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The density-based formulation aims at recasting the clustering problem to a mathematically sound framework, by linking the groups to some features of the density assumed to underlie the data. Even if early probabilistic approaches to cluster analysis can be traced back to fifty years ago, the topic has recently found a renewed and vibrant interest in the scientific community. This may be motivated both by the computational advancements witnessed in the last years and by more conceptual and substantial reasons. The increased availability of mixed type and complex structured data has indeed required a rigorous formalization of the clustering problem. Stemming from the same roots, density-based clustering has been developed following two distinct paths. In its parametric formulation, a connection among groups and unimodal components of a mixture model is drawn. On the other hand, according to the nonparametric paradigm, clusters are seen as the domains of attraction of the modes of the density. Revolving around this approach to clustering, the thesis explores both the formulations by highlighting at the same time contact points and dissimilarities. Moreover differently structured data are considered ranging from the unidimensional setting to more complex three-way structure. Three main contributions can be highlighted. In the first one we derive some asymptotic results to address nonparametric density estimation from a clustering-oriented perspective. In the second contribution we propose an ensemble approach for density-based clustering, which inherits the strengths from both the parametric and the nonparametric formulation, and possibly enhances the robustness and the stability of the partitions. The third contribution addresses the problem of clustering complex multivariate time-dependent data by adopting a parametric approach and proposing a flexible modification of the Latent Block Model.

Model-independent searches in particle physics aim at completing our knowledge of the universe by looking for new possible particles not predicted by the current theories. Such particles, referred to as signal, are expected to behave as a deviation from the background, representing the known physics. Information available on the background can be incorporated in the search, in order to identify potential anomalies. From a statistical perspective, the problem is recasted to a peculiar classification one where only partial information is accessible. Therefore a semisupervised approach shall be adopted, either by strengthening or by relaxing assumptions underlying clustering or classification methods respectively. In this work, following the first route, we semisupervise nonparametric clustering in order to identify a possible signal. The main contribution consists in tuning a nonparametric estimate of the density underlying the experimental data with the aid of the available information on the physical theory. As a side contribution, a variable selection procedure is presented. The whole procedure is tested on a dataset mimicking proton-proton collisions performed within a particle accelerator. While finding motivation in the field of particle physics, the approach is applicable to various science domains, where similar problems of anomaly detection arise.