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

TL;DR

This paper develops a new semi-parametric regression model for bivariate angular data on the torus, using geometric loss and differential geometry, to better analyze cyclone wind-wave directions, demonstrated on Indian cyclone datasets.

Contribution

It introduces the first mathematical framework for bivariate angular regression on the torus, utilizing geometric loss and differential geometry for improved modeling of coupled directional processes.

Findings

Enhanced modeling of cyclone-driven wind-wave directions.

Effective semi-parametric estimation without distributional assumptions.

Successful application to cyclone datasets from India.

Abstract

This study introduces a novel torus-to-torus regression framework to improve the analysis and prediction of cyclone-driven wind-wave directional dynamics. 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 approach enhances the capacity to model coupled directional processes commonly observed in extreme coastal cyclones. 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-wave directional datasets from two major cyclonic events, Amphan and Biparjoy, that impacted the eastern and western coastlines of India, respectively.