Time-varying clustering for skew-symmetric data

We introduce a novel methodological clustering approach tailored for the analysis of three-way two-mode time-dependent skew-symmetric data. The model builds upon the decomposition of a skew-symmetric matrix into within-cluster and between-cluster effects, which are further decomposed into regression components and residuals when external variables are available. To capture the directional imbalances over time, the model jointly identifies optimal partitions of objects and dynamically adjusts weights for different times periods. These weights are linearly associated with the external variables, allowing for the analysis of time-varying effects and dynamic behavior of the objects. By clustering both objects and times, the model provides a comprehensive view of how the relationships with external variables and how imbalances evolve over time. This approach offers insights into the temporal dynamics of the data, enabling the identification of consistent patterns, shifts, and changes in behavior across different periods. An Alternating Least-Squares algorithm is developed to estimate model’s parameters. The model’s effectiveness is demonstrated through applications to artificial data and motivated by the analysis of internal migration flows in Italy, challenging traditional narratives related to north-south migrations.