cluster_validity: Internal Cluster Validity Measures#


Implementation of a number of so-called cluster validity indices critically reviewed in (Gagolewski, Bartoszuk, Cena, 2021). See Section 2 therein and (Gagolewski, 2022) for the respective definitions.

The greater the index value, the more valid (whatever that means) the assessed partition. For consistency, the Ball-Hall and Davies-Bouldin indexes as well as the within-cluster sum of squares (WCSS) take negative values.


calinski_harabasz_index(X, y)

  M = 25L,
  owa_numerator = "SMin:5",
  owa_denominator = "Const"

generalised_dunn_index(X, y, lowercase_d, uppercase_d)

negated_ball_hall_index(X, y)

negated_davies_bouldin_index(X, y)

negated_wcss_index(X, y)

silhouette_index(X, y)

silhouette_w_index(X, y)

wcnn_index(X, y, M = 25L)



numeric matrix with n rows and d columns, representing n points in a d-dimensional space


vector of n integer labels, representing a partition whose quality is to be assessed; y[i] is the cluster ID of the i-th point, X[i, ]; 1 <= y[i] <= K, where K is the number or clusters


number of nearest neighbours

owa_numerator, owa_denominator

single string specifying the OWA operators to use in the definition of the DuNN index; one of: "Mean", "Min", "Max", "Const", "SMin:D", "SMax:D", where D is an integer defining the degree of smoothness


an integer between 1 and 5, denoting \(d_1\), …, \(d_5\) in the definition of the generalised Dunn (Bezdek-Pal) index (numerator: min, max, and mean pairwise intracluster distance, distance between cluster centroids, weighted point-centroid distance, respectively)


an integer between 1 and 3, denoting \(D_1\), …, \(D_3\) in the definition of the generalised Dunn (Bezdek-Pal) index (denominator: max and min pairwise intracluster distance, average point-centroid distance, respectively)


A single numeric value (the more, the better).


Marek Gagolewski and other contributors


Ball G.H., Hall D.J., ISODATA: A novel method of data analysis and pattern classification, Technical report No. AD699616, Stanford Research Institute, 1965.

Bezdek J., Pal N., Some new indexes of cluster validity, IEEE Transactions on Systems, Man, and Cybernetics, Part B 28, 1998, 301-315, doi:10.1109/3477.678624.

Calinski T., Harabasz J., A dendrite method for cluster analysis, Communications in Statistics 3(1), 1974, 1-27, doi:10.1080/03610927408827101.

Davies D.L., Bouldin D.W., A Cluster Separation Measure, IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI-1 (2), 1979, 224-227, doi:10.1109/TPAMI.1979.4766909.

Dunn J.C., A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-Separated Clusters, Journal of Cybernetics 3(3), 1973, 32-57, doi:10.1080/01969727308546046.

Gagolewski M., Bartoszuk M., Cena A., Are cluster validity measures (in)valid?, Information Sciences 581, 620-636, 2021, doi:10.1016/j.ins.2021.10.004; preprint:

Gagolewski M., A Framework for Benchmarking Clustering Algorithms, 2022,

Rousseeuw P.J., Silhouettes: A Graphical Aid to the Interpretation and Validation of Cluster Analysis, Computational and Applied Mathematics 20, 1987, 53-65, doi:10.1016/0377-0427(87)90125-7.

See Also#

The official online manual of genieclust at

Gagolewski M., genieclust: Fast and robust hierarchical clustering, SoftwareX 15:100722, 2021, doi:10.1016/j.softx.2021.100722.


X <- as.matrix(iris[,1:4])
X[,] <- jitter(X)  # otherwise we get a non-unique solution
y <- as.integer(iris[[5]])
calinski_harabasz_index(X, y)  # good
## [1] 486.6681
calinski_harabasz_index(X, sample(1:3, nrow(X), replace=TRUE))  # bad
## [1] 2.836713