Spatial Statistics for Riemannian Data

• Activity Number: 507
• Type: Invited
• Date/Time: Wednesday, August 7, 2013 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Piercesare Secchi*+ and Davide Pigoli
• Companies: Politecnico di Milano and Politecnico di Milano
• Keywords: Riemannian data ; Spatial dependence ; Meteorological data ; universal kriging ; tangent space approximation

Abstract

The statistical analysis of data belonging to Riemannian manifolds is becoming increasingly relevant. The aim of this work is to introduce models for spatial dependence among Riemannian data, with a special focus on the case of covariance matrices. We introduce a semivariogram for a covariance matrices field and an estimator for the mean covariance matrix, taking into account both the non Euclidean nature of the data and their spatial correlation. Simulated data are used to evaluate the performance of the proposed estimator: considering spatial dependence leads to better estimates when observations are irregularly spaced in the region of interest. We apply the proposed methodology to the exploration of covariance matrices between temperature and precipitation in the province of Quebec, Canada. We obtain estimates that are in better agreement with previous analysis of Canadian climate than those obtained ignoring spatial dependence. Finally, we propose a kriging estimator for covariance fields based on a tangent space model. This allows to deal with non stationary fields, the deterministic drift being handled in the tangent space with traditional spatial statistics techniques.