Abstract Estimating the midpoints of interval-valued data poses significant challenges, particularly when uncertainty and variability are inherent. This paper advances midpoint estimation by leveraging both Maximum Likelihood (ML) and Bayesian methodologies, incorporating various forms of additional information through parameter constraints. In practical applications, such information often naturally arises, enabling the imposition of constraints such as order, rectangular, spherical, and bounded-order conditions on the parameter space. By integrating these constraints, the estimators can exploit structural relationships among parameters to achieve improved precision and accuracy. Through comprehensive simulation studies, we assess the performance of ML and Bayesian estimators under these constrained settings, demonstrating that Bayesian estimators consistently outperform ML estimators, particularly when employing noninformative priors. The constrained Bayesian approach not only results in lower mean squared errors (MSEs) but also yields more precise credible intervals compared to the confidence intervals obtained via ML, showcasing its robustness and efficiency. Furthermore, a real-world climatological case study validates our theoretical and simulation findings, demonstrating the practical utility of the proposed methods for interval-valued temperature data. This example illustrates how integrating additional information–often readily available–enhances the accuracy and reliability of midpoint estimation in multivariate contexts. Our work underscores the potential of structural constraints in symbolic data analysis, providing actionable insights across diverse scientific domains.

In multivariate time series (MTS) anomaly detection, existing graph neural network (GNN) methods often neglect multi-level feature representations, relying solely on the final layer output and fixed thresholds, which leads to information loss and high false-positive rates. To address this issue, we propose a MTS adaptive hierarchical attention (MTS-AHA) model that integrates cross-layer features from local to global topological structures across GNN layers. Additionally, we introduce a dynamic threshold module that adaptively adjusts the threshold by fitting the tail features of the distribution. Extensive experiments on three datasets validate the superiority of our method and its components.