A block (or co-) clustering partitions more than one mode of a data array at once, for example the rows and the columns of a matrix. Each mode gets its own partition, and each partition can be scored with a silhouette. extSilhouette() combines the mode-wise results into the extended silhouette of Schepers et al. (2008; see also Bhat Kapu and Kiruthika 2025). If mode \(m\) has \(n_m\) objects and average width \(\bar S_m\), \[ \mathrm{ExS} = \frac{\sum_m n_m \bar S_m}{\sum_m n_m}. \] The weights are the numbers of objects, so ExS is the average width over all clustered objects, each counting once, and a mode with many more objects than another dominates the combined value. extSilhouette(weights = "equal") gives every mode the same weight instead, and a numeric vector sets the weights directly. This vignette shows the workflow and what the weighting means in practice.

library(Silhouette)
library(drclust)

Data with a block structure

We simulate a \(150 \times 24\) matrix whose rows fall into four groups and whose columns fall into three, so that each of the 12 blocks has its own mean.

set.seed(2025)
n <- 150; p <- 24
row_group <- sample(1:4, n, replace = TRUE)
col_group <- sample(1:3, p, replace = TRUE)
block_means <- matrix(c( 2, -1,  0,
                        -2,  1,  1,
                         0,  2, -2,
                         1, -2,  2), nrow = 4, byrow = TRUE)
Xb <- scale(1.5 * block_means[row_group, col_group] + matrix(rnorm(n * p), n, p))

Silhouettes for a double k-means fit

Double k-means (Vichi 2001), implemented in doublekm() of drclust (Prunila and Vichi 2025), approximates the data matrix by \(U C V^\top\), where \(U\) and \(V\) are binary membership matrices for rows and columns and \(C\) holds the block means. The prototype of row cluster \(k\) is row \(k\) of \(C V^\top\), and the prototype of column cluster \(q\) is column \(q\) of \(U C\). Each mode therefore has a matrix of distances to its prototypes, which Silhouette() accepts directly. We recompute \(C\) by least squares from \(U\) and \(V\) so that the prototypes match the fitted memberships exactly.

block_sil <- function(Xs, K, Q, seed) {
  set.seed(seed)
  fit <- doublekm(Xs, K = K, Q = Q, Rndstart = 10, prep = 0)
  U <- fit$U
  V <- fit$V
  C <- solve(crossprod(U), t(U) %*% Xs %*% V) %*% solve(crossprod(V))
  list(rows = Silhouette(proxy::dist(Xs, C %*% t(V))),       # n x K distances
       columns = Silhouette(proxy::dist(t(Xs), t(U %*% C))))  # p x Q distances
}
modes <- block_sil(Xb, K = 4, Q = 3, seed = 1)
extSilhouette(modes, dim_names = names(modes))
#> ------------------------------------------
#> Extended silhouette (size weights): 0.6464
#> ------------------------------------------
#> Dimension summary:
#>  dimension n_obs weight avg_sil_width
#>       rows   150 0.8621        0.6323
#>    columns    24 0.1379        0.7348

The result holds the extended silhouette and a table of the number of objects, the share of the weight, and the average width for each mode.

Choosing the numbers of row and column clusters

We fit a grid of \((K, Q)\) and record the extended silhouette together with the mode-wise averages.

grid <- expand.grid(K = 2:5, Q = 2:4)
fits <- lapply(seq_len(nrow(grid)),
               function(g) block_sil(Xb, grid$K[g], grid$Q[g], seed = g))
ext <- lapply(fits, extSilhouette)
grid$ExS <- sapply(ext, `[[`, "ext_sil_width")
grid$rows <- sapply(ext, function(e) e$dim_table$avg_sil_width[1])
grid$columns <- sapply(ext, function(e) e$dim_table$avg_sil_width[2])
grid[order(-grid$ExS), ][1:4, ]
#>    K Q       ExS      rows   columns
#> 6  3 3 0.6608322 0.6639725 0.6412055
#> 7  4 3 0.6464243 0.6322869 0.7347831
#> 10 3 4 0.6399489 0.6641385 0.4887635
#> 11 4 4 0.6189673 0.6326928 0.5331826

Three heat maps of average width over K from 2 to 5 row clusters and Q from 2 to 4 column clusters, one each for rows, columns and the extended silhouette. The row averages barely depend on Q; the column averages are highest at Q = 3; the extended silhouette follows the row panel closely.

The extended silhouette is largest at \((K, Q) = (3, 3)\), while the data have four row groups and three column groups. The reason is the weighting: with 150 rows and 24 columns, the rows carry 86% of the weight, so the extended silhouette largely repeats the row averages, which barely depend on \(Q\) and here prefer fewer row clusters. The column averages carry the information about the column partition but can only tip close decisions.

When the modes differ greatly in size, report the mode-wise averages alongside the extended silhouette, and consider how much weight each mode should get. In this example the equally weighted extended silhouette, extSilhouette(..., weights = "equal"), is largest at \((K, Q) = (4, 3)\):

grid$equal <- sapply(fits, function(f) extSilhouette(f, weights = "equal")$ext_sil_width)
grid[order(-grid$equal), ][1:3, c("K", "Q", "rows", "columns", "equal")]
#>   K Q      rows   columns     equal
#> 7 4 3 0.6322869 0.7347831 0.6835350
#> 6 3 3 0.6639725 0.6412055 0.6525890
#> 8 5 3 0.5183600 0.7363131 0.6273365

Equal weighting is not generally better; it simply answers a different question, one in which a column is as important as a row.

Soft block clustering

Block clustering methods that return membership probabilities for rows and columns, such as block probabilistic distance clustering (Bhat Kapu and Kiruthika 2025), fit the same workflow: compute a "Silhouette" object for each mode with softSilhouette() (or with Silhouette() on each mode’s distances), and pass the list to extSilhouette(). Use the same width and averaging method for every mode, so that the mode-wise averages are on the same scale.

Session information

sessionInfo()
#> R version 4.6.2 (2026-10-29)
#> Platform: x86_64-pc-linux-gnu
#> Running under: Ubuntu 26.04.1 LTS
#> 
#> Matrix products: default
#> BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
#> LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.32.so;  LAPACK version 3.12.0
#> 
#> locale:
#>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
#>  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
#>  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
#>  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
#>  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
#> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
#> 
#> time zone: Etc/UTC
#> tzcode source: system (glibc)
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] ggplot2_4.0.4    drclust_0.1.1    mclust_6.1.3     Silhouette_0.9.7
#> [5] rmarkdown_2.32  
#> 
#> loaded via a namespace (and not attached):
#>  [1] gtable_0.3.6           xfun_0.62              bslib_0.12.0          
#>  [4] ggrepel_0.9.8          rstatix_1.1.0          lattice_0.23-1        
#>  [7] vctrs_0.7.3            tools_4.6.2            generics_0.1.4        
#> [10] stats4_4.6.2           flexmix_2.3-21         parallel_4.6.2        
#> [13] tibble_3.3.1           proxy_0.4-29           DEoptimR_1.2-2        
#> [16] cluster_2.1.8.3        pkgconfig_2.0.3        pheatmap_1.0.13       
#> [19] RColorBrewer_1.1-3     S7_0.2.2               factoextra_2.2.0      
#> [22] lifecycle_1.0.5        compiler_4.6.2         farver_2.1.2          
#> [25] carData_3.0-6          htmltools_0.5.9        sys_3.4.3             
#> [28] buildtools_1.0.0       class_7.3-24           sass_0.4.10           
#> [31] yaml_2.3.12            Formula_1.2-6          pillar_1.11.1         
#> [34] car_3.1-5              ggpubr_1.0.0           jquerylib_0.1.4       
#> [37] tidyr_1.3.2            prabclus_2.3-5         MASS_7.3-66           
#> [40] diptest_0.77-2         cachem_1.1.0           fpc_2.2-15            
#> [43] abind_1.4-8            robustbase_0.99-7      tidyselect_1.2.1      
#> [46] digest_0.6.39          dplyr_1.2.1            purrr_1.2.2           
#> [49] kernlab_0.9-33         labeling_0.4.3         maketools_1.3.2       
#> [52] RcppArmadillo_15.6.2-1 fastmap_1.2.0          grid_4.6.2            
#> [55] cli_3.6.6              magrittr_2.0.5         broom_1.0.13          
#> [58] withr_3.0.3            scales_1.4.0           backports_1.5.1       
#> [61] otel_0.3.0             nnet_7.3-21            ggsignif_0.6.4        
#> [64] modeltools_0.2-25      evaluate_1.0.5         knitr_1.52            
#> [67] rlang_1.3.0            Rcpp_1.1.2             glue_1.8.1            
#> [70] jsonlite_2.0.0         R6_2.6.1

References

Bhat Kapu, Shrikrishna, and C. Kiruthika. 2025. Block Probabilistic Distance Clustering: A Unified Framework and Evaluation. Research Square preprint, version 1. https://doi.org/10.21203/rs.3.rs-6973596/v1.

Prunila, Ionel, and Maurizio Vichi. 2025. “drclust: An R Package for Simultaneous Clustering and Dimensionality Reduction.” The R Journal 17 (4): 104–33. https://doi.org/10.32614/RJ-2025-046.

Schepers, Jan, Eva Ceulemans, and Iven Van Mechelen. 2008. “Selecting Among Multi-Mode Partitioning Models of Different Complexities: A Comparison of Four Model Selection Criteria.” Journal of Classification 25 (1): 67–85. https://doi.org/10.1007/s00357-008-9005-9.

Vichi, Maurizio. 2001. “Double k-Means Clustering for Simultaneous Classification of Objects and Variables.” In Advances in Classification and Data Analysis. Springer. https://doi.org/10.1007/978-3-642-59471-7_6.