# Combine two, three, (n) metrics for calculating dissimilarity matrix

I have a data set with 9000 instances and 40 attributes of mixed data, that is categorical and numeric. My target is to group them into clusters using whichever clustering algorithm works best. I've heard/read that for such a data set Gower distance is suitable. My question is can I combine two (or n) metrics for calculating distances between instances, for example I would like to use let's say Euclidean distance on numeric attributes and Gower distance on categorical attributes. I could always divide my data set into two data sets, one with numeric attributes and the other with categorical. But how could one interpret each result? Summing them up just sounds ... wrong.

My second question is what exactly does Gower distance do with numeric values? Does my first question even make sense?

Here is a snippet of my code, I am using R and functions daisy, agnesfrom package cluster:

df.diss <- daisy(df, metric = "gower", type = list(numeric = c(1, 4, 6, 8, 9, 11, 12, 13, 14, 17 : 37), symm = c(2, 3, 5, 7)), stand = FALSE)
df.clust <- agnes(df.diss)


Using these functions or even R is not a must.

If you look at Gower in detail, you'll notice it uses Manhattan on numerical attributes. You can easily modify it to use Euclidean.

However, feature weighting will have a major impact on the results. There are a few approaches for supervised weighting of features IIRC, but I have not yet seen anything reliable for automatic weighting that does not require labels.

So in the end, you will have the problem that your distance function looks something (for Euclidean) like this:

$$d(x,y) = \sqrt{\sum_{i\in \text{numerical}} \omega_i(x_i-y_i)^2} + \sum_{i \in \text{categorical}} \omega_i \mathbb{1}_{x_i == y_i}$$

where you will face the challenge of choosing all the $\omega_i$ weights.

• That is one way I still have to try. My initial idea that I'm currently working on, was to binarize all categorical variables. All of my categorical attributes have a rather small domain. Is this a good idea or at least worth a try? – peech Jan 5 '16 at 12:55
• If you binarize them, that is equivalent to put a weight of $\sqrt{2}$ on each categorial attribute... look at the distances of two variables, that differ only in one attribute. You must not ignore feature weighting. – Has QUIT--Anony-Mousse Jan 5 '16 at 13:43
• how about unsupervised feature weighting on categorical data? or maybe if I somehow manage to define similarity matrix for my features of categorical data? – peech Jan 5 '16 at 16:33
• You will need some careful feature weighting. It is best to use your domain knowledge. You can try some of the heuristics like IDF. But you will need some good feature weighting for best results. – Has QUIT--Anony-Mousse Jan 5 '16 at 16:34