibble() |> 
  select(-Species)
n <- n_clusters(data, package = c("easystats", "NbClust", "mclust"))
n |> 
  as_tibble()
#> # A tibble: 29 × 4
#>    n_Clusters Method           Package   Duration
#>         <dbl> <chr>            <chr>        <dbl>
#>  1          2 Elbow            easystats   0.783 
#>  2          2 Silhouette       easystats   0.115 
#>  3          2 Gap_Maechler2012 easystats   1.29  
#>  4          2 Gap_Dudoit2002   easystats   1.10  
#>  5          2 Ch               NbClust     0.0521
#>  6          2 DB               NbClust     0.0367
#>  7          2 Duda             NbClust     0.0644
#>  8          2 Pseudot2         NbClust     0.0465
#>  9          2 Beale            NbClust     0.0458
#> 10          2 Ratkowsky        NbClust     0.0383
#> # ℹ 19 more rows

^{Created on 2024-06-23 with reprex v2.1.0}

However, there is not a direct reference for each method. For example in the case of Elbow I was curious about how this method was implemented because Elbow method is a visual heuristic without a strict mathematical formula.

Checking the source code of n_clusters this is the way it is calculated:

library(factoextra)

data <- scale(iris[, -5])
out <- factoextra::fviz_nbclust(data, stats::kmeans, method = "wss", 
                         k.max = 10, verbose = FALSE)$data
names(out) <- c("n_Clusters", "WSS")
gradient <- c(0, diff(out$WSS))
optim <- out$n_Clusters[which.min(gradient)]
optim
#> [1] 2
#> Levels: 1 2 3 4 5 6 7 8 9 10

^{Created on 2024-06-23 with reprex v2.1.0}

However, in factoextra the following plot is pointed out:

And in the fviz_nbclust documentation the following reference is pointed out but there is not a description about a strict mathematical formula.

I understand in the case of Elbow the logic used in n_clusters but without a description in the function documentation it is difficult to understand the method without checking the source code.

¿Where can I find the references for the rest of the 28 methods used in n_clusters without checking the source code?