Studying hierarchical latent structures in heterogeneous populations with missing information

An ultrametric Gaussian mixture model is a powerful tool for modeling hierarchical relationships among latent concepts, making it ideal for studying complex phenomena in diverse and potentially heterogeneous populations. However, in many cases, only an incomplete set of observations is available on the phenomenon under study. To address this issue, we propose MissUGMM, an ultrametric Gaussian mixture model which takes into account the missing at random mechanism for the unobserved values. Our approach is estimated using the expectation-maximization algorithm and achieves favorable results in comparison to other existing mixture models in simulations conducted with synthetic and benchmark data sets, even without a theorized ultrametric structure underlying the data. Furthermore, MissUGMM is applied to a real-world problem for exploring the sustainable development of cities across countries starting from incomplete information provided by municipalities. Overall, our results demonstrate that MissUGMM is a powerful and versatile model in dealing with missing data and is applicable to a broader range of real-world problems.