I am clustering a mixed geological data set containing numeric (pump pressure, bit speed, mud temperature), nominal (presence or absence of a specific stones), and ordinal data (relative concentration of minerals with 0-absent, to 4-very abundant).
My candidates algorithms are Ward, DBSCAN and BIRCH. I am looking for a good validation criterion to determine the quality of the clustering output. I have read about the Cubic Clustering Criterion, but if you look at how it works it is quite similar to that of Silhouette Score, which measure the within sum-of-squares and between-sum-of squares. It can also be observed that these two have a distinction with that of Calinski-Harabazs.
Any advice on the advantages and disadvantages among these three validation metrics, given that I am clustering a mixed data type composed of numeric, nominal and ordinal data?