Table 2 Validation metrics.

Results of validation metrics [RMSSTD (34), RS (35), determinant ratio index (62), and SD validity index (35)] applied to the data from Fig. 4. The validation metrics indicate the type of best score (max or min). The bold values represent the best score for each structure and validation metric.

CompactnessConnectednessSeparation
Validation metricRMMSTDRSDeterminant ratio indexSD validity index
AlgorithmStructureMinMaxMinMin
K-meansNo structure0.380.234.046.67
WardNo structure0.400.182.587.56
DBSCANNo structure0.440.001.00N/A
Mixture modelsNo structure0.380.234.096.67
K-meansSpherical clusters0.780.81249.080.32
WardSpherical clusters1.180.5626.430.18
DBSCANSpherical clusters0.580.861683.532.52
Mixture modelsSpherical clusters0.780.81249.080.32
K-meansLong parallel clusters0.890.272.770.79
WardLong parallel clusters0.930.212.100.75
DBSCANLong parallel clusters0.840.2122.364.32
Mixture modelsLong parallel clusters0.890.272.790.79
K-meansHalf-moons0.550.354.231.77
WardHalf-moons0.570.303.771.98
DBSCANHalf-moons0.600.213.032.61
Mixture modelsHalf-moons0.560.335.211.93
K-meansNested circles0.520.172.714.04
WardNested circles0.530.131.973.30
DBSCANNested circles0.570.001.004059.71
Mixture modelsNested circles0.520.172.713.99