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In the k-prototypes clustering algorithm, the distance function consists of two dissimilarity components - one for the numerical elements of the observations, and one for their categorical elements. In pseudo-mathematical terms, we have:

$$d(x, y) = d_1(x_{num}, y_{num}) + \gamma d_2(x_{cat}, y_{cat})$$

where $\gamma > 0$ is a weight that allows to favor more or less one of the two data types when building centroids. (Setting $\gamma$=0 boils down to performing clustering on numerical variables only, whereas the greater the weight, the greater the impact of categorical variables on the final result.)

In his introduction to k-modes and k-prototypes from 1998, Huang says the following: "Average standard deviation of numeric attributes may be used as a guidance in specifying $\gamma$. [...] However, it is too early to consider this as a general rule."

Have some clear mathematical results been found since then? How can I pick a $\gamma$ value so that my clustering procedure doesn't favor one data type over the other?

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As you already mentioned, and according to the kmodes library, the standard deviation of numeric variables is calculated using the mean:

if gamma is None:
    gamma = 0.5 * np.mean(Xnum.std(axis=0))

This is in accordance with Huang [1997] enter image description here

See this.


Reference:

Huang, Z.X. (1997) Clustering Large Data Sets with Mixed Numeric and Categorical Values. Proceedings of the First Pacific Asia Knowledge Discovery and Data Mining Conference, Singapore, World Scientific, 21-34.

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  • $\begingroup$ On second thought, when I use k-prototypes, I usually end up using standardization, which would generate a gamma = 1, considering the mean of the stardart deviations of the numeric data. One approach I have used is to calculate gamma as the ratio between the total numeric dissimilarity and the total categorical dissimilarity $\endgroup$ Aug 24, 2023 at 16:30

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