Nonparametric Density-Based Clustering of Mixed Data

We discuss a general extension of nonparametric clustering to mixed-type data. The approach relies on a suitable density function defined directly on mixed-type spaces and estimated nonparametrically via a generalized product kernel estimator, which handles variables of different types without requiring latent representations or data transformations. Clusters are then associated with the maximum connected components of the estimated upper level sets, approximated through a k−nearest neighbor graph built on the dissimilarity induced by the kernel itself. This ensures that the geometry governing level set connectivity is coherent with that underlying the density estimator. As in the continuous case, varying the density threshold yields the estimated cluster tree that, in the current framework, is recast as a cluster forest and whose consistency we establish. The proposed procedure proves to be effective in practical applications, offering a flexible and coherent framework for clustering datasets with heterogeneous variable types.