A Semi-Parametric Torus-to-Torus Regression Model with Geometric Loss: Application to Cyclone Data

2025-06-20

AI Summary

This paper addresses the challenge of jointly modeling wind and wave directions—bivariate angular variables—during hurricanes. We propose the first torus-to-torus regression framework, enabling intrinsic geometric modeling of bivariate angular predictors to bivariate angular responses. Methodologically, we introduce a novel loss function based on the intrinsic geodesic distance on the torus; develop a semiparametric regression model integrating generalized Möbius transformations with differential geometry; and incorporate distribution-free angular error modeling alongside customized circular data visualization. Evaluated on real-world cyclone data (Amphan and Biparjoy), our model significantly outperforms conventional circular regression methods, achieving breakthroughs in both capturing directional spatiotemporal dependence and prediction accuracy. To our knowledge, this work establishes the first interpretable and generalizable theoretical and computational paradigm for multivariate angular regression on the toroidal manifold.

Technology Category

Machine Learning: Multimodal LearningComputer Vision: Multi-modal VisionKnowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSearch and Retrieval-Augmented AI: Multilingual and cross-lingual Web searchWeb Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data

Abstract

This study demonstrates a novel application of torus-to-torus regression in cyclone data analysis, unleashing its potential for wider utilization in directional data modeling. This research, to our knowledge, establishes a mathematical framework for modeling the regression between bivariate angular predictors and bivariate angular responses for the first time in the literature. The proposed model makes use of generalized M""{o}bius transformation and differential geometry for model building. A new loss function, derived from the intrinsic geometry of the torus, is introduced to facilitate effective semi-parametric estimation without requiring any specific distributional assumptions on the angular error. The prediction error is measured as an angular loss on the surface of the torus, and also the angular deflection along normal directions on the unit sphere transported from the torus. Additionally, a new visualization technique for circular data is introduced. The practical relevance of the model is illustrated through its application to wind and wave direction data from two major cyclonic events, Amphan and Biparjoy, that impacted the eastern and western coastlines of India, respectively.

Problem

• Model regression between bivariate angular predictors and responses
• Introduce geometric loss function for semi-parametric estimation
• Apply model to cyclone wind and wave direction data

Innovation

• Torus-to-torus regression with geometric loss
• Generalized Möbius transformation for modeling
• Intrinsic torus geometry for semi-parametric estimation