Abstract

Directional and circular measurements frequently arise in environmental sciences, for example in the analysis of wind directions recorded at multiple monitoring stations, ocean wave directions, or geological orientation data. When several angular variables are observed jointly, the data naturally lie on a torus, and appropriate probabilistic models must account for their periodic structure. Among the available distributions, the multivariate von Mises sine model is one of the most widely used for describing dependence among circular variables.
In practice, however, environmental directional datasets often contain anomalous observations caused by measurement errors, extreme meteorological events, or heterogeneous generating mechanisms. Such atypical circular observations can strongly affect likelihood-based inference. In particular, maximum likelihood estimation for multivariate von Mises models is known to be highly sensitive to contamination, potentially leading to distorted parameter estimates and unreliable conclusions about directional dependence.

In this work we address robust inference for the multivariate von Mises sine distribution; see Singh et al. (2002, Biometrika). The von Mises sine density depends on parameters μ (mean directions), κ (concentration parameters), and Λ (a matrix describing dependence between angular components). The normalizing constant that appears in the density of p-variate von Mises sine distribution is, unfortunately, intractable for p larger than 2. Thus, we focus on the high-concentration regime (large κs), where the normalization constant can be approximated, leading to the concentrated multivariate sine model described by Mardia et al. (2012, Journal of Applied Statistics).

To mitigate the effect of anomalous observations, we adopt the weighted likelihood framework, see Markatou et al. (Jasa, 1998), where observations that disagree with the assumed model are automatically downweighted. The estimator is defined through weighted likelihood estimating equations, where the score contribution of each observation carries a data-dependent weight derived from Pearson residuals, which compare a nonparametric density estimate on the torus with the fitted model.
This approach provides a probabilistic measure of outlyingness and avoids the need to define ad hoc geometric distances on the torus. Observations showing large discrepancies between empirical and model densities receive small weights, allowing the method to automatically identify anomalous directional patterns or substructures.

Simulation experiments investigate the finite-sample behavior of the estimator under several contamination scenarios, including clustered and scattered outliers, and across different dimensions. The results indicate that the proposed methodology effectively identifies outliers and provides robust estimates for location, concentration, and shape parameters. We also show that, for robust inference, formulations based on the Wrapped Normal approximation may yield improved performance, offering a better balance between robustness and efficiency.