Derivation of distance in two-step clustering

Question

I am working with the two step cluster process in SPSS Modeler (Clementine) and trying to get a sense for the distance function used. It is a log-likelihood function (as stated in docs) but I am unsure for even the continuous variables (the function handles continuous and nominal variable) how this is a log likelihood (it is missing most of the elements of a Gaussian). Below is a screen shot of the documentation describing the distance formula.

Has anyone seen the derivation of this distance function?

Answer 1

SPSS two step cluster model algorithm is described in more detail in:

Chiu, Tom, DongPing Fang, John Chen, Yao Wang, and Christopher Jeris (2001), "A robust and scalable clustering algorithm for mixed type attributes in large database environment," Proceedings of the seventh ACM SIGKDD international conference on Knowledge discovery and data mining KDD '01.

More generally, if you look at the model-based clustering literature and the latent class literature you should get an understanding of how continuous and nominal variables enter into the likelihood. The various documentations for Latent Gold are pretty useful and available on the web.

Most of the published literature uses a slightly different model to that in SPSS. The difference relates to the treatment of the class sizes (the priors). The SPSS modification, which is a simplification, seems to be aimed at reducing computing costs (as opposed to increasing rigor).