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

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 Möbius 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.

Figures (7)

• Figure 1: Top-left: scatter plots depicting the predictor values on the torus $\mathbb{T}_2$. Top-right: Graph illustrating the expected responses via the proposed Möbius map on the torus $\mathbb{T}_2$. Bottom-right: A plot illustrating the responses with angular error, $\arg\boldsymbol{\epsilon}=(\arg \epsilon_1, \arg \epsilon_2)$, derived from a von Mises sine model with zero mean direction.
• Figure 2: (a) torus-to-torus regression model. (b) Circular-circular regression model.
• Figure 3: (a) Area between $(0,0)$, and $(\theta, \theta)$ on the curved torus. (b) Area between $(0,0)$, and $(\theta, \theta)$ in a flat torus. (c) The normals at observed and predicted points. (d) Geodesic distance between two normals on a sphere. (e) Area between $(0,0)$, and $(\theta, \theta)$ on sphere. (f) Area between $(0,0)$, and $(\theta, \theta)$ in a flat sphere.
• Figure 4: Scatter plot obtained from the model Equation-\ref{['tor_reg_curve']} (red box), simulated data in Equation-\ref{['tor_reg_model_full']} (black small circle), and the predicted data (green cross) on the flat torus (a) for the horizontal angle $\phi$, and (b) for the vertical angle $\theta.$ (c) and (d) is the density plot of the same, respectively.
• Figure 5: QQ plot of the observed and predicted (a) for the horizontal angle $\phi$, and (b) for the vertical angle $\theta.$ (c) and (d) is the plot of the residuals (restricted to the range $[-\pi, \pi]$ for enhanced visual clarity) with a reference line at $0$ (radian).

Theorems & Definitions (5)

• Definition 1
• lemma 1
• proof
• Theorem 2
• proof