Covariance models through difference between Matérn families

It is well known that Whittle–Matérn family represents one of the most utilized class of covariance functions because of some special features which are capable to describe several correlation structures. However, one of the main drawbacks of this last family concerns its failure to model negative correlation structures. In this paper, utilizing the Whittle–Matérn family, new classes of covariance models, suitable to model negative correlations, have been presented and their properties have been analyzed. The new families are flexible enough because they allow to choose covariance models which always present positive values in their domain, as well as covariance models which are negative in a subset of their domain. On the other hand, all the traditional hole effect models, proposed in the literature, essentially stem from the Bessel family and they are characterized by the same features, i.e., they present a countable infinity of zeros and a parabolic behaviour near the origin. Conversely, the models proposed in this paper are characterized by a complementary behaviour, which cannot be detailed by all the classical hole effect correlation models: they are essentially characterized by just one zero, moreover they are able to describe various behaviours near the origin (linear and parabolic), as well as different behaviours concerning the concavity in proximity of the origin (upwards and downwards). In summary, they can describe correlation structures which cannot be detailed by the Whittle–Matérn family.