On the use of difference-based covariance models

There exist applications for which the phenomenon under study can have not always positive correlation structures. The traditional Whittle-Matern family of covariance functions is distinguished by being positive, hence this class is not able to model correlation structures with negative values. In the very recent years, theoretical results regarding families of covariance functions developed through the difference of covariance functions and characterized by negative values in a subset of the corresponding domain, have been obtained in the literature, both in the complex and in the real domain. In this paper, some models belonging to these families are reviewed and their flexibility, with respect to the traditional covariance models, is underlined from a practical point of view. Various case studies referred to distinct temporal and spatial datasets whose correlation structures are featured by positive and negative values, are discussed in the paper with the aim of highlighting the pro of fitting difference-based covariance models with respect to the traditional ones. The advantages of the difference-based covariance models are assessed in terms of predictive performances. Finally, an application on a 2-dimensional simulated dataset is also furnished.