Multivariate Wrapped Normal estimation with missing values

L. Greco${}^a$ and C. Agostinelli${}^b$

${}^a$University Giustino Fortunato Benevento, Italy, ${}^b$Department of Mathematics, university of Trento, Italy

This contribution addresses maximum likelihood estimation estimation and model-based imputation for multivariate circular data lying on a $p$-dimensional torus in the presence of missing values, when the missing data mechanism is ignorable (Little and Rubin, 2019). Actually, the periodic nature of the sample space invalidates conventional imputation techniques designed on the Euclidean space. The multivariate Wrapped Normal distribution (WN) is a flexible and computationally tractable model for torus data, with pdf ${\varphi }_p^{\circ }(y;\mu ,\mathrm{\Sigma })={\sum }_{j\in {\mathbb{Z}}^p}{\varphi }_p(y+2\pi j;\mu ,\mathrm{\Sigma }),$ where ${\varphi }_p(\cdot ;\mu ,\mathrm{\Sigma })$ denotes the $p$-variate normal density function, $y\in {[0,2\pi )}^p$, $\mu \in {[0,2\pi )}^p$ is the mean direction, $\mathrm{\Sigma }\in {ℝ}^{p\times p}$ is a positive definite scatter matrix. In practice, the infinite sum is truncated to a finite set ${𝒞}_J={\{-J,-J+1,\mathrm{\dots },J\}}^p$ for a sufficiently large $J$, as the terms decay rapidly for concentrated distributions (Nodehi et al., 2021; Greco et al., 2023). The methodology leverages the conditional properties of the normal distribution on the unwrapped space, embedding the imputation of missing torus data into an Expectation-Maximization algorithm that treats both the wrapping coefficients and the missing entries as latent variables.

Keywords: EM algorithm, MAR, MCAR.